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COHOMOLOGY OF VECTOR BUNDLES AND SYZYGIES The central theme of this book is an exp...

Author:
Jerzy Weyman

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COHOMOLOGY OF VECTOR BUNDLES AND SYZYGIES The central theme of this book is an exposition of the geometric technique of calculating syzygies. It is written from the point of view of commutative algebra; without assuming any knowledge of representation theory, the calculation of syzygies of determinantal varieties is explained. The starting point is a deﬁnition of Schur functors, and these are discussed from both an algebraic and a geometric point of view. Then a chapter on various versions of Bott’s theorem leads to a careful explanation of the technique itself, based on a description of the direct image of a Koszul complex. Applications to determinantal varieties follow. There are also chapters on applications of the technique to rank varieties for symmetric and skew symmetric tensors of arbitrary degree, closures of conjugacy classes of nilpotent matrices, discriminants, and resultants. Numerous exercises are included to give the reader insight into how to apply this important method.

CAMBRIDGE TRACTS IN MATHEMATICS General Editors

B. BOLLOBAS, W. FULTON, A. KATOK, F. KIRWAN, P. SARNAK

149

Cohomology of Vector Bundles and Syzygies

Jerzy Weyman Northeastern University

Cohomology of Vector Bundles and Syzygies

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge , United Kingdom Published in the United States by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521621977 © Jerzy Weyman 2003 This book is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2003 ISBN-13 ISBN-10

978-0-511-06601-6 eBook (NetLibrary) 0-511-06601-5 eBook (NetLibrary)

ISBN-13 978-0-521-62197-7 hardback ISBN-10 0-521-62197-6 hardback

Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

To Katarzyna

Contents

Preface

page xi

1 Introductory Material 1.1 Multilinear Algebra and Combinatorics 1.2 Homological and Commutative Algebra 1.3 Determinants of Complexes 2 Schur Functors and Schur Complexes 2.1 Schur Functors and Weyl Functors 2.2 Schur Functors and Highest Weight Theory 2.3 Properties of Schur Functors. Cauchy Formulas, Littlewood– Richardson Rule, and Plethysm 2.4 The Schur Complexes Exercises for Chapter 2 3 Grassmannians and Flag Varieties 3.1 The Pl¨ucker Embeddings 3.2 The Standard Open Coverings of Flag Manifolds and the Straightening Law 3.3 The Homogeneous Vector Bundles on Flag Manifolds Exercises for Chapter 3 4 Bott’s Theorem 4.1 The Formulation of Bott’s Theorem for the General Linear Group 4.2 The Proof of Bott’s Theorem for the General Linear Group 4.3 Bott’s Theorem for General Reductive Groups Exercises for Chapter 4 5 The Geometric Technique 5.1 The Formulation of the Basic Theorem 5.2 The Proof of the Basic Theorem 5.3 The Proof of Properties of Complexes F(V)• ix

1 1 12 27 32 32 49 57 66 78 85 85 91 98 104 110 110 117 123 132 136 137 141 146

x

6

7

8

9

Contents

5.4 The G-Equivariant Setup 5.5 The Differentials in Complexes F(V)• . 5.6 Degeneration Sequences Exercises for Chapter 5 The Determinantal Varieties 6.1 The Lascoux Resolution 6.2 The Resolutions of Determinantal Ideals in Positive Characteristic 6.3 The Determinantal Ideals for Symmetric Matrices 6.4 The Determinantal Ideals for Skew Symmetric Matrices 6.5 Modules Supported in Determinantal Varieties 6.6 Modules Supported in Symmetric Determinantal Varieties 6.7 Modules Supported in Skew Symmetric Determinantal Varieties Exercises for Chapter 6 Higher Rank Varieties 7.1 Basic Properties 7.2 Rank Varieties for Symmetric Tensors 7.3 Rank Varieties for Skew Symmetric Tensors Exercises for Chapter 7 The Nilpotent Orbit Closures 8.1 The Closures of Conjugacy Classes of Nilpotent Matrices 8.2 The Equations of the Conjugacy Classes of Nilpotent Matrices 8.3 The Nilpotent Orbits for Other Simple Groups 8.4 Conjugacy Classes for the Orthogonal Group 8.5 Conjugacy Classes for the Symplectic Group Exercises for Chapter 8 Resultants and Discriminants 9.1 The Generalized Resultants 9.2 The Resultants of Multihomogeneous Polynomials 9.3 The Generalized Discriminants 9.4 The Hyperdeterminants Exercises for Chapter 9

References Notation Index Subject Index

149 152 154 156 159 160 168 175 187 195 209 213 218 228 228 234 239 245 251 252 263 278 283 296 309 313 314 318 328 332 355 359 367 369

Preface

This book is devoted to the geometric technique of calculating syzygies. This technique originated with George Kempf and was ﬁrst used successfully by Alain Lascoux for calculating syzygies of determinantal varieties. Since then it has been applied in studying the deﬁning ideals of varieties with symmetries that play a central role in geometry: determinantal varieties, closures of nilpotent orbits, and discriminant and resultant varieties. The character of the method makes it comparable to the symbolic method in classical invariant theory. It works in only a limited number of cases, but when it does, it gives a complete answer to the problem of calculating syzygies, and this answer is hard to get by other means. Even though the basic idea is more than 20 years old, this is the ﬁrst book treating the geometric technique in detail. This happens because authors using the geometric method have usually been interested more in the special cases they studied than in the method itself, and they have used only the aspects of the method they needed. Therefore the basic theorems from chapter 5 stem from the efforts of several mathematicians. The possibilities offered by the geometric method are not exhausted by the examples treated in the book. The method can be fruitfully applied to any representation of a linearly reductive group with ﬁnitely many orbits and with actions such that the orbits can be described explicitly. The varieties treated in chapters 7 and 9 show that the scope of the method is not limited to such actions. The book is written from the point of view of a commutative algebraist. We develop the rudiments of representation theory of general linear group in some detail in chapters 2–4. At the same time we assume some knowledge of commutative algebra and algebraic geometry, including sheaves and their cohomology—for example, the notions covered in chapters II and III of the book [H1] of Hartshorne. The notions of Cohen–Macaulay and Gorenstein

xi

xii

Preface

rings and rational singularities are also used, and their deﬁnitions are brieﬂy recalled in chapter 1. Some parts of the book demand more advanced knowledge. One statement in chapter 5 is an application of Grothendieck duality, whose statement is brieﬂy recalled in chapter 1. We advise less experienced readers to just accept Theorem (5.1.4) and look ﬁrst at its applications. In some sections of chapters 5 and 8 we assume familiarity with highest weight theory and some facts on linear algebraic groups. At the same time, in exercises we use the geometric technique to develop some of that theory for the classical groups. Still, the reader not familiar with these notions should be able to understand all the remaining chapters. Let us describe brieﬂy the contents of the book. The ﬁrst chapter discusses preliminaries. We recall elementary notions from multilinear algebra and combinatorics. The section on commutative and homological algebra covers brieﬂy the deﬁnitions of depth, Koszul complexes, the Auslander– Buchsbaum–Serre theorem, and Cohen–Macaulay and Gorenstein rings. There is also a section on de Jong’s algorithm for explicit calculation of normalization, on the exactness criterion of Buchsbaum and Eisenbud, and on Grothendieck duality. The ﬁnal section is devoted to a brief review of the notion of the determinant of a complex. Chapters 2, 3, and 4 are devoted to developing the representation theory of general linear groups and to the proof of Bott’s theorem on the cohomology of line bundles on homogeneous spaces. This provides the basic tools for the calculations to be performed in later chapters. Our approach is based on Schur and Weyl functors, introduced in the ﬁrst section of chapter 2. They are deﬁned by generators and relations. The relation to highest weight theory and Schur–Young theory is discussed. This is followed by a discussion of Cauchy formulas, the Littlewood–Richardson rule, and plethysm. The ﬁnal section of chapter 2 discusses Schur complexes. In chapter 3 we relate Schur functors to geometry by realizing them as multihomogeneous components of homogeneous coordinate rings of ﬂag varieties. This is followed by a proof of the Cauchy formula based on restriction of the straightening from the Grassmannian to its afﬁne open subset, and by a section on tangent bundles of Grassmannians and ﬂag varieties. In chapter 4 we prove Bott’s theorem on cohomology of line bundles on ﬂag varieties. We follow the approach of Demazure. In the last section we formulate the theorem for arbitrary reductive groups and give explicit interpretations for classical groups.

Preface

xiii

Chapter 5 is devoted to the description of terms and properties of the direct images of Koszul complexes. We study the basic setup, i.e., the diagram Z ⊂ ↓ q Y ⊂

X×V ↓q X

where X is an afﬁne space, V is a nonsingular projective variety, Z is a total space of a vector subbundle S of the trivial bundle X × V , and Y = q(Z ). The variety Z can be described as the vanishing set of a cosection p ∗ (ξ ) → O X ×V , and it is a locally complete intersection. Here ξ is the dual of the factor (X × V )/S. The original idea of Kempf was that the study of a direct image of the resulting Koszul complex can be used to prove results about the deﬁning equations and syzygies of the subvariety Y . We give the basic properties of these direct images in the general case and in important special cases, for example when Z is a desingularization of Y . We also treat the more general case of twisted Koszul complexes. The remaining chapters are devoted to examples and applications. In each of these chapters a different aspect of the method is illustrated. In chapter 6 we study the case of determinantal varieties. Here we show how to handle basic calculations in simple cases when the bundle ξ is a tensor product of tautological bundles. Apart from the proof of Lascoux’s theorem and the calculation of syzygies of determinantal ideals for symmetric and skew symmetric matrices, we also give results on the equivariant modules supported in determinantal varieties. Chapter 7 is devoted to the rank varieties for tensors of degree higher than two. This illustrates that the method can be applied in cases when the variety X or even Y does not have ﬁnitely many orbits with respect to some action of the reductive group. In chapter 8 the study of nilpotent orbit closures allows us to understand how to handle the situation when the cohomology groups needed to get the syzygies cannot be calculated directly, but still partial results can be recovered by estimating the terms in a spectral sequence associated to ﬁltrations on a basic bundle ξ . We also prove the Hinich–Panyushev theorem on rational singularities of normalizations of nilpotent orbits for a general simple group. Chapter 9 illustrates the use of twisted modules supported in resultant and discriminant varieties, which allow one to get natural determinantal expressions for the deﬁning equation. Each chapter is followed by exercises. They should allow readers to learn how to apply the geometric method on their own. At the same time they

xiv

Preface

illustrate further applications of the method. In particular the exercises to chapter 6 deal with the analogues of the determinantal varieties for the symplectic and orthogonal groups. In exercises to chapter 7 we give some calculations of minimal resolutions of Pl¨ucker ideals. The book can be read on several levels. For the reader who is not familiar with representation theory and/or derived categories and Grothendieck duality, we suggest ﬁrst reading chapters 2 through 4. Then one can proceed with the proof of the statement of Theorem 5.1.2 given in section 5.2. At that point most of the following chapters (with the exception of section 8.3) can be understood using only that statement. In such a way the book could be used as the basis of an advanced course in commutative algebra or algebraic geometry. It could also serve as the basis of a seminar. A lot of general notions and theories can be nicely illustrated in the special cases treated in later chapters by the methods given in the book. A reader familiar with representation theory can just skim through chapters 2 through 4 to get familiar with the notation, and then proceed straight to chapter 5 and study the applications. I am indebted to many people who introduced me to the subject, especially to David Buchsbaum, Corrado De Concini, Jack Eagon, David Eisenbud, Tadeusz J´ozeﬁak, Piotr Pragacz, and Joel Roberts. I also beneﬁted from conversations on some aspects of the material with Kaan Akin, Giandomenico Bofﬁ, Michel Brion, Bram Broer, Andrzej Daszkiewicz, Steve Donkin, Toshizumi Fukui, Laura Galindo, Wilberd van der Kallen, Jacek Klimek, Hanspeter Kraft, Witold Kra´skiewicz, Alain Lascoux, Steve Lovett, Olga Porras, Claudio Procesi, Rafael Sanchez, Mark Shimozono, Alex Tchernev, and Andrei Zelevinsky. Throughout my work on the book I was partially supported by grants from the National Science Foundation.

1 Introductory Material

1.1. Multilinear Algebra and Combinatorics 1.1.1. Exterior, Divided, and Symmetric Powers; Multiplication and Diagonal Maps Let K be a commutative ring, and let E be a free K-module with a basis {e1 , . . . , en }. We deﬁne the r -th exterior power r E of E to be the r -th tensor power E ⊗r of E divided by the submodule generated by the elements: u 1 ⊗ . . . ⊗ u r − (−1)sgn σ u σ (1) ⊗ . . . ⊗ u σ (r ) for all σ ∈ r , u 1 , . . . , u r ∈ E. We denote the coset of u 1 ⊗ . . . ⊗ u r by u 1 ∧ . . . ∧ ur . The following basic properties of exterior powers are proved in [L, chapter XIX, section 1]. (1.1.1) Proposition. (a) Let {e1 , . . . , en } be an ordered basis of E. Then the elements ei1 ∧ . . . ∧ eir for 1 ≤ i 1 < . . . < ir ≤ n form abasis of r E. In particular, r E is a free K-module of dimension nr . (b) (Universality property of exterior powers) We have a functorial isomorphism θ M : Alt (E , M) → HomK r

r

r

E, M

where Altr (E r , M) denotes the set of multilinear alternating maps from r ( f )(u 1 ∧ . . . ∧ u r ) = f (u 1 , . . . , u r ). E ×r to M, given by the formula θ M

1

2

Introductory Material

(c) We have natural isomorphisms α : r

r

∗

(E ) →

r

∗ E

sending the exterior product l1 ∧ . . . ∧ lr to the linear function l on e E deﬁned by the formula l(u 1 ∧ . . . ∧ u r ) = (−1)sgn σ lσ (1) (u 1 ) . . . lσ (r ) (u r ). σ ∈ r

The r -th exterior power is an endofunctor on the category of free Kmodules and linear maps. More precisely, for two free K-modules E, F and a linear map φ : E → F we have a well-deﬁned linear map r

r

φ:

r

E→

r

F

φ(u 1 ∧ . . . ∧ u r ) = φ(u 1 ) ∧ . . . ∧ φ(u r ). Let us deﬁned by the formula denote m = dim F. Let {e1 , . . . , en } be a basis of E and let { f 1 , . . . , f m } be a basis of F. In these bases φ correspond to the m × n matrix (φ j,i ) where φ(ei ) =

m

φ j,i f j .

j=1

The map r φ can be written in the corresponding bases of r E, r F as follows: r φ(ei1 ∧ . . . ∧ eir ) = M( j1 , . . . , jr | i 1 , . . . , ir ; φ) f j1 ∧ . . . ∧ f jr , 1≤ j1 0, and satisfying η(1) = 1. deﬁned to be zero on all spaces The following proposition is an elementary calculation.

(1.1.2) Proposition. (a) The maps m, , , η deﬁne on • (E) the structure of commutative, cocommutative bialgebra. (b) The map α : • (E ∗ ) → ( • E)∗ deﬁned in (1.1.1) (c) is an isomorphism of bialgebras. Part (b) of the proposition means that the dual map to the multiplication map m on • (E) is the diagonal map on • (E ∗ ) and vice versa. We deﬁne the r -th symmetric power Sr E of E to be the r -th tensor power E ⊗r of E divided by the submodule generated by the elements u 1 ⊗ . . . ⊗ u r − u σ (1) ⊗ . . . ⊗ u σ (r ) for all σ ∈ r , u 1 , . . . , u r ∈ E. We denote the coset of u 1 ⊗ . . . ⊗ u r by u 1 . . . ur . The following basic properties of symmetric powers are proved in [L, chapter XVI, section 8].

4

Introductory Material

(1.1.3) Proposition. (a) Let {e1 , . . . , en } be an ordered basis of E. Then the elements e1i1 . . . enin for i 1 + . . . + i n = r form a basis of Sr E. In particular Sr E is a free n+r −1 K-module of dimension . r (b) (Universality property of symmetric powers) We have a functorial isomorphism θ M : Symr (E r , M) → HomK (Sr E, M) where Symr (E r , M) denotes the set of multilinear symmetric maps r ( f )(u 1 . . . u r ) = f (u 1 , . . . , u r ). from E ×r to M, given by the formula θ M The r -th symmetric power is an endofunctor on the category of free Kmodules and linear maps. More precisely, for two free K-modules E, F and a linear map φ : E → F we have a well-deﬁned linear map Sr φ : Sr E → Sr F deﬁned by the formula Sr φ(u 1 . . . u r ) = φ(u 1 ) . . . φ(u r ). Let us denote m = dim F. Let {e1 , . . . , en } be a basis of E, and let { f 1 , . . . , f m } be a basis of F. In these bases φ correspond to the m × n matrix (φ j,i ) where φ(ei ) =

m

φ j,i f j .

j=1

The map Sr φ can be written in the corresponding bases of Sr E, Sr F as follows: Sr φ(ei1 . . . eir ) = P( j1 , . . . , jr | i 1 , . . . , ir ; φ) f j1 . . . f jr , 1≤ j1 0, and satisfying η(1) = 1. We have the following analogue of (1.1.2) (a). (1.1.4) Proposition. The maps m, , , η deﬁne on Sym(E) the structure of a commutative, cocommutative bialgebra. Let us investigate the duality. The algebra Sym(E) = r ≥0 Sr E is not ﬁnite dimensional, so instead of the dual we have to work with the graded dual (Sr E)∗ . Sym(E)∗gr := r ≥0

6

Introductory Material

The module map E ∗ = (S1 E)∗ → Sym(E)∗gr induces by universality an algebra map β : Sym(E ∗ ) → Sym(E)∗gr . This map β is an isomorphism only when K contains a ﬁeld of rational numbers. In fact it is given by the formula lσ (1) (u 1 ) . . . lσ (r ) (u r ). β(l1 . . . lr )(u 1 . . . u r ) = σ ∈r

In particular, when l1 = . . . = lr , u 1 = . . . = u r we see that β(l1r ) = r !(u r1 )∗ . In order to describe the graded dual of the symmetric algebra we introduce the divided powers. We deﬁne the r -th divided power Dr (E) as the dual of the symmetric power. Dr (E) := (Sr (E ∗ ))∗ . Its basis is the dual basis to the natural basis of the symmetric power. If {e1 , . . . , en } is a basis of E, we deﬁne e1(i1 ) . . . en(in ) to be the element of the dual basis to the basis {(e1∗ ) j1 . . . (en∗ ) jn }, dual to (e1∗ )i1 . . . (en∗ )in . For every u ∈ E we can deﬁne its r -th divided power u (r ) ∈ Dr E. It is given by the formula (r ) n p (p ) u i ei = u 1 1 . . . u npn e1 1 . . . en( pn ) . i=1

p1 +...+ pn =r

It is easy to check that this deﬁnition does not depend on the choice of basis {e1 , . . . , en }. (1.1.5) Proposition. The divided powers have the following properties: (a) (b) (c) (d) (e)

u (0) = 1, u(1) = u, u (r ) ∈ Dr E, u ( p) u (q) = p+q u ( p+q) , q

p (u + v)( p) = k=0 u (k) v ( p−k) , ( p) ( p) ( p) (uv) = u v , (u ( p) )(q) = [ p, q]u ( pq) for u ∈ E; [ p, q] = [( pq)!]/(q! pq !).

(1.1.6) Remark. In the notation used above, e1(i1 ) . . . en(in ) has a double meaning. It is the element of the dual basis to the basis in the symmetric power,

1.1. Multilinear Algebra and Combinatorics

7

and it is the product of divided powers. It is not difﬁcult to see that the two elements coincide. The r -th divided power is an endofunctor on the category of free K-modules and linear maps. More precisely, for two free K-modules E, F and a linear map φ : E → F we have a well-deﬁned linear map Dr φ : Dr E → Dr F which is best described as the transpose of the map Sr (φ ∗ ) : Sr (F ∗ ) → Sr (E ∗ ). This also gives the description of the matrix coefﬁcients for Dr φ as polynomials in the entries of φ, which we leave to the reader. Dr (E) on E is a commutative, The divided power algebra D(E) := cocommutative algebra because it is a graded dual of the symmetric algebra on E ∗ . Again we denote the components of the multiplication map by m : Dr E ⊗ Ds E → Dr +s E, and the components of the comultiplication by : Dr +s E → Dr E ⊗ Ds E. Let us record the duality statements. (1.1.7) Proposition. (a) The multiplication map m : Dr E ⊗ Ds E → Dr +s E is the dual of the diagonal map : Sr +s E ∗ → Sr E ∗ ⊗ Ss E ∗ . (b) The diagonal map : Dr +s E → Dr E ⊗ Ds E is the dual of the multiplication map m : Sr E ∗ ⊗ Ss E ∗ → Sr +s E ∗ . (c) The diagonal map : Dr +s E → Dr E ⊗ Ds E is given by the formula (e1(i1 ) . . . en(in ) ) =

j1 +...+ jn =r, 0≤ js ≤i s for s=1,...,n

(j )

(i − j1 )

e1 1 . . . en( jn ) ⊗ e1 1

. . . en(in − jn ) .

8

Introductory Material

1.1.2. Partitions, Skew Partitions. Combinatorics of Z2 -Graded Tableaux. Let n be a natural number. A partition λ of n is a sequence λ = (λ1 , . . . , λs ) of natural numbers such that λ1 ≥ λ2 ≥ . . . ≥ λs ≥ 0 and λ1 + λ2 + . . . + λs = n. We identify the partitions (λ1 , . . . , λs ) and (λ1 , . . . , λs , 0). To each partition λ we associate its Young frame (or Ferrers diagram) D(λ). It can be deﬁned as D(λ) = {(i, j) ∈ Z × Z |(1 ≤ i ≤ s, 1 ≤ j ≤ λi }. To represent the Young frames graphically we think of them as contained in the fourth quadrant. A Young frame is a set of boxes with λi boxes in the i-th row from the top. Formally it could be achieved by considering the point ( j, −i) instead of (i, j). (1.1.8) Example. λ = (4, 2, 1): D(λ) =

.

Formally the boxes of D((4, 2, 1)) correspond to the set of points {(1, −1), (2, −1), (3, −1), (4, −1), (1, −2), (2, −2), (1, −3)} in the grid Z × Z. Let λ be a partition. We say that λ has a Durfee square of size r (or rank λ = r ) if λr ≥ r , λr +1 ≤ r , i.e., if the biggest square ﬁtting inside of λ is an r × r square. Let λ be a partition, and let X be a box in λ. The set of boxes to the right of X (including X ) is called an arm of X . The set of boxes below X (including X ) is called the leg of X . The arm length (leg length) of X are deﬁned as the numbers of boxes in the arm (leg) of X . The arm and leg of X form a hook of X . The number of boxes in the hook of X is called the hook length of X . Let λ be a partition of rank r . Let ai (bi ) be the arm length (leg length) of the i-th box on the diagonal of λ. The partition λ is uniquely determined by its rank r and the numbers ai , bi (1 ≤ i ≤ r ). These numbers satisfy the conditions a1 > . . . > ar > 0, b1 > . . . , br > 0.

1.1. Multilinear Algebra and Combinatorics

9

We will sometimes denote by λ = (a1 , . . . , ar |b1 , . . . , br ) the partition with diagonal arm lengths ai and diagonal leg lengths bi . We refer to this as a Frobenius (or hook) notation for λ. (1.1.9) Example. The partition λ = (4, 3, 2) in the hook notation is (4, 2|3, 2). ¯ The boxes in the arm (leg) of the i-th diagonal box are ﬁlled with symbol i (i): X 1¯ 1¯

1 1 1 . X 2 ¯2

Let λ be a partition. The conjugate (or dual) partition λ is deﬁned by setting λi = card{t |λt ≥ i}. The Young frame of λ is obtained from the Young frame of λ by reﬂecting in the line y = −x. (1.1.10) Example. λ = (4, 2, 1), λ = (3, 2, 1, 1):

D(λ ) =

.

Let λ and µ be two partitions. We say that µ is contained in λ (denoted µ ⊂ λ) if for each i we have µi ≤ λi . Let λ and µ be two partitions with µ ⊂ λ. We refer to such a pair as a skew partition λ/µ. We associate to a skew partition λ/µ the skew Young frame D(λ/µ) := D(λ) \ D(µ). Graphically we can represent it as a Young frame of λ with the boxes corresponding to µ missing. (1.1.11) Example. λ = (4, 2, 2, 1, 1), µ = (3, 1):

D(λ/µ) =

.

10

Introductory Material

Let A = (A0 , A1 ) be a Z2 -graded set, i.e. the pair of sets indexed by {0, 1}. Assume that the set A is ordered by a total order . A tableau of shape λ/µ with values in A is a function T : D(λ/µ) → A. (1.1.12) Definition. (a) A tableau T of shape λ/µ with values in A is row standard if for each (u, v) we have T (u, v) T (u, v + 1) with equality possible if T (u, v) ∈ A1 . (b) We say that a tableau T of shape λ/µ with values in A is column standard if T (u, v) T (u + 1, v) with equality possible when T (u, v) ∈ A0 . (c) A tableau T of shape λ/µ with values in A is standard if it is both column standard and row standard. (1.1.13) Notation. We denote by RST(λ/µ, A) (CST(λ/µ, A), ST(λ/µ, A)) the set of row standard (column standard, standard) tableaux of shape λ/µ with values in A. We denote by [1, m] ∪ [1, n] the Z2 -graded set A with A0 = [1, m], A1 = [1 , n ] and with the order deﬁned to be the natural order on A0 and A1 with A0 preceeding A1 . Similarly we deﬁne the Z2 -graded set [1, n] ∪ [1, m] (here A1 preceeds A0 ). (1.1.14) Examples. Let λ = (4, 2, 2, 1, 1), µ = (2, 1). Let A = [1, 2] ∪ [1, 3] . (a) The tableau 1 2

1 T1 = 1 1 2 2 is row standard but not column standard. (b) The tableau 1 1

1 T2 = 1 2 2 3 is column standard but not row standard.

1.1. Multilinear Algebra and Combinatorics

11

(c) The tableau 1 2 1 T3 = 1 2 2 3 is standard. Let λ/µ be a skew partition, and let A = (A0 , A1 ) be a Z2 -graded set ordered by the total order . We deﬁne the orders (relative to ) on the sets of row standard (column standard, standard) tableaux as follows. Consider the set RST(λ/µ, A). Given two tableaux T, U , we have T U if T = U . Assume that T = U . Let us write them as T = (T1 , . . . , Ts ), U = (U1 , . . . , Us ) with Ti (Ui ) being the part of T (U ) from the i-th row of λ/µ. Let j be the minimal i for which Ti = Ui . We have T j = (T ( j, 1), . . . , T ( j, λ j − µ j )), U j = (U ( j, 1), . . . , U ( j, λ j − µ j )). Now let k be the smallest index for which T ( j, k) = U ( j, k) (such a k exists by the choice of j). We say that T U if and only if T ( j, k) U ( j, k). The order on ST(λ/µ, A) is deﬁned to be the restriction of from RST(λ/µ, A). Finally we deﬁne the order on CST(λ/µ, A). Given two tableaux T, U from CST(λ/µ, A), then T = U implies T ≤ U . Assume T = U . We write T = (T 1 , . . . , T s ), U = (U 1 , . . . , U s ) with T i (U i ) being the part of T (U ) from the i-th column of λ/µ. Let j be the minimal i for which T i = U i . We have T j = (T (1, j), . . . , T (λj − µj , j)), U j = (U (1, j), . . . , U (λj − µj , j)). Now let k be the smallest index for which T (k, j) = U (k, j) (such k exists by the choice of j). We say that T U if and only if T (k, j) U (k, j). Note. The order on ST(λ/µ, A) is the restriction of the order on RST(λ/µ, A). It is different from the restriction of on CST(λ/µ, A). (1.1.15) Examples. Let λ = (2, 1), µ = (0). Set A = [1, 2] ∪ [1, 2] . In RST (λ/µ, A) we have 1 2 1 1 2 1 . 1 2 1

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In CST(λ/µ, A) we have 1 1 1 2 2 1 . 2 1 1 In ST(λ/µ, A) we have 1 2 1 1 . 1 2

1.2. Homological and Commutative Algebra 1.2.1. Regular Sequences, Koszul Complexes, Depth Let R be a commutative Noetherian ring. Let M be an R-module. The dimension dim M of M is deﬁned to be the Krull dimension of R/Ann(M), where Ann(M) = {x ∈ R | x M = 0 } is the annihilator of M. Let I be an ideal in R. If I M = M, we deﬁne the I -depth of M as depth R (I, M) = min {i | ExtiR (I, M) = 0 }. In the case I M = M we deﬁne depth R (I, M) = ∞. For a ﬁnitely generated R-module M we have I M = M if and only if depth R (I, M) < ∞ if and only if depth R (I, M) ≤ dim M. A sequence a = (a1 , . . . , an ) of elements from R is an M-sequence (or a regular sequence on M) if M = (a1 , . . . , an )M and the multiplication ai : Mi−1 → Mi−1 is injective for i = 0, 1, . . . , n − 1, where Mi := M/ (a1 , . . . , ai )M. The connection between these notions is expressed in (1.2.1) Theorem. Let R be a Noetherian ring, and M a ﬁnitely generated R-module. Let I be an ideal in R. The following conditions are equivalent: (a) depth R (I, M) ≥ n. (b) ExtiR (R/I, M) = 0 for i < n. (c) There exists an M-sequence a = (a1 , . . . , an ) of length n with ai ∈ I for i = 1, . . . , n. A regular sequence (a1 , . . . , an ) is a maximal regular M-sequence if there is no b such that (a1 , . . . , an , b) is an M-sequence. In particular the theorem

1.2. Homological and Commutative Algebra

13

implies that two maximal regular M-sequences with terms from I must have the same length, equal to depth(I, M). Let M be an R-module, and let a = (a1 , . . . , an ) be a sequence of elements from R. We deﬁne the Koszul complex K (a, M)• as follows. For an n-dimensional free R-module E = R n with a basis e1 , . . . , en we set K (a, M)i = i E ⊗ R M, and the differential d:

i

E ⊗R M →

i−1

E ⊗R M

is deﬁned by the formula d(e j1 ∧ . . . ∧ e ji ⊗ m) =

i

(−1)u+1 e j1 ∧ . . . ∧ eˆ ju ∧ . . . ∧ e ji ⊗ a ju m.

u=1

Let M be a ﬁnitely generated R-module. We deﬁne the codimension of M, codim R (M) := ht Ann(M), where ht denotes the height of an ideal. We also deﬁne the grade of M, grade R (M) = depth R (Ann(M), R). The homological properties of Koszul complex include the information about the depth. (1.2.2) Theorem. Let M be a ﬁnitely generated R-module, and let a = (a1 , . . . , an ) be a sequence of elements from R. Denote I = (a1 , . . . , an ). Then depth R (I, M) = n − max{ i |Hi (K (a, M)) = 0 }. (1.2.3) Corollary. Let R be a commutative ring. Assume that I = (a1 , . . . , an ) is an ideal generated by a regular sequence. Then the Koszul complex K (a, R)• is a free resolution of the R-module R/I . The ideal I is a complete intersection ideal of codimension n if there exists a regular sequence (a1 , . . . , an ) such that I = (a1 , . . . , an ). Thus the ﬁnite free resolutions of complete intersection ideals are provided by Koszul complexes. The projective dimension, codimension, and grade of an R-module are related.

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Introductory Material

(1.2.4) Theorem. For an R-module M = 0 we have pd R (M) ≥ codim(M) ≥ grade R (M). A ﬁnitely generated R-module N is perfect if pd R (N ) = grade(N ). In that case the inequalities in (1.2.4) become equalities. We call codim(N ) the codimension of N . Sometimes by abuse of notation we call the grade of R/I the grade of the ideal I . Let us note the following consequence of Theorem (1.2.4) applied to N = R. (1.2.5) Proposition. Let I be an ideal of codimension n. The functor N → ExtnR (N , R) is an exact contravariant involution on the category of perfect modules N with Ann(N ) = I up to radical. If R/I is a perfect module, we call I a perfect ideal. An ideal I is Gorenstein if I is perfect and Extn (R/I, R) ∼ = R/I for n = codim(R/I ). 1.2.2. Cohen–Macaulay Rings and Modules, Gorenstein Rings Let (R, m) be a local ring. The depth of a module M is deﬁned as depth R (M) := depth R (m, M). We have the following inequalities: (1.2.6) Proposition. Let M be a ﬁnitely generated module over a local ring R. Then depth(M) ≤ dim M ≤ dim R. An R-module M is Cohen–Macaulay if depth(M) = dim M. If depth (M) = dim R, we say that M is maximal Cohen–Macaulay. The zero module is by deﬁnition maximal Cohen–Macaulay. The ring R is Cohen–Macaulay if it is Cohen–Macaulay as a module over itself. The projective dimension and depth of a module over a local ring are complementary to each other. (1.2.7) Theorem (Auslander–Buchsbaum Formula). Let R be a Noetherian local ring. Assume that pd R (M) < ∞. Then we have pd R (M) + depth(M) = depth(R).

1.2. Homological and Commutative Algebra

15

It follows that if R is Cohen–Macaulay and M is a maximal Cohen– Maculay module of ﬁnite projective dimension over R, then M is R-free. If R is a Cohen–Macaulay local ring and M is a ﬁnitely generated R-module of ﬁnite projective dimension, then M is Cohen–Macaulay if and only if it is perfect. If R is a Cohen–Macaulay local ring, then dim R = dim R/P for every associated prime P of R. This means that R is equidimensional. A local ring (R, m) is Gorenstein if an only if R has a ﬁnite injective dimension as an R-module. (1.2.8) Theorem. Let (R, m) be a local ring of dimension d. The following conditions are equivalent: (a) (b) (c) (d) (e)

R is Gorenstein, for i = d we have ExtiR (K , R) = 0, Extd (K , R) = K , there exists i > d such that ExtiR (K , R) = 0, ExtiR (K , R) = 0 for i < d, ExtdR (K , R) = K , R is Cohen–Macaulay and ExtdR (K , R) = K .

Recall that the embedding dimension of a local ring is emdim (R) = dim K m/m 2 . A local ring R is regular if emdim(R) = dim R. We denote by gl dim R the global dimension of R. (1.2.9) Theorem (Auslander and Buchsbaum, Serre). Let (R, m) be a local ring of dimension d. Then the following are equivalent: (a) (b) (c) (d) (e)

R is a regular local ring, gl.dim R < ∞, gl.dim R = d, pd R K = d, m is generated by a regular sequence of length d.

The connection between the notions of Cohen–Macaulay (Gorenstein) ring and perfect (Gorenstein) ideal is stated in the next proposition. (1.2.10) Proposition. (a) Let R be a Cohen–Macaulay local ring. Then the ring R/I is Cohen– Macaulay if and only if I is perfect, (b) Let R be a Gorenstein local ring. Then R/I is Gorenstein if and only if I is a Gorenstein ideal.

16

Introductory Material

The theory outlined above for local rings has an analogue for graded rings and graded modules. Let us state the corresponding statements. Let R be a graded ring R = i≥0 Ri where R0 = K is a ﬁeld and Ri are ﬁnite dimensional vector spaces over K . We assume that R is generated as a K -algebra by elements of degree 1, which implies that R is Noetherian. We denote by m the maximal ideal m = R+ = i>0 Ri . For a graded R-module M we denote depth R (M) := depth R (m, M). Then the following statements hold. (1.2.6) Proposition. Let M be a ﬁnitely generated graded module over a graded ring R. Then depth R (M) ≤ dim M ≤ dim R. (1.2.7) Theorem (Auslander–Buchsbaum formula). Let R be a graded ring, and let M be a graded R-module. Assume that pd R (M) < ∞. Then we have pd R (M) + depth R (M) = depth R (R). (1.2.8) Theorem. Let R be a graded ring of dimension d with the maximal ideal m = R + . The following conditions are equivalent: (a) (b) (c) (d) (e)

R is Gorenstein, for i = d we have ExtiR (K , R) = 0, Extd (K , R) = K , there exists i > d such that ExtiR (K , R) = 0, ExtiR (K , R) = 0 for i < d, ExtdR (K , R) = K , R is Cohen–Macaulay and ExtdR (K , R) = K .

The theorem characterizing the regular rings differs because the only graded regular ring is a polynomial ring. The embedding dimension of a graded ring R is emdim(R) = dim K m/m 2 . A graded ring R is regular if emdim(R) = dim R. (1.2.9) Theorem. Let R be a graded ring of dimension d with the maximal ideal m = R + . Then the following are equivalent: (a) (b) (c) (d) (e) (f)

R is a regular graded ring, gl.dim R < ∞, gl.dim R = d, pd R K = d, m is generated by a regular sequence of length d, R is a polynomial ring over K in d variables.

1.2. Homological and Commutative Algebra

17

The deﬁnitions of Cohen–Macaulay, Gorenstein, and regular rings generalize to the global case. We deﬁne a commutative ring R to be regular (Cohen–Macaulay, Gorenstein) if for every prime ideal P ∈ P in R(R) the localization R P is regular (Cohen–Macaulay, Gorenstein). 1.2.3. Minimal Resolutions We will be working throughout the book with complexes dn

d1

F• : . . . → Fn → . . . → F1 → F0 . . . of free R-modules. We use the following notation. The rank of the free module Fi will be denoted by f i . Sometimes we will choose bases {e(i) j }1≤ j≤ f i in free modules Fi . Then the homomorphism di can be identiﬁed with f i−1 × f i (i) matrix (φk, j )1≤k≤ f i−1 ,1≤ j≤ f i where di (e(i) j )=

f i−1

(i) (i−1) φk, . j ek

k=1

We deﬁne the rank ri of di to be the biggest integer r such that there exists a nonzero r × r minor of φ (i) . We denote by Iri (di ) the ideal of ri × ri minors (i) of the matrix φk, j. Assume that R is a local (resp. graded) ring, and let m denote the maximal ideal (resp. m = R+ ). A complex dn

d1

F• : . . . → Fn → . . . → F1 → F0 of free R-modules is minimal if for each i, 1 ≤ i ≤ n, we have di (Fi ) ⊂ m Fi−1 . Equivalently, after choosing bases in Fi we can identify the differential di with a matrix with entries in R. The minimality condition says that all entries of matrices of differentials di are in the maximal ideal m. (1.2.11) Proposition. Let (R, m) be a local ring. Denote K = R/m. Let M be an R-module. (a) The module M has a minimal free resolution F• . The resolution F• is unique up to isomorphism. (b) Fi ⊗ R R/m = ToriR (R/m, M), so rank Fi = dim K ToriR (R/m, M) . Let us notice that one can construct the minimal free resolution of a ﬁnitely generated module M from short exact sequences πi

0 → i+1 (M) → Fi → i (M) → 0

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Introductory Material

where the modules i (M) and maps π1 are constructed inductively as follows. We take 0 (M) = M and consider the vector space V (M) = M/m M. We choose a basis {v1(0) , . . . v (0) f 0 } of the vector space V (M) and take as F0 a (0) (0) free R-module with a basis {e1(0) , . . . e(0) f 0 }. We deﬁne π0 (e j ) := m j where (0) m (0) j ∈ 0 (M) is an element whose representative modulo m M is v j . The homomorphism π0 is onto by Nakayama’s lemma. Next we deﬁne 1 (M) = Ker π0 and continue the procedure with 1 (M). The module i (M) is the i-th syzygy module of M. It is unique up to isomorphism. The uniqueness of minimal free resolution occurs also in the case of graded rings and graded modules. Let R = i≥0 Ri be a graded ring, as deﬁned in the previous section. We denote by m = R+ the maximal ideal of elements of positive degree. Let M = ⊕i≥0 Mi be a graded R-module. Recall that we deﬁne the shifted module M(n) by setting M(n)i := Mn+i . The analogue of (1.2.11) is true in the graded case. Then one can construct the minimal free resolution as sketched above, as the analogue of Nakayama’s lemma holds, and the analogue of (1.2.11) is true. Let F• be a minimal resolution of M. We will write (i, j) R(− j) f . Fi = j≥i

The numbers f (i, j) are the dimensions of the graded pieces of graded K -vector spaces ToriR (K , M). Sometimes they are called the graded Betti numbers of M. We have a very useful exactness criterion for acyclicity of a ﬁnite complex of free R-modules. (1.2.12) Theorem (Buchsbaum–Eisenbud acyclicity criterion, [BE1]). Let R be a Noetherian ring, and let dn

d1

F• : 0 → Fn → . . . → F1 → F0 be a complex of free ﬁnitely generated R-modules. Then F• is acyclic (i.e. Hi (F• ) = 0 for i > 0) if and only if the following two conditions hold: (a) ri + ri−1 = f i for 1 ≤ i ≤ n + 1 (with the convention rn+1 = 0), (b) depth(I (di )) ≥ i for 1 ≤ i ≤ n. (1.2.13) Proposition. Let F• be a complex from (1.2.12). √ √ (a) We have I (di ) ⊂ I (di+1 ) for 1 ≤ i ≤ n − 1,

1.2. Homological and Commutative Algebra

19

(b) Assume that F• is a resolution of a perfect module M, and let I be the deﬁning ideal of the support of M. Then for every i, 1 ≤ i ≤ n, we √ have I (di ) = I . Proof. Assume that I (di ) = R. Then the map di splits and it follows that all maps d j have to split for j > i. This means that I (d j ) = R for j > i. Applying localization, we get (a). To prove (b) we notice that if F• is a resolution of a perfect module M, then F ∗ is also a resolution of the perfect module with the same support by (1.2.5). Applying (a), we see that all radicals of ideals I (di ) are equal and therefore have to be equal to I . This statement implies the following result of Eagon and Northcott. (1.2.14) Theorem (Generic Perfection Theorem, [EN2]). Let K be a commutative ring, and let R = K[X 1 , . . . , X n ] be a polynomial ring over K. Let F• : 0 → Fm → Fm−1 → . . . → F1 → F0 be a free resolution of an R-module M. Assume that the complex F• is perfect, i.e., m = depth Ann R (M). Assume that M is free as a K-module. Then for every ring homomorphism φ : R → S such that m = depth Ann S (M ⊗ R S), the complex F• ⊗ R S is a free resolution over S of an S-module M ⊗ R S. Proof. Applying Proposition (1.2.13) (b) to the complex F• , we see that all ideals I (di ) for this complex are equal up to a radical to Ann R (M). We deduce that the same is true for the complex F ⊗ R S, and the depth assumption implies that the complex F ⊗ R S is acyclic by the Buchsbaum–Eisenbud criterion.

1.2.4. Effective Calculation of Normalization We describe a very useful algorithm due to Grauert and Remmert and to de Jong ([GRe], [dJ]) for calculating the normalization of a reduced afﬁne ring. The algorithm is based on the following criterion of normality. Consider a radical ideal J containing a nonzero divisor, whose zero set contains the nonnormal locus of R. Then we have canonical inclusions R ⊂ Hom R (J, J ) ⊂ R¯

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Introductory Material

given by the maps r → ψr , ψ →

ψ(x) , x

where ψr is a multiplication by r and x ∈ J is a nonzero divisor. (1.2.15) Proposition ([DGJP]). Let R be a reduced Noetherian ring. Let J be a radical ideal in R containing a nonzero divisor, whose zero set V (J ) contains the nonnormality locus of R. Then R is normal if and only if R = Hom R (J, J ). The proposition implies that if R is not normal, then Hom R (J, J ) is an ¯ different from R. The algorithm consists intermediate ring between R and R, of ﬁnding J and then replacing R by the bigger ring Hom R (J, J ). The authors ¯ prove that after ﬁnitely many steps we reach R. One also has an interesting presentation of Hom R (J, J ) as a ring. One starts with the R-module generators u 0 = x, u 1 , . . . , u s of Hom R (J, J ). Since Hom R (J, J ) is an algebra, we have the quadratic relations s ui u j i, j u k ak = . x x x k=0

We also have the linear relations between u 0 , . . . , u s . Let us assume that the

j relations sk=0 bk u k ( j = 1, . . . , m) are the generators of the ﬁrst syzygies between u 0 , . . . , u s . We deﬁne an epimorphism of commutative R-algebras θ : R[T1 , . . . , Ts ] → Hom R (J, J ). (1.2.16) Proposition ([DGJP]). The kernel of the homomorphism θ is gen

j erated by the linear relations sk=1 bk T j ( j = 1, . . . , m) and the quadratic

s i, j relations Ti T j − k=0 ak Tk (1 ≤ i, j ≤ s). The above algorithm and presentation allow to write down the normalization quite explicitly.

1.2.5. Duality for Proper Morphisms and Rational Singularities In this subsection we use the notions related to derived categories. Our principal references are [H2], [GM]. Apart from the notion of rational singularities, these results will be used only in the proof of the duality statement for complexes F• (V) (Theorem (5.1.4)) and in the proof of the Hinich–Panyushev theorem in section 8.3. Thus this subsection can skipped in the ﬁrst reading.

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21

Let X be a locally Noetherian scheme. We denote by D ∗ (X ) (where ∗ = ∅, +, −, b) the derived category D ∗ (A), where A is the category of ∗ ∗ O X -modules. By DQco X (X ) we denote the thick subcategory DA (A) where A is the category of O X -modules and A = Qco X is a category of quasicoherent O X -modules. Note that by [H2, Proposition I.4.8] the natural embedding ∗ D ∗ (Qco X ) → DQco X (X ) is an equivalence of categories, for a quasicompact ∗ (X ). scheme X and for ∗ = +, ∅. We will use the abbreviation Dqc We start with the discussion of dualizing complexes. Let X be a Noetherian scheme. A complex of quasicoherent O X -modules I • is a dualizing complex of X if I • is bounded, each term of I • is an injective module, each cohomology group is coherent, and the canonical map O X → Hom•O X (I • , I • ) is a quasiisomorphism. A dualizing complex is treated as an object in the derived category, so any complex isomorphic to a dualizing complex in + DQco X (X ) is also called a dualizing complex. If I • is a dualizing complex of X , then for any complex F • of O X -modules with coherent cohomology groups, the canonical map F • → Hom•O X (Hom•O X (F • , I • ), I • ) is a quasiisomorphism. Dualizing complexes are unique in the following sense. (1.2.17) Theorem ([H2, Theorem V.3.1]). Let I • be a dualizing complex on X , and I • a complex of O X -modules bounded above with coherent cohomology groups. Then I • is dualizing if and only if there exists an invertible sheaf + L and an integer n such that I • is isomorphic to I • ⊗O X L[n] in DQco X (X ). In this case L and n are determined by L[n] = R Hom•O X (I • , I • ). Let X be a Noetherian scheme with a dualizing complex I X• . Let r := min{i ∈ Z | H i (I X• ) = 0}. We deﬁne the canonical sheaf ω X := H r (I X• ). The coherent sheaf ω X is deﬁned up to tensoring with an invertible sheaf. If the scheme X is afﬁne, this means that the canonical module ω X is deﬁned uniquely up to isomorphism as an O X -module. (1.2.18) Proposition. Let X be a Noetherian scheme. (a) If X is a Cohen–Macaulay scheme, then the dualizing complex I X• has only one cohomology, so ω X = I X• . (b) If X is a Gorenstein scheme, then ω X = O X .

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Introductory Material

It follows that in the case of afﬁne Cohen–Macaulay schemes we recover the theory of canonical modules as described in [HKu]. Let X, Y be Noetherian schemes. The following theorem was proved by Nagata [N]. (1.2.19) Theorem. Let f : X → Y be a morphism of ﬁnite type between Noetherian schemes. Then f is compactiﬁable, i.e., there exists a scheme X˜ , a proper morphism p : X˜ → Y , and an open immersion i : X → X˜ such that pi = f . The factorization f = pi is called a compactiﬁcation of f . The following theorem is known as a global duality theorem for proper morphisms. (1.2.20) Theorem. Let p : Y → X be a proper morphism. Then the de+ + + (Y ) → Dqc (X ) has the right adjoint p ! : Dqc (X ) → rived functor R + p∗ : Dqc + (Y ). Dqc Let Fin(X ) denote the category of X -schemes of ﬁnite type. A morphism f : Y → Y in Fin(X ) has a compactiﬁcation f = pi by (1.2.19). We deﬁne + + f ! := i ∗ ◦ p ! : Dqc (Y ) → Dqc (Y ),

where p ! is the right adjoint of R + p∗ . The functors f ! have the following properties. (1.2.21) Proposition. With the above notation, the following hold: (a) The deﬁnition of f ! is independent of a compactiﬁcation f = pi. (b) For any two morphisms f, g in Fin(X ) we have (g ◦ f )! = f ! ◦ g ! , provided g ◦ f is deﬁned. (c) If h : Y → Y is a smooth morphism in Fin(X ), of relative dimension d, then h ! is isomorphic to the functor h , where L h (F) = h ∗ F ⊗O ωY /Y [d]. Y

(d) If g : Y → Y is a ﬁnite morphism in Fin(X ), then g ! is isomorphic to g , where g (F) = g¯ ∗ R Hom•OY (g∗ OY , F), where g¯ : (Y, OY ) → (Y , g∗ OY ) is the canonical morphism of ringed spaces associated to g.

1.2. Homological and Commutative Algebra

23

(e) Let f : Y → Y be a morphism from Fin(X ), and let g : Z → Y be a ﬂat morphism of Noetherian schemes. Let Z = Y ×Y Z with the commutative square f

Z ↓g

→

Y

→

f

Z ↓g . Y

Then we have a canonical isomorphism (g )∗ ◦ f ! = ( f )! ◦ g ∗ . (f) Let f : X → Y be a morphism of ﬁnite type. Let IY• be a dualizing complex on Y . Then I X• := f ! (IY• ) is a dualizing complex on X . (1.2.22) Theorem (Duality for Proper Morphisms). Let p : Y → X be a − + (Y ), G • ∈ Dqc (X ). proper morphism between Noetherian schemes, F• ∈ Dqc Then there is an isomorphism θ p : R p∗ R Hom•OY (F• , p ! G • ) ∼ = R Hom•O X (R p∗ F• , G • ). (1.2.23) Remark. The proof of Theorem (1.2.22) is the main subject of Hartshorne’s lecture notes [H2]. The existence of the adjoint p ! is proven in the appendix by Deligne. In our application we will use only the case when p is projective, and therefore one needs only the contents of ﬁrst three chapters of [H2]. The proofs in [H2] are rather complicated, and the signs are not always correct. These questions are addressed in recent lecture notes of Brian Conrad [C], where the fully rigorous version of the duality theorem is developed. An alternative approach based on techniques from algebraic topology was developed by Neeman [Ne]. Still, this approach depends on unbounded derived functors and techniques of Thomason [TT]. The reader should also compare the notes [Ha7] of Hashimoto for fuller treatment of this material. One of the applications of duality for proper morphisms is the notion of rational singularities. We follow the approach of Kempf from [KKMSD]. (1.2.24) Corollary. Let p : Y → X be a proper morphism of smooth varieties. Denote by m, n the dimensions of X, Y respectively. Let L be an invertible sheaf on Y . Assume that R i f ∗ (ωY ⊗ L−1 ) = 0 for i > 0. Then there are natural isomorphisms of sheaves on X given by R i p∗ L = Extn−m+i ( p∗ (ωY ⊗ L−1 ), ω X ). OX Proof. We apply Theorem (1.2.22) to the morphism f , and to the sheafs F = ωY ⊗ L−1 , G = ω X . Then we take the cohomology sheaves of complexes on both sides.

24

Introductory Material

Let X be a smooth scheme of dimension m. A quasicoherent sheaf F on m− j X is Cohen–Macaulay of pure dimension k if ExtO X (F, ω X ) is zero unless j = k is the dimension of the support of F. (1.2.25) Remark. The above deﬁnition is related to the deﬁnition of Cohen– Macaulay modules. Assume that X is an afﬁne smooth variety. Let A be the ˜ For a sheaf F = M ˜ the coordinate ring of X . Then we have ω X = O X = A. m− j condition on vanishing of ExtO X (F, ω X ) is equivalent to the vanishing of m− j modules Ext A (M, A). This translates to the depth condition by Theorem (1.2.1), which, by Proposition (1.2.6) and the following deﬁnition, is equivalent to the Cohen–Macaulayness of the module M. (1.2.26) Corollary. Let p : Y → X be a proper map of smooth varieties. Assume that R i f ∗ (ωY ⊗ L−1 ) = 0 for i > 0 and additionally R i p∗ L = 0 for i > 0. Then the sheaves p∗ (ωY ⊗ L−1 ), p∗ L are Cohen–Macaulay of pure dimension y. In fact the duality F → Extm−n (−, ω X ) interchanges them. We refer to the duality F → Extm−n (−, ω X ) as the Ext duality. It is an exact involution on the category of Cohen–Macaulay sheaves supported in p(Y ). (1.2.27) Definition. Let f : Z → Y be a proper birational morphism, with Z smooth. We call such f a resolution of singularities. The resolution f is a rational resolution if the following conditions are satisﬁed: (a) Y is normal, i.e., the natural map OY → f ∗ O Z is an isomorphism, (b) R i f ∗ O Z = 0 for i > 0, (c) R i f ∗ ω Z = 0 for i > 0. Condition (c) is not needed over a ﬁeld of characteristic 0 because of the following relative version of the Kodaira vanishing theorem. (1.2.28) Theorem (Grauert–Riemenschneider, [GR], [Ke4]). Let K be a map of characteristic 0. Let f : Z → Y be a proper map of smooth varieties deﬁned over K, with k = dim Z , n = dim Y . Then Ri f∗ω Z = 0 for i > k − n.

1.2. Homological and Commutative Algebra

25

(1.2.29) Proposition. Let f : Z → Y be a resolution of singularities. Assume that condition (c) from (1.2.27) holds. Assume that Y can be embedded in a smooth variety X . Then the resolution f is a rational resolution if and only if the following conditions are satisﬁed: (d) OY is Cohen–Macaulay. (e) The natural morphism f ∗ ω Z → ωY (where ωY = Extm−n O X (OY , O X ) is a dualizing sheaf on Y ) is an isomorphism. Proof. Let j : Y → X be an embedding. Let g = j ◦ f . Condition (c) is satisﬁed, so by (1.2.24) we get the isomorphisms j∗ R i f ∗ O Z = Extm−n+i (g∗ ω Z , ω X ). OX Let us assume that conditions (a) and (b) are satisﬁed. Condition (b) implies that g∗ ω Z or f ∗ ω Z are Cohen–Macaulay. If conditions (a) and (b) are satisﬁed, we have that OY = f ∗ O X is the Ext-dual of the Cohen–Macaulay sheaf g∗ ω Z . Thus OY is Cohen–Macaulay. The homomorphism in (e) is the Ext-dual of the isomorphism OY → f ∗ O Z . Thus (d) and (e) hold. Assume that (d) and (e) are true. Then OY is Cohen–Macaulay with Extdual Cohen–Macaulay sheaf ωY by (d). Using (e), we see that f ∗ ω Z is Cohen– Macaulay. The isomorphisms j∗ R i f ∗ O Z = Extm−n+i (g∗ ω Z , ω X ) OX imply condition (b) and that f ∗ O Z is the Ext-dual of f ∗ ω Z . Further, the Extdual of the homomorphism OY → f ∗ O Z of Cohen–Macaulay sheaves is an isomorphism by (e). This implies that the homomorphism OY → f ∗ O Z is an isomorphism. This implies (a), and the proposition is proven. (1.2.30) Remark. Let us assume that we work over a ﬁeld K of characteristic zero. By [GR] the sheaf f ∗ ω Z does not depend on the choice of desingularization Z , only on the variety Y . Therefore conditions (d) and (e) are just conditions on the variety Y . The assumption of embeddability in a smooth variety is locally satisﬁed for any variety Y . (1.2.31) Definition. The variety Y deﬁned over a ﬁeld K of characteristic zero has rational singularities if one of the equivalent conditions holds: (a) Conditions (d) and (e) are (locally) satisﬁed. (b) There exists a desingularization f : Z → Y which is a rational resolution. (c) Every desingularization f : Z → Y is a rational resolution.

26

Introductory Material

We ﬁnish this section with a nice criterion, due to Hinich [Hi], for a scheme to have rational singularities and be Gorenstein. (1.2.32) Proposition. Let f : X → Y be a proper birational morphism of ﬁnite type schemes over a ﬁeld K, with X smooth and Y normal. Suppose that there exists a morphism of sheaves φ : O X → ω X such that f ∗ (φ) : f ∗ (O X ) → f ∗ (ω X ) is an isomorphism. Then Y is Gorenstein and it has rational singularities. Proof. Consider the diagram f

−→

X p

Y !q

.

Spec K By (1.2.21) (b), p (K) = f q (K). Applying (1.2.21) (c) to p : X → Spec (K), one obtains p ! (K) = L p ∗ (K) ⊗ L ω X [n] = ω X [n], where n = dim X = dim Y . Denoting ωY := q ! (K)[−n], we have f ! (ωY ) = ω X . By (1.2.21) (f), ωY is the dualizing complex on Y . Applying the duality theorem (1.2.22) to the morphism f and complexes ω X , ωY , we get !

! !

R f ∗ R Hom X (ω X , ω X ) = R HomY (R f ∗ (ω X ), ωY ).

(∗)

Since H i (R Hom X (ω X , ω X )) = O X for i = 0 and 0 for i > 0, the left hand side can be identiﬁed with R f ∗ O X . By the Grauert–Riemenschneider theorem (1.2.28), R i f ∗ (ω X ) = 0 for i > 0. Since f is proper, birational, X is smooth, and Y is normal, we have f ∗ O X = OY . Therefore by the assumption of the proposition we have the isomorphism α : R f ∗ (ω X ) → OY , α = f (φ)−1 , and (∗) gives the isomorphism β : R f ∗ (O X ) → ωY . Recall that we have a morphism φ : O X → ω X . It induces the morphism ψ = αφβ −1 : ωY → OY . By assumption of the proposition we know that ψ induces an isomorphism H 0 (ψ) in zero cohomology, since R 0 f ∗ (φ) = f ∗ (φ). Let i : U → Y be an open immersion. Then by (1.2.21) (f), the complex i ∗ (ωY ) is a dualizing complex on U , and i ∗ (ψ) induces an isomorphism in zero cohomology. Since being Gorenstein is a local property, it is enough

1.3. Determinants of Complexes

27

to prove the assertion in the case when Y = Spec(R) for some commutative Noetherian ring R. We can therefore work in the category of R-modules. We are given a morphism ψ : ω R → R in D + (R) which induces an isomorphism H 0 (ψ) : H 0 (ω R ) → R. By deﬁnition ψ is represented by a diagram

s

ω R ← C • → R, where s is a quasiisomorphism. The morphism is given by a diagram ... ...

→ C −1 ↓ → 0

d −1

→

C0 ↓ 0 → R

d0

→ C1 ↓ → 0

→

...

→

...

with 0 d −1 = 0. Since H 0 () : H 0 (C • ) → R is an isomorphism, there exists a cycle z ∈ C 0 such that 0 (z) = 1. Therefore the R-module map a → az from R to C 0 extends to a morphism σ : R → C • of complexes. One obviously has σ = 1 R . Thus C • = R ⊕ Ker(). Since C • has a ﬁnite injective dimension (it is a dualizing complex), R also has a ﬁnite injective dimension. Therefore R is Gorenstein. Now, since Y is Gorenstein, ωY has only one nonzero cohomology, so ψ : ωY → OY is an isomorphism. We conclude that R i f ∗ (O X ) = H i (ωY ) = 0 for i > 0.

1.3. Determinants of Complexes In this section we collect the facts we need about determinants of complexes of vector spaces and modules. Let us start with the complex of vector spaces dn

dn−1

dm+1

V• : 0 → Vn →Vn−1 → . . . → Vm+1 → Vm over a ﬁeld K. For a vector space of dimension n we deﬁne its determinant to be the one dimensional vector space det(V ) := n V . Similarly we deﬁne the inverse of the determinant of V by setting det(V )−1 := n (V ∗ ). We deﬁne the determinant of a complex V• to be a one dimensional vector space det(V• ) =

n i=m

i

det(Vi )(−1) .

28

Introductory Material

(1.3.1) Proposition. Let 0 → V → V → V → 0 be an exact sequence of complexes. Then we have a canonical isomorphism det(V ) ⊗ det(V ) → det(V ). Proof. Let {u 1 , . . . , u m } be a basis of V . Let {v1 , . . . , vn } be a basis of V . Denote by g the linear map from V to V , and let f denote the linear map from V to V . We choose the elements w1 , . . . , wn in V so f (wi ) = vi for i = 1, . . . , n. We deﬁne the isomorphism j : det(V ) ⊗ det(V ) → det(V ) by setting j(u 1 ∧ . . . ∧ u m ⊗ v1 ∧ . . . ∧ vn ) := g(u 1 ) ∧ . . . ∧ g(u m ) ∧ w1 ∧ . . . ∧ wn . One checks directly that this isomorphism does not depend on the choice of vectors w1 , . . . , wn or on the choice of bases {u 1 , . . . , u m }, {v1 , . . . , vn }. (1.3.2) Proposition. Let V• be a complex of vector spaces. Let H (V• ) be a complex of vector spaces whose i-th term is Hi (V• ), with zero differential. Then we have a canonical isomorphism Eud : det(V• ) = det(H (V• )). Proof. Decompose the complex V• to short exact sequences 0 → Ker(di ) → Vi → Im(di ) → 0, 0 → Im(di+1 ) → Ker(di ) → Hi (V• ) → 0, and use isomorphisms from Proposition (1.3.1) We have the following properties of determinants of complexes of vector spaces. They easily follow from deﬁnitions. (1.3.3) Proposition. (a) Let V [i]• be a complex V• shifted to the left (i.e. V [i] j := Vi+ j ). Then i det(V [i]• ) = det(V• )(−1) . (b) Let 0 → V• → V• → V• → 0

1.3. Determinants of Complexes

29

be an exact sequence of complexes of vector spaces. Then det(V• ) = det(V• ) ⊗ det(V• ). (c) If the complex V• is exact, we have the isomorphism Eud : det(V• ) → K. Assume that (V• , v) is a based exact complex, i.e., V• is exact and v denotes the choice of bases {v1(i) , . . . , vn(i)i } of vector spaces Vi . Then we have a natural basis element (denoted also v) in one dimensional vector space (−1)i given by the tensor product of volume elements (v1(i) ∧ vn(i)i ) i det(Vi ) and their duals. We deﬁne a determinant of the based complex (V• , v) to be a nonzero scalar det(V• , v) = Eud (v). Let R be a commutative domain, and let dm+1

dn

F• : Fn →Fn−1 → . . . → Fm be a complex of free R-modules. Assume that F• is generically exact. This means that, tensoring with the ﬁeld of fractions K := R(0) , we get an exact sequence of vector spaces. Let us ﬁx bases { f 1(i) , . . . , f n(i) } in the R-modules i Fi . Then we can talk about the element det(F• , f ) ∈ K∗ , where f is the corresponding volume element. For another choice of bases (over R) in Fi corresponding to a volume element f we see that det(F• , f ) = det(F• , f ) u where u is a unit in R. Thus a determinant of a generically exact complex of free modules is well deﬁned as an element of K∗ /R ∗ . We want to investigate the numerator and denominator of this function. Notice that the construction of the determinant of a complex of free modules commutes with the localization. The properties (1.3.3) of determinants of complexes are also true for generically exact complexes of free modules over R. Let us assume that R is a unique factorization domain. We can write

det(F• ) = f ord f (det(F• )) . f ∈Irr(R)

where Irr(R) denotes the set of irreducible elements in R. In order to understand the determinant of F• it is enough to understand the numbers ord f (det(F• )) Let us ﬁx the irreducible element f . We recall that the ideal ( f ) is prime and that the localization R( f ) is a discrete valuation ring. For each ﬁnitely generated R-module M we deﬁne its f -multiplicity mult f (M) as the length of the localization M ⊗ R( f ) . The multiplicity of a ﬁnitely generated R-module

30

Introductory Material

M is ﬁnite if and only if the localization M( f ) is annihilated by some power of f . (1.3.4) Theorem. Let us assume that R is a unique factorization domain. Let F• be a complex of free R-modules that is generically exact. Then the order ord f (det(F• )) of the irreducible element f in the determinant of F• is given by the formula (−1)i mult f (Hi (F• )). ord f (det(F• )) = i

Proof. By localizing we may assume that the ring R is a discrete valuation ring, so we need just to calculate the determinant of a free complex over such ring. However, each homology module Hi (F• ) is torsion, because F• is generically exact. Therefore each Hi (F• ) is a direct sum of cyclic modules (i) R/( f j ). Let G (i) : 0 → G (i) 1 → G 0 be a free resolution of Hi (F• ) (which is f

j

a direct sum of complexes 0 → R →R). We use the decreasing induction on j, starting with j = n + 1 to deﬁne complexes F•( j) such that Hi (F• ) if i < j , Hi (F•( j) ) = 0 otherwise. The complex F•(n+1) = F• . Suppose we constructed the complex F•( j+1) . Its top nonzero homology is H j (F•( j+1) ) = H j (F• ). Therefore we can construct a map ψ j : G ( j) [ j]• → F•( j+1) lifting the identity map on homology in degree j. We deﬁne F•( j) to be the cone of the map ψ j . The complex F•(m−1) will be exact, and thus its determinant will be a unit in R. Using multiplicativity of the determinant with respect to short exact sequences and using exact sequences associated to the cone construction, we see that it is enough to prove the statement of the theorem for the complexes G (•j) . jThis means it is enough to f check the statement for each summand 0 → R →R. Here both sides of the equality give j, so Theorem (1.3.4) is proved. (1.3.5) Corollary. Assume that m = 0 and the complex F• is acyclic in codimension 2, i.e., all modules Hi (F• ) are supported in codimension 2. Then the determinant of F• is the greatest common divisor of the maximal minors of the map d1 . Proof. Using the statement (1.3.4), we see that the only factors that can occur in det(F• ) are the equations of codimension 1 components of the support of

1.3. Determinants of Complexes

31

H0 (F• ). This means after localizing at ( f ) the complex will be acyclic. This again reduces the statement to the case of complexes of length 1 resolving a torsion module, in which case we can check it directly. (1.3.6) Remark. The last statement is closely related to the so-called ﬁrst structure theorem of Buchsbaum and Eisenbud [BE3]. It states that if rank (di ) = ri , then there exists a sequence of maps ai : R → ri Fi−1 such that ri ∗ di = ai ai+1 . In fact one knows (Theorem 3, Chapter 7.) that a complex satisfying the assumptions of (1.3.5) always satisﬁes this theorem. One knows also that for i > 1 the ideal generated by entries of ai has depth ≥ 2. This means that the determinant of F• is the entry of a1 . But the ideal of maximal minors of d1 equals a1 I (a2 ), where I (a2 ) is the ideal of entries of a2 , which has depth ≥ 2. Thus in this case a1 is the greatest common divisor of maximal minors of d1 .

2 Schur Functors and Schur Complexes

In this chapter we develop the representation theory of general linear groups. We follow the approach from [ABW2] based on the explicit characteristic free deﬁnition of Schur and Weyl functors. This approach is sufﬁcient for our goals, and it seems to be easier to grasp for the reader not familiar with representation theory. In section 2.1 we deﬁne Schur and Weyl functors and prove the standard basis theorems. In section 2.2 we discuss the connection of Schur functors with highest weight theory, and provide the alternate deﬁnition using Young symmetrizers in characteristic 0. In section 2.3 we derive various formulas from representation theory, including the Littlewood–Richardson rule and Cauchy formulas. Finally in section 2.4 we give the deﬁnition and basic properties of Schur complexes. 2.1. Schur Functors and Weyl Functors Let us ﬁx a free module E of dimension n over a commutative ring K. Let λ = (λ1 , . . . , λs ) be a partition of a number m. We consider the module Lλ E =

λ1

E⊗

λ2

E ⊗ ... ⊗

λs

E/R(λ, E),

where the submodule R(λ, E) is the sum of submodules: λ1

E ⊗ ... ⊗

λ a−1

E ⊗ Ra,a+1 (E) ⊗

λ a+2

E ⊗ ... ⊗

λs

E

for 1 ≤ a ≤ s − 1, where Ra,a+1 (E) is the submodule spanned by the images of the following maps θ(λ, a, u, v; E) with u + v < λa+1 : u E ⊗ λa −u+λa+1 −v E ⊗ v E ↓ 1⊗⊗1 u E ⊗ λa −u E ⊗ λa+1 −v E ⊗ v E ↓ m 12 ⊗m 34 λa E ⊗ λa+1 E.

32

2.1. Schur Functors and Weyl Functors

33

Let e1 , . . . , en will be a ﬁxed ordered basis of E. We introduce the ordered set [1, n] = {1, 2, . . . , n}, which is the set indexing our basis. We refer to section 1.1.2. for notions related to tableaux used in this section. Let T be a tableau of shape λ with the entries in [1, n]. We associate to T the element in L λ E which is a coset of the tensor eT (1,1) ∧ . . . ∧ eT (1,λ1 ) ⊗ eT (2,1) ∧ . . . ∧ eT (2,λ2 ) ⊗ . . . ⊗ eT (s,1) ∧ . . . ∧ eT (s,λs ) in L λ E. In the sequel we will identify these two objects and we will call both of them the tableaux of shape λ corresponding to the basis {e1 , e2 , . . . , en }. (2.1.1) Remark. It is convenient to think about the relations R(λ, E) in graphical terms using the Young frames. Let us deﬁne the Young scheme of shape λ to be the Young frame of shape λ with some boxes empty and some ﬁlled. We associate to the map θ (λ, a, u, v; E) its Young scheme which is empty in all rows except the a-th and (a + 1)-st, and its restriction to these rows is ... • ... • • ... • • ... • • • ... • • ... • ... with u empty boxes, followed by λa − u ﬁlled boxes in the a-th row, and λa+1 − v ﬁlled boxes, followed by v empty boxes in the (a + 1)-st row. Notice that the condition u + v < λa+1 assures that there will be at least one column in this frame with two boxes ﬁlled. The image of typical element U1 ⊗ . . . ⊗ Ua−1 ⊗ V1 ⊗ V2 ⊗ V3 ⊗ Ua+2 ⊗ . . . ⊗ Us , where U j = eT ( j,1) ∧ . . . ∧ eT ( j,λ j ) , V1 = ex1 ∧ . . . ∧ exu , V2 = e y1 ∧ . . . ∧ e yλa +λa+1 −u−v , V3 = ez1 ∧ . . . ∧ ezv , is a sum of tableaux, where we put in each tableau element T ( j, t) in the t-th box in the j-th row for j = a, a + 1. In the a-th and (a + 1)st row we put x 1 , . . . , xu in the empty u boxes in the a-th row, put z 1 , . . . , z v in the empty v boxes in the (a + 1)st row, and shufﬂe the elements y1 , . . . , yλa +λa+1 −u−v between the ﬁlled boxes in the a-th and (a + 1)st rows, with the appropriate signs coming from exterior diagonal. (2.1.2) Example. Take λ = (3, 3), u = v = 1. The corresponding Young scheme is • • . • •

34

Schur Functors and Schur Complexes

Take x1 = 1, z 1 = 6, {y1 , y2 , y3 , y4 } = {2, 3, 4, 5}. The image of the corresponding vector by θ(λ, 1, u, v; E) is 1 2 3 1 2 4 1 2 5 1 3 4 1 3 5 1 4 5 − + + − + . 4 5 6 3 5 6 3 4 6 2 5 6 2 4 6 2 3 6 (2.1.3) Example. (a) If λ = (t) then L λ E = t E. Indeed, by deﬁnition L λ E = t E/ R((t), E), but R((t), E) = 0, since the partition (t) has only one part. (b) If λ = (1t ) then L λ E = St E. Indeed, the relations Ra,a+1 (E) express the symmetry between the a-th and (a + 1)st row. (c) Let λ = (2, 1). In that case there is only one choice of u, v, namely u = v = 0. The corresponding Young scheme is • • . • The image of θ(1, 0, 0; E) on the typical element e y1 ∧ e y2 ∧ e y3 is y1 y2 y y y y − 1 3 + 2 3. y3 y2 y1 We conclude that L 2,1 E is the cokernel of the diagonal map :

3

E→

2

E ⊗ E.

(d) Let λ = (2, 2). There are three types of relations θ(1, u, v). The choices of u, v are (u, v) = (0, 0), (1, 0), (0, 1). The corresponding Young schemes are • • , • •

• , • •

• • . •

It is easy to see that the relations coming from θ(1, 1, 0; E) are the consequences of two other types of relations. Therefore we have two types of relations. For the map θ (1, 0, 0; E) the image of the typical element e y1 ∧ e y2 ∧ e y3 ∧ e y4 is y1 y2 y y y y y y y y y y − 1 3 + 1 4 + 2 3 − 2 4 + 3 4. y3 y4 y2 y4 y2 y3 y1 y4 y1 y3 y1 y2

2.1. Schur Functors and Weyl Functors

35

For the relation θ (1, 0, 1; E) the image of the typical element e y1 ∧ e y2 ∧ e y3 ⊗ ez is y1 y2 y1 y3 y2 y3 − + . y3 z y2 z y1 z We conclude that L 2,2 E is the factor of 2 E ⊗ 2 E by the images of two maps 3 E ⊗ E → 2 E ⊗ 2 E (corresponding to u = 0, 4 2 v = 1) and the diagonal E→ E ⊗ 2 E (corresponding to u = v = 0). (e) Let λ = (3, 2). There are three choices of pairs u, v: (u, v) = (0, 0), (1, 0), (0, 1). The Young schemes are • • • , • •

• • , • •

• • • . •

It follows that L 3,2 E is a factor of the module 3 E ⊗ 2 E by the images of three maps: 4 E ⊗ E → 3 E ⊗ 2 E (corresponding to 4 3 2 u = 0, v = 1), E ⊗ E→ E⊗ E (corresponding to u=1, v = 0), and 5 E → 3 E ⊗ 2 E (corresponding to u = v = 0). (f) We will show below that for two rowed partitions there are two ways of choosing smaller set of relations θ (1, u, v; E) that still sufﬁce to deﬁne the Schur functor. One choice is to take all pairs (u, v) with u = 0. The other choice is to take all pairs (u, v) with one overlap, i.e. all pairs (u, v) for which the Young scheme has exactly one column with two ﬁlled boxes (equivalently, u + v = λ2 − 1). (g) Let λ = (2, 2, 1). We have two types of relations corresponding to the ﬁrst pair of rows (described in example (d)), and one type corresponding to the second and third rows (described in example (c)). Choosing the relations with one overlap the Young schemes are • • , •

• • • ,

• • . •

(h) Let λ be a hook i.e. a partition of the form λ = ( p, 1q−1 ). Graphically, ... .. λ= .

,

36

Schur Functors and Schur Complexes

with p boxes in the ﬁrst row and q boxes in the ﬁrst column. The relations between two rows of length 1 express the symmetry (cf. example (b)). There is only one type of relations corresponding to the ﬁrst two rows, for the pair u = v = 0. It follows that the Schur functor L ( p,1q−1 ) E is the cokernel of the composition map p+1

⊗1

E ⊗ Sq−2 E −→

p

1⊗m

E ⊗ E ⊗ Sq−2 E −→

p

E ⊗ Sq−1 E.

We recall from section 1.1.2 that a tableau T is standard if the numbers in each row of T form an increasing sequence and the numbers in each column of T form a nondecreasing sequence. This notion plays a key role in representation theory thanks to the following (2.1.4) Proposition. Let E be a free K-module of dimension n. Let e1 , . . . , en be a basis of E. The set ST(λ, [1, n]) of standard tableaux of shape λ with entries from [1, n] form a basis of L λ E. In particular, L λ E is also a free module. Proof of Proposition (2.1.4). First we prove that the standard tableaux generate L λ E. It is clear that the set RST(λ, [1, n]) of row standard tableaux with entries from [1, n] generate L λ E. Let us order the set of such tableaux by the order deﬁned in section 1.1.2. We will prove that if the tableau T is not standard then we can express it modulo R(λ, E) as a combination of earlier tableaux. Let us assume ﬁrst that T has two rows. Since T is not standard, we can ﬁnd w for which T (1, w) > T (2, w). We consider the map θ (λ, 1, u, v; E) for u = w − 1 and v = λ2 − w. The key observation is that image of the tensor V1 ⊗ V2 ⊗ V3 , where V1 = eT (1,1) ∧ eT (1,2) ∧ . . . ∧ eT (1,w−1) , V2 = eT (1,w) ∧ . . . ∧ eT (1,λ1 ) ∧ eT (2,1) ∧ . . . ∧ eT (2,w) , V3 = eT (2,w+1) ∧ . . . ∧ eT (2,λ2 ), contains the tableau T with the coefﬁcient 1, and all the other tableaux occurring in this image are earlier than T in the order . Indeed, in all summands other than T we shufﬂe the smaller numbers from the second row to the ﬁrst one, replacing bigger numbers. Therefore T can be expressed modulo R(λ, E) as a combination of earlier tableaux. Let us consider the general case. If T is not standard, then we can ﬁnd such a and w that T (a, w) > T (a + 1, w). Now we apply the previous argument

2.1. Schur Functors and Weyl Functors

37

to the tableau S which consists of the a-th and (a+1)st rows of T . Notice that the relations R(λ, E) we are using do not do anything to the other rows of T , so we can express T as a sum of earlier tableaux in the order . It remains to prove that the standard tableaux are linearly independent in L λ E. Consider a map φλ :

λ1

E⊗

λ2

E ⊗ ... ⊗

λs

α

E −→ ⊗(i, j)∈λ E(i, j)

β

−→ Sλ1 E ⊗ Sλ2 E ⊗ . . . ⊗ Sλt E, where α is the tensor product of exterior diagonals :

λj

E → E( j, 1) ⊗ E( j, 2) ⊗ . . . ⊗ E( j, λ j )

and β is the tensor product of multiplications m : E(1, i) ⊗ E(2, i) ⊗ . . . ⊗ E(λi , i) → Sλi E. If we imagine the copies of E correspond to the boxes of λ with λ j E corresponding to the boxes in the ﬁrst row and Sλi E corresponding to boxes in the i-th column of λ, we can think of the image φλ (T ) of the tableau T as ﬁrst shufﬂing (with signs) the terms of T in each row, and then multiplying the terms in each column of the tableaux we obtain. The map φλ is called the Schur map associated to the partition λ. (2.1.5) Example. (a) Let λ = (2, 2). Then the map φλ :

2

E⊗

2

E −→ S2 E ⊗ S2 E

is given by the formula φλ (x ∧ y ⊗ u ∧ v) = xu ⊗ yv−yu ⊗ xv−xv ⊗ yu+yv ⊗ xu. (b) It is useful to think about the map φλ in graphical terms as follows. We consider the case λ = (3, 2), but the general case will become clear. The tensor product 3 E ⊗ 2 E has a basis corresponding to the set RST(λ, [1, n]). It can be thought of as a set of standard tableaux of shapes (3) and (2), corresponding to rows of λ. The tensor product S2 E ⊗ S2 E ⊗ E has a basis consisting of triples of costandard

38

Schur Functors and Schur Complexes

tableaux of shapes (2), (2), (1), corresponding to columns of λ. The map φλ acts according to the scheme

↓ , and the image of a tableau T is the sum (with signs) of tableaux obtained from T by shufﬂing each of its rows. The role of the map φλ is explained in the next statement. (2.1.6) Proposition. The image φλ (R(λ, E)) equals 0. Proof. We want to show that the spaces λ1

E ⊗ ... ⊗

λ a−1

E ⊗ Ra,a+1 (E) ⊗

λ a+2

E ⊗ ... ⊗

λs

E

are in the kernel of φλ . Since Ra,a+1 is the span of images of the maps of type θ (λ, a, u, v; E), we choose one such map (i.e., we choose a and u, v such that u + v < λa+1 ). Let us consider the element U = U1 ⊗ U2 ⊗ . . . ⊗ Ua−1 ⊗ U ⊗ Ua+2 ⊗ . . . ⊗ Us , where Ui = xi,1 ∧ xi,2 ∧ . . . ∧ xi,λi ∈ λi E and U is the image under θ(λ, a, u, v; E) of the element x1 ∧ . . . ∧ xu ⊗ y1 ∧ . . . ∧ yλa +λa+1 −u−v ⊗ z 1 ∧ . . . ∧ z v . Let us consider the tensor φλ U . It is formally a sum of tensors in Sλ1 E ⊗ Sλ2 E ⊗ . . . ⊗ Sλt E each of which is the tensor product of products of xi, j ’s, x j ’s, ym ’s, and z p ’s shufﬂed in some way. Let us write our image formally as such a sum by writing products in each Sλj E in the order they get multiplied by β. Let us consider a summand T in our sum. Since u + v < λa+1 , two of the ym ’s have to occur in the same symmetric power Sλj E. Let us choose the smallest such j. Let yb and yc occur in Sλj E. Let us consider another summand T in our sum φλ U which differs from T by changing places of yb and yc when applying the map 1 ⊗ ⊗ 1 from θ(λ, a, u, v; E). We can easily check that the correspondence T → T deﬁnes an involution on the summands in φλ . Moreover, each pair of such summands cancels out in

2.1. Schur Functors and Weyl Functors

39

Sλ1 E ⊗ Sλ2 E ⊗ . . . ⊗ Sλt E, because they come with different signs and the product yb yc is symmetric. This means φλ U = 0. (2.1.7) Example. Let us take λ = (3, 3), u = v = 1. Consider U = x ⊗ y1 ∧ y2 ∧ y3 ∧ y4 ⊗ z. Then if T = x y1 ⊗ y3 y4 ⊗ y2 z then T = x y1 ⊗ y4 y3 ⊗ y2 z. Now T occurs as a summand in φλ (x ∧ y3 ∧ y2 ⊗ y1 ∧ y4 ∧ z), and T occurs as a summand in φλ (x ∧ y4 ∧ y2 ⊗ y1 ∧ y3 ∧ z). One checks easily that T and T cancel out. Proposition (2.1.6) means that φλ induces a surjective map from L λ E to Im φλ . Next we will show that the map φλ maps standard tableaux to linearly independent elements of Sλ1 E ⊗ Sλ2 E ⊗ . . . ⊗ Sλt E. This will prove (2.1.4), and at the same time it will prove that L λ E = Im φλ . In order to see that the images of standard tableaux are linearly independent, we notice that the module Sλ1 E ⊗ Sλ2 E ⊗ . . . ⊗ Sλt E has a basis corresponding naturally to the set RST(λ , [1, n] ) of row costandard tableaux of shape λ . We order this set by the order deﬁned in section 1.1.2. If T is a standard tableau of shape λ, then the smallest element (with respect to the order ) occurring in φλ (T ) is eT (1,1) . . . eT (λ1 ,1) ⊗ eT (1,2) . . . eT (λ2 ,2) ⊗ . . . ⊗ eT (1,t) . . . eT (λt ,t) . Indeed, if in applying the map α we make an exchange of elements in some row, we put bigger elements to the earlier columns, so we get the later (with respect to ) elements. Moreover, one sees instantly that eT (1,1) . . . eT (λ1 ,1) ⊗ eT (1,2) . . . eT (λ2 ,2) ⊗ . . . ⊗ eT (1,t) . . . eT (λt ,t) occurs in α(T ) with coefﬁcient 1. It is also obvious that the initial elements eT (1,1) . . . eT (λ1 ,1) ⊗ eT (1,2) . . . eT (λ2 ,2) ⊗ . . . ⊗ eT (1,t) . . . eT (λt ,t) are different for different standard tableaux T . This proves that the images φλ T of standard tableaux T are linearly independent.• Since the exterior and symmetric powers are GL(E)-modules, and the diagonal and multiplication maps are GL(E)-equivariant, it is clear that the group GL(E) acts on L λ E in a natural way. The space L λ E becomes a GL(E)module, which is called the Schur module corresponding to the partition λ. The notion of a Schur module can be generalized to skew partitions. Let λ/µ be a skew partition. We deﬁne the Schur map φλ/µ :

λ1 −µ1

E⊗

λ2 −µ2

E ⊗ ... ⊗

→ Sλ1 −µ1 E ⊗ . . . ⊗ Sλt −µt E

λ s −µs

E

40

Schur Functors and Schur Complexes

as a composition φλ/µ :

λ1 −µ1

E⊗

λ2 −µ2

E ⊗ ... ⊗

λ s −µs

α

E −→ ⊗(i, j)∈λ/µ E(i, j)

β

−→ Sλ1 −µ1 E ⊗ Sλ2 −µ2 E ⊗ . . . ⊗ Sλt −µt E, where α is the tensor product of exterior diagonals :

λ j −µ j

E → E( j, µ j + 1) ⊗ E( j, µ j + 2) ⊗ . . . ⊗ E( j, λ j )

and β is the tensor product of multiplications m : E(µi + 1, i) ⊗ E(µi + 2, i) ⊗ . . . ⊗ E(λi , i) → Sλi E. If we imagine the copies of E correspond to the boxes in a skew Young frame λ/µ, with λ j −µ j E corresponding to the boxes in the j-th row, and Sλi −µi E corresponding to boxes in the i-th column, we can think of the image φλ/µ (T ) of a tableau T as ﬁrst shufﬂing (with signs) the terms of T in each row, and then multiplying the terms of each column of each summand. (2.1.8) Example. Let λ = (3, 2), µ = (1). The tensor product 2 E ⊗ 2 E has a basis corresponding to the set RST(λ/µ, [1, n]). It can be thought of as a pair of standard tableaux of shapes (2) and (2) (corresponding to the rows of (3, 2)/(1)). The tensor product E ⊗ S2 E ⊗ E has a basis consisting of triples of costandard tableaux of shapes (1), (2), (1) (corresponding to columns of (3, 2)/(1)). The map φ(3,2)/(1) acts according to the scheme

↓

↓

2.1. Schur Functors and Weyl Functors

41

We deﬁne the skew Schur module L λ/µ E to be the image of φλ/µ . The description of the relations and of the standard basis of skew Schur modules is the same as for the Schur modules. (2.1.9) Proposition. (a) L λ/µ E = λ1 −µ1 E ⊗ λ2 −µ2 E ⊗ . . . ⊗ λs −µs E/R(λ/µ, E) where R(λ/µ, E) is spanned by the subspaces: λ1 −µ1

E ⊗ ... ⊗

⊗ ... ⊗

λ s −µs

λa−1 −µa−1

E ⊗ Ra,a+1 (E) ⊗

λa+2 −µa+2

E

E

for 1 ≤ a ≤ s − 1, where Ra,a+1 (E) is the vector space spanned by the images of the following maps θ (λ/µ, a, u, v; E) with u + v < λa+1 − µa : u E ⊗ λa −µa −u+λa+1 −µa+1 −v E ⊗ v E ↓ 1⊗⊗1 λa −µa −u u E⊗ E ⊗ λa+1 −µa+1 −v E ⊗ v E ↓ m 12 ⊗m 34 λa −µa E ⊗ λa+1 −µa+1 E. (b) The standard tableaux of shape λ/µ form a basis of L λ/µ . Proof. First of all one can check by direct calculation that the map φλ/µ sends the elements from R(λ/µ, E) to zero. This is done in the same way as in the case µ = ∅, so we leave it to the reader. Then it is enough to check two things: (1) The standard tableaux of shape λ/µ generate the factor λ1 −µ1

E⊗

λ2 −µ2

E ⊗ ... ⊗

λ s −µs

E/R(λ/µ, E).

(2) The images φλ/µ (T ) for the standard tableaux T of shape λ/µ are linearly independent. Facts (1) and (2) allow us to identify the image of φλ/µ with the factor λ1 −µ1

E⊗

λ2 −µ2

E ⊗ ... ⊗

λ s −µs

E/R(λ/µ, E).

The proof of (1) and (2) is exactly the same as for Schur functors, so we leave it to the reader as an exercise.

42

Schur Functors and Schur Complexes

We can associate to each of the relations θ(λ/µ, a, u, v; E) its Young scheme, as we did in the case µ = ∅. The only difference is that there are missing boxes. Again the condition u + v < λa+1 − µa assures at least one overlap in the Young scheme of the relation. (2.1.10) Example. Let us take λ = (4, 3), µ = (1, 0). The Young scheme of the relation θ((4, 3)/(1, 0), 1, 1, 0; E) is • • . • • • (2.1.11) Remark. We can interpret the relations R(λ/µ, E) in terms of tableaux as we did in the case µ = ∅. Again we start with the case of two rows. The image of typical element V1 ⊗ V2 ⊗ V3 with V1 = ex1 ∧ . . . ∧ exu , V2 = e y1 ∧ . . . ∧ e yλa −µa +λa+1 −µa+1 −u−v , V3 = ez1 ∧ . . . ∧ ezv is a sum of tableaux, where we put in each tableau x 1 , . . . , xu in the u empty boxes in the ﬁrst row, put z 1 , . . . , z v in the v empty boxes in the second row, and shufﬂe the elements y1 , . . . , yλa +λa+1 −u−v between the ﬁlled boxes in the ﬁrst and second rows, with the appropriate signs coming from exterior diagonal. If the number of parts of λ is bigger than 2, the relations R(λ/µ, E) can be interpreted in terms of tableaux as follows. Fix a, u, v. The Young scheme of the map θ (λ/µ, a, u, v; E) has all the boxes empty except the a-th and (a + 1)st, where the scheme is the same as in the case of two rows. The image of the typical element U1 ⊗ . . . ⊗ Ua−1 ⊗ V1 ⊗ V2 ⊗ V3 ⊗ Ua+2 ⊗ . . . ⊗ Us , where U j = ei( j,1) ∧ . . . ∧ ei( j,λ j −µ j ) , V1 = ex1 ∧ . . . ∧ exu , V2 = e y1 ∧ . . . ∧ e yλa −µa +λa+1 −µa+1 −u−v , and V3 = ez1 ∧ . . . ∧ ezv , is the same as in the case of two rows, except that in each summand we put element i ( j, t) in the t-th box in the j-th row for j = a, a + 1. (2.1.12) Example. Let λ = (3, 2), µ = (1, 0), u = v = 0. The corresponding Young scheme is • • . • • Take {y1 , y2 , y3 , y4 } = {2, 3, 4, 5}. The image of the corresponding vector e2 ∧ e3 ∧ e4 ∧ e5 by θ((3, 2)/(1, 0), 1, 0, 0; E) is 2 3 2 4 2 5 3 4 3 5 4 5 − + + − + . 4 5 3 5 3 4 2 5 2 4 2 3

2.1. Schur Functors and Weyl Functors

43

(2.1.13) Example. (a) Consider the skew shape λ = (3, 2), µ = (1, 0). The only possible relation in this case is θ ((3, 2)/(1, 0), 1, 0, 0; E). Its Young scheme is • • . • • The Schur module L (3,2)/(1,0) E is a factor of 2 E ⊗ 2 E by the image 4 E embedded by the exterior diagonal. of (b) Take λ = (4, 2), µ = (2, 0). In this case there are no possible relations, and L (4,2)/(2,0) E is just the tensor product 2 E ⊗ 2 E. (c) The previous example generalizes as follows. Assume that the skew Young frame λ/µ is disconnected, i.e., it can be written as a union of two skew Young frames λ(1)/µ(1) and λ(2)/µ(2) with no boxes in the same row or column. Then L λ/µ E = L λ(1)/µ(1) E ⊗ L λ(2)/µ(2) E. Proof of (c). The relations of type θ (λ/µ, a, u, v; E) between two rows of λ(i)/µ(i) (i = 1, 2) are the same as the relations between corresponding rows of λ/µ. Since λ/µ is disconnected, there are no relations between rows of λ(1)/µ(1) and of λ(2)/µ(2). We conclude this section with the deﬁnition of the skew Weyl modules K λ/µ . They are the duals of the skew Schur modules. To deﬁne the modules K λ/µ E we take the deﬁnition of L λ/µ E, but instead of exterior powers we use divided powers, and instead of symmetric powers we use exterior powers. Thus we deﬁne the Weyl map ψλ/µ : Dλ1 −µ1 E ⊗ Dλ2 −µ2 E ⊗ . . . ⊗ Dλs −µs E →

λ1 −µ1

E ⊗ ... ⊗

λ t −µt

E

as a composition α

ψλ/µ : Dλ1 −µ1 E ⊗ . . . ⊗ Dλs −µs E −→

E(i, j)

(i, j)∈λ−µ β

−→

λ1 −µ1

E ⊗ ... ⊗

λ t −µt

E,

where α is the tensor product of divided diagonals : Dλ j −µ j E → E( j, µ j + 1) ⊗ E( j, µ j + 2) ⊗ . . . ⊗ E( j, λ j ),

44

Schur Functors and Schur Complexes

and β is the tensor product of multiplications m:

E(µi

+ 1, i) ⊗

E(µi

+ 2, i) ⊗ . . . ⊗

E(λi , i)

→

λ i −µi

E.

We deﬁne the skew Weyl module K λ/µ E to be the image of ψλ/µ . The description of the relations and of the standard basis of skew Weyl modules is analogous to that for the skew Schur modules. Before we state it, let us deﬁne the tableaux. As before we ﬁx an ordered basis e1 , . . . , en of E. Let r ( j,1) r ( j,n) . . . en , where of course us choose the tensors U j ∈ Dλ j −µ j , U j = e1 r ( j, 1) + . . . + r ( j, n) = λ j − µ j . Then the image ψλ/µ (U1 ⊗ . . . ⊗ Us ) will be denoted by a tableau T of shape λ/µ which in the j-th row has r ( j, 1) 1’s, r ( j, 2) 2’s, . . . , r ( j, n) n’s. The order of these elements will not matter, because we will assume each tableau to be symmetric in the symbols in every row. We will identify the tableau T with the tensor ψλ/µ (U1 ⊗ . . . ⊗ Us ). (2.1.14) Example. Take λ = (3, 2), µ = ∅. The tableau T =

1 1 2 2 2

is identiﬁed with the image ψ(3,2) (e1(2) e2 ⊗ e2(2) ). (2.1.15) Proposition. (a) K λ/µ E = Dλ1 −µ1 E ⊗ Dλ2 −µ2 E ⊗ . . . ⊗ Dλs −µs E/U (λ/µ, E), where U (λ/µ, E) is the sum of subspaces Dλ1 −µ1 E ⊗ . . . ⊗ Dλa−1 −µa−1 E ⊗ Ua,a+1 (E) ⊗ Dλa+2 −µa+2 E ⊗ . . . ⊗ Dλs −µs E for 1 ≤ a ≤ s − 1, where Ua,a+1 (E) is the module spanned by the images of the following maps θ (λ/µ, a, u, v; E) with u + v < λa+1 − µa : Du E ⊗ Dλa −µa −u+λa+1 −µa+1 −v E ⊗ Dv E ↓ 1⊗⊗1 Du E ⊗ Dλa −µa −u E ⊗ Dλa+1 −µa+1 −v E ⊗ Dv E ↓ m 12 ⊗m 34 Dλa −µa E ⊗ Dλa+1 −µa+1 E. (b) The costandard tableaux ST(λ, µ, [1, n] ) of shape λ/µ (cf. section 1.1.2) with the entries from [1 , n ] form a basis of K λ/µ E.

2.1. Schur Functors and Weyl Functors

45

Proof. For the remainder of the section we write j instead of j for j [1, n]. First of all, one can check by direct calculation that the map ψλ/µ sends the elements from U (λ/µ, E) to zero. This is done in the same way as in the case of Schur modules. Then it is enough to check two things: (1) The costandard tableaux of shape λ/µ generate the factor Dλ1 −µ1 E ⊗ Dλ2 −µ2 E ⊗ . . . ⊗ Dλs −µs E/U (λ/µ, E). (2) The images ψλ/µ (T ) for the costandard tableaux T of shape λ/µ are linearly independent. Facts (1) and (2) allow us to identify the image of ψλ/µ with the factor Dλ1 −µ1 E ⊗ Dλ2 −µ2 E ⊗ . . . ⊗ Dλs −µs E/U (λ/µ, E). The proof of (2) is exactly the same as for Schur modules, so we leave it to the reader. However, we prove fact (1) because the proof in the case of Weyl modules is slightly different. The reason is that the map θ (λ/µ, a, u, v; E) involves the multiplication in the divided powers algebra, which involves some integer coefﬁcients. It is clear that the row costandard tableaux RST(λ/µ, [1, n] ) with entries from [1, n] generate K λ/µ E. Let us order the set of such tableaux by the order deﬁned in section 1.1.2. We will prove that if the tableau T is not costandard, then we can express it modulo U (λ/µ, E) as a combination of earlier tableaux. Let us assume ﬁrst that T has two rows. Since T is not costandard, we can ﬁnd w for which T (1, w) ≥ T (2, w). Let us ﬁnd the smallest w with this property. Let w be the biggest number for which T (2, w ) = T (2, w). We consider the map θ (λ/µ, a, u, v, E) for u = w − µ1 − 1 and v = λ2 − w . The key observation is that image of the tensor V1 ⊗ V2 ⊗ V3 where V1 = eT (1,µ1 +1) ∪ eT (1,µ1 +2) ∪ . . . ∪ eT (1,w−1) , V2 = eT (1,w) ∪ . . . ∪ eT (1,λ1 ) ∪ eT (2,µ2 +1) ∪ . . . ∪ eT (2,w ), V3 = eT (2,w +1) ∪ . . . ∪ eT (2,λ2 ), where ∪ indicates we take the corresponding tensor in the divided power, contains the tableau T with the coefﬁcient 1, and all the other tableaux occurring in this image are earlier than T . The coefﬁcients of multiplication in the divided algebra do not spoil anything, because by choice of w and w we have T (1, w − 1) < T (1, w) and T (2, w ) < T (2, w + 1). Therefore T can be expressed modulo U (λ/µ, E) as a combination of earlier tableaux.

46

Schur Functors and Schur Complexes

Let us consider the general case. If T is not costandard, then we can ﬁnd a and w such that T (a, w) ≥ T (a + 1, w). Now we apply the previous argument to the tableau S which consists of the a-th and (a+1)st rows of T . Notice that the relations U (λ/µ, E) we are using do nothing to the other rows of T , so in the same way we can express T as a sum of earlier tableaux. (2.1.16) Example. Let us standardize the tableau 1 4 5 2 3 3 in K (3,3) E. We have to use the relation θ (λ, a, u, v; E) with u = 1, v = 0: 1 4 5 1 2 3 1 2 4 1 2 5 1 3 3 =− − − − 2 3 3 3 4 5 3 3 5 3 3 4 2 4 5 −

1 3 4 1 3 5 − . 2 3 5 2 3 4

All the tableaux on the right hand side except the third one and the last one are standard, and we have 1 2 5 1 2 3 1 2 4 =− − . 3 3 4 3 4 5 3 3 5 Similarly, 1 3 5 1 2 3 1 3 3 1 3 4 =− −2 − . 2 3 4 3 4 5 2 4 5 2 3 5 Putting all these expressions together, we get 1 4 5 1 2 3 1 3 3 = + . 2 3 3 3 4 5 2 4 5 In the case when µ = 0, i.e. in the case of the partition, we denote K λ/µ by K λ and we call it a Weyl module. (2.1.17) Example. (a) For λ = (t), K λ E = Dt E. Indeed, by deﬁnition K λ E = Dt E/U ((t), E), but U ((t), E) = 0, since the partition (t) has only one part. (b) For λ = (1t ), K λ E = t E. Indeed, the map ψλ is onto in this case. The relations θ (a, u, v; E) between two rows of length 1 correspond to u = v = 0, and they express antisymmetry in the rows.

2.1. Schur Functors and Weyl Functors

47

(c) Let λ = (2, 1). The only relation that occurs in K (2,1) E is θ (1, 0, 0; E). The corresponding Young scheme is • • . • The image by θ (1, 0, 0; E) of the typical element e y1 ∪ e y2 ∪ e y3 is y1 y2 y y y y + 1 3 + 2 3 y3 y2 y1 if y1 , y2 , y3 are all different, y1 y1 y1 y3 + y3 y1 if y1 = y2 = y3 , and y1 y1 y1 if y1 = y2 = y3 . (d) Let λ = (2, 2). There are three types of relations θ (1, u, v; E). The possible pairs (u, v) are (0, 0), (1, 0), (0, 1). The corresponding Young schemes are • • , • •

• , • •

• • . •

Therefore we have three types of relations. For the map θ (1, 0, 0; E) the image of typical element e y1 ∪ e y2 ∪ e y3 ∪ e y4 is y1 y2 y1 y3 y1 y4 y2 y3 y2 y4 y3 y4 + + + + + y3 y4 y2 y4 y2 y3 y1 y4 y1 y3 y1 y2 when all numbers yi are different, with easy adjustments when repetitions occur. For the map θ (1, 0, 1; E) the image of the typical element e y1 ∪ e y2 ∪ e y3 ⊗ ez is y1 y2 y1 y3 y2 y3 + + y3 z y2 z y1 z if all numbers are different, with easy adjustments when repetitions occur. Finally, for the map θ (1, 1, 0; E) the image of typical element

48

Schur Functors and Schur Complexes

ex ⊗ e y1 ∪ e y2 ∪ e y3 is x y1 x y2 x y3 + + y2 y3 y1 y3 y1 y2 if all numbers are different, with easy adjustments when repetitions occur. We note a slight difference between the cases of Schur and Weyl modules. For Schur modules we could eliminate the relation θ (1, 0, 0; E). This is impossible for Weyl modules. Indeed, let us consider the case when all four numbers are the same and they equal y. The relations coming from θ (1, 1, 0; E) and θ (1, 0, 1; E) give the relation 2

y y =0 y y

in K 2,2 E. To get the relation y y =0 y y we need the relation θ (1, 0, 0; E). (e) Let λ = (3, 2). There are three choices of pairs u, v. (u, v) = (0, 0), (1, 0), (0, 1). The Young schemes are • • • , • •

• • , • •

• • • . •

It follows that K 3,2 E is a factor of the module D3 E ⊗ D2 E by the images of three maps: D4 E ⊗ E → D3 E ⊗ D2 E (corresponding to u = 0, v = 1), E ⊗ D4 E → D3 E ⊗ D2 E (corresponding to u = 1, v = 0), and D5 E → D3 E ⊗ D2 E (corresponding to u = v = 0). (f) We will show below that for two rowed partitions one can choose a smaller set of relations θ (1, u, v; E) with u = 0 that still sufﬁce to deﬁne the Weyl functor. Example (c) above shows that the other choice that worked for Schur functors (choosing relations with one overlap) does not deﬁne a Weyl functor. (g) Let λ = (2, 2, 1). We have three types of relations corresponding to the ﬁrst pair of rows (described in example (d)), and one type corresponding to the second and third row (described in example (c)). The Young schemes are • • • • ,

• • , •

• • • ,

• • . •

2.2. Schur Functors and Highest Weight Theory

49

(h) Let λ be a hook, i.e. a partition of the form λ = ( p, 1q−1 ). Graphically ... .. λ= .

,

with p boxes in the ﬁrst row and q boxes in the ﬁrst column. The relations between two rows of length 1 express the antisymmetry (cf. example (b)). There is only one type of relations corresponding to the ﬁrst two rows, for the pair u = v = 0. It follows that the Weyl functor K ( p,1q−1 ) E is the cokernel of the map D p+1 E ⊗

q−2

⊗1

E −→ D p E ⊗ E ⊗

q−2

1⊗m

E −→ D p E ⊗

q−1

E.

We ﬁnish this section by stating the obvious functoriality property of all above constructions. (2.1.18) Proposition. (a) The constructions of modules L λ/µ E, K λ/µ E are functorial with respect to the free module E. They deﬁne the endofunctors of the category of free K-modules. We refer to these functors as Schur functors and Weyl functors respectively. (b) The functors L λ/µ , K λ/µ are polynomial, homogeneous of degree |λ/µ|. (c) We have the functorial isomorphisms K λ/µ E = (L λ /µ E ∗ )∗ . Proof. We start with the proof of (a). The modules L λ/µ E, K λ/µ E are deﬁned as images of natural transformations φλ/µ , ψλ/µ of endofunctors of a category of free K-modules. They are therefore functors themselves. Their values are again in the category of free K-modules by the standard basis theorem. Part (b) follows because the exterior, symmetric, and divided powers are homogeneous polynomial functors. Statement (c) is a consequence of (1.1.7), because that statement implies that the map ψλ /µ is the dual of the map φλ/µ for E ∗ . 2.2. Schur Functors and Highest Weight Theory The modules L λ E play a crucial role in the representation theory of the general linear group. In this section we describe this connection. We assume that the commutative ring K is an inﬁnite ﬁeld of arbitrary characteristic.

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Let us denote by T ( by U ) the subgroup of GL(E) of all diagonal matrices (all upper triangular matrices with 1’s on the diagonal) with respect to a ﬁxed basis e1 , . . . , en of E. We recall that a rational representation of GL(E) is a vector space V together with a homomorphism of algebraic groups ρ : GL(E) → GL(V ). In terms of coordinate rings this means that we have a homomorphism ρˆ : K[GL(V )] −→ K[GL(E)] satisfying the conditions dual to the conditions for homomorphism. A rational representation V is called polynomial if ρ extends to the algebraic map ρ : EndK (E) → EndK (V ). (2.2.1) Proposition. Let V be a rational representation of GL(E), and let n E be a determinant representation of GL(E). Then for m >> 0 the rep resentation V ⊗K ( n E)⊗m is polynomial. Proof. Indeed, for each k, l (1 ≤ k, l ≤ dim V ) the image ρ(Y ˆ k,l ) is an element of K[GL(E)] = K[{X i, j }1≤i, j≤n , T ]/(T det(X i, j ) − 1), where X i, j is the (i, j)-th entry function on GL(E). Multiplying the representation ρ by the ˆ k,l ) will be muldeterminant n E means that the corresponding image ρ(Y tiplied by det(X i, j ). In this way, by multiplying by a sufﬁciently high power of det(X i, j ) we can clear the denominators of all elements ρ(Yi, j ). The following fact is well known in representation theory (cf. [B, section 8]). (2.2.2) Proposition. (a) Every character χ : T → GL(1) = K∗ is of the form (t1 , . . . , tn ) → χ χ χ t1 1 t2 2 . . . tn n for some integers χ1 , χ2 , . . . , χn . Here we denote by (t1 , . . . , tn ) the diagonal n × n matrix with the entries t1 , . . . , tn . (b) Every rational representation V of GL(E) has a decomposition Vχ , V = χ∈char(T)

where Vχ = {v ∈ V |ρ(t)v = χ (t)v} for all t ∈ T. The characters of T are called weights. The subspace Vχ of V is called the weight space of V corresponding to the weight χ . We denote by i the weight i (t1 , . . . , tn ) = ti .

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51

The next step is to investigate how the elements of U change weights of vectors from V . We denote by Ai, j (x) the elementary endomorphism Ai, j (x)(es ) = es + xδs j ei . (2.2.3) Proposition. (a) Let V be a polynomial representation of GL(E), and W a subrepresentation. Let v ∈ Vχ . Then there exists a natural number r and elements v0 , . . . , vr in V such that for every x ∈ K Ai, j (x)v = v0 + xv1 + ... + x r vr . Moreover, the vector vs is a weight vector of weight χ + s(i − j ) and if v ∈ W , then v0 , . . . , vr ∈ W . (b) Every nonzero polynomial representation V of GL(E) contains a nonzero U-invariant weight vector. Proof. The existence of the vectors vi follows at once from the fact that the representation V is a polynomial representation. To calculate the weight of vi we notice that if t = (t1 , . . . , tn ) is a diagonal matrix, then t Ai, j (x) = Ai, j (xti t −1 j )t. Applying both sides to v yields that the weight of vs is χ + s(i − j ). To prove the last statement we notice that we can use the assumption that K is inﬁnite. Then we can ﬁnd r + 1 values x1 , . . . , xr +1 for which the corresponding Vandermonde determinant is nonzero. This means that if W is a subrepresentation and the vectors Ai, j (x1 )v, . . . , Ai, j (xr +1 )v are in W , then v0 , . . . , vr +1 are also in W . This completes the proof of the ﬁrst part. To prove the second part we order the weights χ = (χ1 , χ2 , . . . , χn ) lexicographically with respect to the sequence χ1 , χ2 , . . . , χn . If V is a ﬁnite dimensional rational representation of GL(E), then there exists the earliest weight χ in this order for which Vχ = 0. By part (a) we see that every element of Vχ is a U-invariant. Indeed, if v ∈ Vχ and i < j, then the weights of elements vs from part (a) for s > 0 are earlier than χ. This means that vs = 0 for s > 0, so Ai, j (x)v = v. Since the elements Ai, j (x) (i < j, x ∈ k) generate U, v is a U-invariant. (2.2.4) Example. The canonical tableau cλ (i, j) = j for all (i, j) ∈ λ is a U-invariant in L λ E.

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The following fact is crucial for the whole theory. (2.2.5) Lemma. The vector space (L λ E)U of U-invariants in L λ E is one dimensional (i.e., it is spanned by cλ ). Proof. Let U− be the subgroup of lower triangular matrices with 1’s on the diagonal in GL(E). We will actually prove the opposite result stating that the space of U− -invariants in L λ E is one dimensional, spanned by the anticanonical tableau cˆ λ deﬁned by cˆ λ (i, j) = n − λi + j. We start with two combinatorial results. Let T be a standard tableau, and let us take 1 ≤ i < j ≤ n. Deﬁne the j tableau Si (T ) to be the tableau obtained from T by replacing i by j in every j row in T containing i but not j. Let h i (T ) denote the number of rows in T j containing i but not j. The tableau Si (T ) does not have to be standard. In j some situations it turns out Si (T ) is standard. (2.2.6) Lemma. Let us ﬁx 1 ≤ i < j ≤ n. Assume that T is a standard tableau of shape λ, with entries from [1, n]. Assume that every row of T containing j an integer ≤ i contains also all integers i, i + 1, . . . , j − 1. Then Si (T ) is j j standard, and T is determined by h i (T ) and Si (T ). Proof. From our assumptions it follows that the rows of T not containing i j must follow the rows of T containing i. Thus T is obtained from Si (T ) by j replacing j by i in the ﬁrst h i (T ) rows that contain j but not i. j To show that Si (T ) is standard it is enough to do it in the case when T has j two rows. The action of Si on a row satisfying our assumption can at most replace the entry p with p + 1 for i ≤ p < j. Therefore the only possible j violation of standardness in Si (T ) arises when i occurs in the ﬁrst row of T and, for some p with i ≤ p < j, p occurs in the same positions in the ﬁrst and second row of T . Since the ﬁrst row of T must contain i, i + 1, . . . , p and since the entries in the second row are strictly increasing and T is standard, i must occur in the second row of T in the same position as in the ﬁrst row. If j does not occur in the second row of T , then p in the second row will be replaced by p + 1 and standardness will be preserved. But if j occurs in the second row, then it must occur exactly j − i positions after i, since all integers i + 1, . . . , j − 1 must occur there by assumption. However the ﬁrst row of T

2.2. Schur Functors and Highest Weight Theory

53

must have an element > j that is j − i positions after i, because it contains i + 1, . . . j − 1 but not j. This contradicts the standardness of T . We can pass from any standard tableau T to the anticanonical tableau by applying the composite operator n−1 n n Sn−2 Sn−2 . . . S13 S12 . Sn−1 j

Denote by h i the number of substitutions of j for i made by the application j of Si in that sequence. (2.2.7) Corollary. The standard tableau T of shape λ is determined by the j numbers h i deﬁned above. Proof. The corollary follows from Lemma (2.2.6) by induction. Now we conclude the proof of (2.2.5). For 1 ≤ i < j ≤ n and for x ∈ K we denote by A j,i (x) the matrix from U− which has entries on the diagonal equal to 1, the entry in the ( j, i)th position equal to x, and all other entries equal to 0. Consider the expansion from Proposition (2.2.3). It is clear that j j for v = T we have vr = Si (T ) and that r = h i (T ). Consider a nonzero linear combination m as Ts y= s=1

of standard, distinct tableaux Ts with all as = 0. To prove (2.2.5) it is enough to show that the anticanonical tableau cˆ λ is contained in the span of U− y. Consider A j,i (x)(y). This can be expanded as a polynomial in x, as in (2.2.3). The expansion gives j as Si (Ts ) + {terms of lower degree in x}. A j,i (x)(y) = x h j

h i (Ts )=h

The element

y =

j

as Si (Ts )

j

h i (Ts )=h

is nonzero by (2.2.6) and by the assumption as = 0. The element y is contained in the linear span of U− y by the Vandermonde determinant argument used in the proof of Proposition (2.2.3).• The statement (2.2.5) has important consequences.

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(2.2.8) Proposition. The submodule Mλ E of L λ E generated by the canonical tableau cλ is an irreducible GL(E)-module. Every irreducible polynomial representation of GL(E) is isomorphic to Mλ E for some λ. Proof. A nonzero submodule W of Mλ E contains a U-invariant element, so it contains cλ . This means that W = Mλ E. This proves the ﬁrst statement. To prove the second part of the proposition let us consider the irreducible polynomial representation V of GL(E). By the second part of (2.2.3) we see that V contains for some weight λ a nonzero U-invariant vector vλ of weight λ. We will show that V and Mλ are isomorphic. Let us consider the submodule W in V ⊕ Mλ generated by (vλ , cλ ). We consider two projections p : W → V and q : W → Mλ . Both maps p, q are nonzero, so they are epimorphisms. Let us assume that one of those maps, say p, has nonzero kernel. This means that the module Ker p is isomorphic to Mλ , so W = V ⊕ Mλ . This is however impossible, because the weight space Wλ is one dimensional. Indeed, if U− denotes the group of lower triangular matrices with 1’s on the diagonal, then the subset U− TU is a Zariski dense subset of GL(E). Therefore W is spanned by U− TU(vλ , cλ ), which is the span of U− (vλ , cλ ). By part (b) of (2.2.3) we see that the only weight vector in the last span is (vλ , cλ ). The same reasoning works for the projection q. Thus p and q are isomorphisms and we are done. If K is a ﬁeld of characteristic zero, then it is well known (see [Hu1] for the proof) that GL(E) is linearly reductive, i.e., all ﬁnite dimensional representations are direct sums of simple ones. Then it follows instantly from (2.2.7) that L λ E is irreducible. Let us state these facts. (2.2.9) Theorem. All irreducible rational representations of GL(E) are iso morphic to Mλ E ⊗K ( n E)⊗m for some partition λ = (λ1 , . . . , λn−1 ) and m ∈ Z. This correspondence is bijective. (2.2.10) Theorem. Assume that K is a ﬁeld of characteristic zero. (a) We have Mλ E = L λ E. (b) Every rational representation of GL(E) is a direct sum of irreducibles. Theorem (2.2.9) is the main statement of the highest weight theory. The irreducible representations are parametrized by the sequences (λ1 , . . . , λn ) with λi ∈ Z, λ1 ≥ . . . ≥ λn . These are dominant integral weights for the group GL(E). They can be deﬁned as the weights whose values on the roots of GL(E)

2.2. Schur Functors and Highest Weight Theory

55

are integral, and whose values on positive roots are nonnegative. The integral weights are the weights whose values on the roots of GL(E) are integral. Let V be a rational representation of GL(E). We deﬁne the character of V χ char(V ) = dimVχ x1 1 . . . xnχn , where x1 , . . . , xn are indeterminates. Obviously we have char(V ⊕ V ) = char(V ) + char(V ), char(V ⊗ V ) = char(V )char(V ). (2.2.11) Remark. (a) When K is a ﬁeld of characteristic 0, then every representation is determined by its character. This follows easily from linear reductivity (2.2.10) (b) and from (2.2.5). (b) The function char(V ) is a symmetric function of x 1 , . . . , xn , because for each χ1 , . . . , χn and for each permutation σ ∈ n we have dim Vχ1 ,...,χn = dim Vχσ (1) ,...,χσ (n) . Indeed, the permutation matrix in GL(N ) corresponding to σ carries one space isomorphically into another. (c) The function char(L λ E) is called the Schur function (cf. [MD, chapter I]). The Schur functions play an important role in combinatorics. (d) The correspondence E → L λ E gives rise to a functor from the category VectK to itself. We will refer to it as a Schur functor. We ﬁnish our discussion with the simple example showing that (2.2.10) is false in positive characteristics. (2.2.12) Example. Let char K = p > 0. Let λ = (1 p ). Then L λ E = S p E, p but Mλ E is the span of the elements ei for 1 ≤ i ≤ n. This subspace is GL(E)-invariant, because in characteristic p we have (x + y) p = x p + y p . For the remainder of this section we assume that the commutative ring K has characteristic 0, i.e., it is a Q-algebra. We give an alternate description of Schur modules using Young idempotents (cf. [DC]). Consider the natural action of the symmetric group m on the tensor product E ⊗m given by σ (v1 ⊗ . . . ⊗ vm ) = vσ −1 (1) ⊗ . . . ⊗ vσ −1 (m) . Let λ be a partition of m and let D be a tableau of shape λ, with entries from [1, m] of weight (1m ) (i.e. with distinct entries). We deﬁne the Young

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symmetrizer e(D) ∈ K[m ] as follows. Denote by R(D) (by C(D)) the subgroup of m of permutations preserving the rows (columns) of D. We set e(D) = sgn(σ ) τ σ. τ ∈R(D) σ ∈C(D)

We deﬁne the representation of GL(E) depending on the tableau D by L D (E) = e(D)E ⊗m . (2.2.13) Lemma. (a) If D, D are the tableaux of the same shape λ, then L D (E) and L D (E) are isomorphic as GL(E)-modules. (b) If D is a tableau of shape λ, then L D (E) is isomorphic to the Schur module L λ (E). Proof. We start with part (a). Let σ be a permutation such that σ (D) = D , i.e., for every (i, j) ∈ D(λ) we have D (i, j) = σ (D(i, j)). Then we have R(D ) = σ R(D)σ −1 , C(D ) = σ C(D)σ −1 , and therefore e(D ) = σ e(D)σ −1 , which implies L D (E) = σ (L D (E)). Indeed, the isomorphism is given by acting by σ on a tensor. To prove (b) let us choose D which is a row standard tableau, minimal with respect to the order deﬁned in section 1.1.2 or, more precisely, D(i, j) = λ1 + . . . + λi−1 + j. We can identify representations L λ E and L D (E) as follows. L λ E can be interpreted as Im φλ . We embed Im φλ into the tensor product ⊗(i, j)∈D(λ) E by symmetrizing in each column. This can be done, because in characteristic 0 the symmetric power can be identiﬁed with the set of symmetric tensors (which in a characteristic free way is isomorphic to the divided power). Call the symmetrization map η. Then the image ηφλ (v) can be identiﬁed with e(D)(α(v)), where α is the product of exterior diagonals used to deﬁne φλ . The approach based on the use of the action of m on E ⊗m , due to Schur, can also give the irreducibility of Schur modules in characteristic 0. We sketch the basic steps in the proof. The interested reader may consult [DC]. The actions of m and of GL(E) on E ⊗m commute. Let us denote the span of all endomorphisms of type g ⊗m (g ∈ GL(E)) in EndK (E ⊗m ) by S(m, E). We also denote by (m, E) the endomorphisms of E ⊗m that are induced by the elements of the group ring K[m ].

2.3. Properties of Schur Functors

57

(2.2.14) Lemma (Schur Commutation Lemma). The algebras S(m, E) and (m, E) are their own commutants in EndK (E ⊗m ). More precisely, S(m, E) = {h ∈ EndK (E ⊗m ) | gh = hg for all g ∈ (m, E) }, (m, E) = {h ∈ EndK (E ⊗m ) | gh = hg for all g ∈ S(m, E) }. The algebra (m, E) is semisimple by Maschke’s theorem. This means the action of S(m, E) on E ⊗m is also semisimple and the Schur modules are precisely the irreducible representations. This follows from the following facts, proven in [DC], in the appendix on non-commutative algebra, part IV. (2.2.15) Proposition. Let B be a semisimple subalgebra in the matrix algebra Mn (K). Let C be the commutant of B, i.e., C = {x ∈ M N (K) | x y = yx for all y ∈ B }. Then the subalgebra C is also semisimple. Denote by VC the vector space V = K N with the structure of a C-module. Every simple module in VC is isomorphic to a module bV where b is an element of a minimal left ideal in B. If b, b generate the same left ideal in B, then bVC and b VC are isomorphic as C-modules. (2.2.16) Proposition. The isomorphism classes of minimal left ideals in K [m ] are in one to one correspondence with partitions of m. The correspondence is given by associating to λ the left ideal generated by e(D), where D is an arbitrary tableau of shape λ with entries from [1, m], with distinct entries.

2.3. Properties of Schur Functors. Cauchy Formulas, Littlewood–Richardson Rule, and Plethysm In this section we discuss some formulas from the representation theory of general linear groups. They will be used in the calculations involving vector bundles in chapters 6 through 9. We try to give both characteristic 0 and characteristic free statements. We start with the direct sum decompositions. Let E and F be two free modules over a commutative ring K. Our formulas express the modules L λ/µ (E ⊕ F) and K λ/µ (E ⊕ F) through the corresponding modules for E and F.

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(2.3.1) Proposition. (a) There is a GL(E) × GL(F)-equivariant ﬁltration on L λ/µ (E ⊕ F) with the associated graded object L λ/ν E ⊗ L ν/µ F. ν|µ⊂ν⊂λ

If K is a commutative ring of characteristic 0, then we have a GL(E) × GL(F)-equivariant isomorphism L λ/µ (E ⊕ F) = L λ/ν E ⊗ L ν/µ F. ν|µ⊂ν⊂λ

(b) There is a GL(E) × GL(F)-equivariant ﬁltration on K λ/µ (E ⊕ F) with the associated graded object K λ/ν E ⊗ K ν/µ F. ν|µ⊂ν⊂λ

If K is a commutative ring of characteristic 0, then we have a GL(E) × GL(F)-equivariant isomorphism K λ/µ (E ⊕ F) = K λ/ν E ⊗ K ν/µ F. ν|µ⊂ν⊂λ

Proof. Since the proofs of both parts of the proposition are the same, we will just prove part (a). We observe that it is enough to prove the ﬁrst statement, since from it one deduces that the characters of the left and right hand sides of the second formula are the same. Let us denote dim E = n, dim F = m. Let us choose the bases e1 , . . . , en of E, f 1 , . . . , f m of F. Then e1 , . . . , en , f 1 , . . . f m is the basis of E ⊕ F, and we order it so f 1 < . . . < f m < e1 < . . . < en . We can consider the tableaux corresponding to this basis. For each such tableau T we deﬁne its F-part f (T ) to be the sequence a1 , . . . , as , where ai is the number of elements f j in the i-th row of T . We order sequences f (T ) lexicographically and denote this order by . Now for each ν such that µ ⊂ ν ⊂ λ we deﬁne the subspace Fν to be the span of all tableaux T such that (ν1 − µ1 , . . . , νs − µs ) f (T ). It is clear that Fν is a GL(E) × GL(F)-submodule. We also order all possible partitions ν by saying that ν ξ if (ξ1 , . . . , ξs ) (ν1 , . . . , νs ). It is clear by deﬁnition that if ξ ν then Fξ ⊂ Fν . Claim. Fν /

ξ ≺ν

Fξ is a factor of L ν/µ F ⊗ L λ/ν E.

It is obvious from the deﬁnition that the factor Fν / ξ ≺ν Fξ is spanned by the tableaux T such that f (T ) = (ν1 − µ1 , . . . , νs − µs ). Each tableau T such

2.3. Properties of Schur Functors

59

that f (T ) = (ν1 − µ1 , . . . , νs − µs ) can be considered as a pair of tableaux: the tableau T (1) of shape ν/µ with the entries from the set { f 1 , . . . , f m }, and the tableau T (2) of shape λ/ν with the entries from the set {e1 , . . . en }. We will denote such T by (T (1), T (2)). We show that the standard relations deﬁning L ν/µ F ⊗ L λ/ν E are satisﬁed

in Fν / ξ ≺ν Fξ .

Let us consider the element in Fν / ξ ≺ν Fξ which is the relation of type θ (λ/ν, a, u, v; E) (cf. section 2.1) on the tableau T (2). More precisely, let us denote by T (1) j the j-th row of T (1), and let us choose the rows T (2) j for j different than a, a + 1, and ﬁnally let us choose V1 ∈ u E, λa −νa +λa+1 −νa+1 −u−v v V2 ∈ E and V3 ∈ E. We consider the relation which shufﬂes the entries of V2 into the last λa − νa − u spots in the a-th row of λ/ν and the ﬁrst λa+1 − νa+1 − v spots in the (a + 1)st row of λ/ν and leaves all other entries in their spots. Let us call this relation R1 . We want to show that

this relation is zero in Fν / ξ ≺ν Fξ . Let us consider the relation R2 of type θ (λ/µ, a, νa − µa + u, v; E ⊕ F) which shufﬂes the entries of V2 ∧ T (1)a . This relation, being the deﬁning relation of L λ/µ (E ⊕ F), is identically zero

in Fν / ξ ≺ν Fξ . The relation R2 has more summands than R1 . However, all the summands occurring in R2 and not in R1 involve shufﬂing some basis elements from F into the earlier rows of λ/µ. Such elements are automati

cally contained in ξ ≺ν Fξ , so they are automatically zero in Fν / ξ ≺ν Fξ . Similarly we deal with the relations on the F-side. This proves our claim. Now the statement of the proposition follows, because by the standard basis theorem the left and right hand sides of the second formula have the same dimension, so in fact Fξ = L ν/µ F ⊗ L λ/ν E. Fν / ξ ≺ν

This completes the proof of (2.3.1). Next we discuss the Cauchy formulas. Let E and F be two free modules. We are interested in the modules Sm (E ⊗ F) and m (E ⊗ F). We want to express these modules in terms of Schur and Weyl modules. (2.3.2) Theorem. (a) There is a natural ﬁltration on Sm (E ⊗ F) whose associated graded object is L λ E ⊗ L λ F. |λ|=m

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(b) There is a natural ﬁltration on object is

m

(E ⊗ F) whose associated graded

L λ E ⊗ K λ F.

|λ|=m

We will prove part (a) of this theorem in chapter 3. We also give there the description of the ﬁltration giving part (b). For the proof we refer to [ABW2], or exercises 4, 5, 6 in chapter 3. Now we state the characteristic zero consequence. (2.3.3) Corollary. Let K be a commutative ring of characteristic 0. We have natural isomorphisms L λ E ⊗ L λ F, Sm (E ⊗ F) = |λ|=m m L λ E ⊗ L λ F. (E ⊗ F) = |λ|=m

Proof. Without loss of generality we can assume that K is a ﬁeld. Indeed, if the theorem is true over the ﬁeld Q of rational numbers, then the result extends to any commutative ring of characteristic 0 by base change. By Theorem (2.3.2) the characters of both sides of our formulas are the same (we notice that since char K = 0, K λ F = L λ F). Using the remark (2.2.11) (a), we get our statement. The formulas from Corollary (2.3.3) are special cases of the problem of outer plethysm. The general problem is to ﬁnd the multiplicities v(µ, ν, λ) in the decomposition L λ (E ⊗ F) = v(µ, ν, λ) L µ E ⊗ L ν F. |µ|=|ν|=|λ|

This is a very difﬁcult problem, solved in very few cases. Notice that substituting in (2.3.3) (a) F ⊗ G for F, we see that Sm (E ⊗ F ⊗ G) = v(µ, ν, λ) L λ E ⊗ L µ F ⊗ L ν G, |λ|=|µ|=|ν|=m

so the multiplicities v(µ, ν, λ) are symmetric in λ, µ, and ν.

2.3. Properties of Schur Functors

61

This interpretation explains why the problem of outer plethysm is so complicated. Let us consider the action of the group SL (E) × SL(F) × SL(G) on E ⊗ F ⊗ G. The dimension of E ⊗ F ⊗ G is in general bigger than the dimension of the acting group SL(E) × SL(F) × SL(G). This means that the structure of the ring of invariants is very complicated. Finding the expression for the Hilbert function of the ring of invariants is, however, equivalent to ﬁnding some of the multiplicities v(µ, ν, λ). This does not exclude the existence of a combinatorial formula for v(µ, ν, λ), but it means that such a formula will not lead to an easy calculation of our multiplicities. Next we state the Littlewood–Richardson rule. It describes a decomposition of the tensor product of Schur functors into Schur functors. Let K be a commutative ring of characteristic 0. Let λ, µ be two partitions. In the case where K is a ﬁeld we have by (2.2.10) u(λ, µ; ν)L ν E, Lλ E ⊗ Lµ E = |ν|=|λ|+|µ|

where u(λ, µ; ν) are some multiplicities. This decomposition carries over to the case of an arbitrary ring K of characteristic 0. Indeed, the explicit isomorphism over Q remains an isomorphism when tensored with K. The Littlewood–Richardson rule gives a beautiful combinatorial description of these multiplicities. In order to state the rule, we need one combinatorial notion. A word w = w1 . . . wt , with w1 , . . . , wt being positive integers, is a lattice permutation if for each s(1 ≤ s ≤ t) and each positive integer i, the number of occurrences of i in w1 , . . . , ws is not smaller than the number of occurrences of i + 1. Let T be a tableau of skew shape ν/λ. From such T we form a word w(T ) by reading T column by column, starting in each column with the lowest entry. In other words, w(T ) = (T (ν1 , 1), T (ν1 − 1, 1), . . . , T (ν1 − λ1 + 1, 1), T (ν2 , 2), . . . , T (νs − λs + 1, s)). We say that the tableau T satisﬁes the condition LP if the word w(T ) is a lattice permutation. Let us denote by P(λ, µ; ν) the set of all standard tableaux of shape ν/λ of weight µ satisfying the condition LP. Then we have (2.3.4) Theorem (Littlewood–Richardson Rule). u(λ, µ; ν) = card P(λ, µ; ν).

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Again there is a characteristic free statement of the Littlewood–Richardson rule involving ﬁltrations (cf. [Bo2]), but we will not need it in the applications. Since we will use the rule sporadically, we give a combinatorial proof as a series of exercises at the end of this chapter. The reader might also look for the proof in [MD, chapter I]. MacDonald proves the statement about the symmetric functions, but we notice that his statement means that the representations L λ E ⊗ L µ E and |ν|=|λ|+|µ| u(λ, µ; ν)L ν E, with u(λ, µ; ν) deﬁned by (2.3.4), have the same characters. Let us state two important special cases of the Littlewood–Richardson rule, known as Pieri’s formulas. We recall that a skew partition λ/µ is called a vertical strip if it contains at most one box in each row, i.e., λi ≤ µi + 1 for all i. We denote the set of all vertical strips by VS. Similarly, the skew partition λ/µ is a horizontal strip if it contains at most one box in each column, i.e. when λ /µ is a vertical strip. We denote the set of all horizontal strips by HS. (2.3.5) Corollary (Pieri’s Formulas). Let K be a commutative ring of characteristic 0. Then we have the natural isomorphisms L ν E, Lλ E ⊗ Sj E = {ν|λ⊂ν,|ν/λ|= j,ν/λ∈VS}

Lλ E ⊗

j

E=

L ν E.

{ν|λ⊂ν,|ν/λ|= j,ν/λ∈HS}

The Littlewood–Richardson rule has an analogue for skew Schur functors. Assuming that K has characteristic 0, we can write L ν/µ E = w(ν/µ; λ)L λ E |λ|=|ν/µ|

Then we have (2.3.6) Theorem (Littlewood–Richardson Rule for Skew Schur Functors). w(ν/µ; λ) = u(λ, µ; ν) = card P(λ, µ; ν). Again we refer for the proof of this fact to [MD, chapter I]. The characteristic free statement involving ﬁltrations is also true (cf. [Bo2]). We state the analogues of Pieri’s formulas for skew shapes.

2.3. Properties of Schur Functors

63

(2.3.7) Corollary. Let K be a commutative ring of characteristic 0. (a)

L ν/(1 j ) E =

L λ E.

{λ | λ⊂ν, |ν/λ|= j, ν/λ∈VS }

(b)

L ν/( j) E =

L λ E.

{λ | λ⊂ν, |ν/λ|= j, ν/λ∈HS }

We conclude this section with a brief discussion of the problem of inner plethysm. The general problem is to decompose the functor L λ (L µ E) into Schur functors. This problem is probably even more difﬁcult than the outer plethysm. To see why, let us look at the situation from the point of view of invariant theory. We look at the special case λ = (1m ), µ = (1n ). Decomposing Sm (Sn E) into Schur functors involves the formula for the dimension of the homogeneous components of the ring of invariants of SL(E) acting on Sn E. Such rings are extremely complicated, at least according to nineteenth century invariant theorists. In view of this remark it is logical to expect nice formulas for Sn (S2 E) and Sn ( 2 E), because the action of GL(E) on S2 E or 2 E has ﬁnitely many orbits, so the rings of SL(E)-invariants are very simple. Such formulas are given in the next proposition. (2.3.8) Proposition. Let K be a commutative ring of characteristic 0. (a)

Sm (S2 E) =

L λ E,

|λ|=2m, λi even for all i

(b) 2 Sm E =

L λ E.

|λ|=2m, λi even for all i

Proof. It is enough to prove the proposition when K is a ﬁeld. The corresponding formulas for characters are given in [MD, chapter I]. However, it is convenient for later applications to give a proof based on U-invariants. Let us ﬁx an ordered basis e1 , . . . , en of the vector space E. We ﬁrst prove the formula (a). We identify the symmetric algebra S = Sym(S2 E) with the polynomial ring K[X i, j ], where X i, j = ei e j . We consider the generic n × n symmetric matrix X = (X i, j )1≤i, j≤n over S. For each r , 1 ≤ r ≤ n, we choose

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Schur Functors and Schur Complexes

the r × r minor of X by deleting rows and columns with numbers > r . Let gr be the determinant of this minor. Then an easy calculation shows that gr is an U-invariant of weight (2r , 0n−r ) in Sr (S2 E). Let λ be a partition of 2m such that λj = 2µ j for 1 ≤ j ≤ n. This means that for each i > 0, λ2i−1 = λ2i . Therefore the product gλ = i>0 gλ2i is a nonzero U-invariant of weight λ in Sm (S2 E). This means that the left hand side of (a) contains the right hand side of (a). Now we prove (a) by induction on n = dim E. For n = 1 the formula is obvious. Let us assume that the formula (a) is true for dim E = n. We will prove that the left and right hand sides have the same dimension for dim E = n + 1. In view of the above construction, it is enough to prove (a). To prove the statement about the dimensions, let us consider the space E ⊕ K of dimension n + 1 over K. We will actually prove that the left and right hand sides of (a) for E ⊕ K are isomorphic as GL(E)-modules. The left hand side decomposes in the following way: S2 (E ⊕ K) = S2 E ⊕ E ⊕ K. Therefore Sm (S2 (E ⊕ K)) = Si (S2 E) ⊗ S j E ⊗ Sl K i+ j+l=m

=

Si (S2 E) ⊗ S j E.

i+ j≤m

Applying the formula (a) for i ≤ n, we get Sm (S2 (E ⊕ K)) =

L ν E ⊗ S j E.

i+ j≤m |ν|=2i, νl even for all l

On the other hand, for the right hand side we get L λ (E ⊕ K) = |λ|=2m, λi even for all i

|λ|=2m, λi even for all i

L µ E.

µ⊂λ, λ/µ∈VS

It remains to show that for any partition α, L α E occurs in the right hand sides of both formulas with the same multiplicity. Let us ﬁx m and α. The multiplicity of L α E in the ﬁrst formula is equal to the cardinality of the set Aα1 = {(ν, i, j) | |ν| = 2i, νl is even for all l, ν ⊂ α, α/ν ∈ VS, |α/ν| = j, i + j ≤ m}. The multiplicity of L α E in the second formula is equal to the cardinality of the set Aα2 = {λ | |λ| = 2m, λl even for all l, α ⊂ λ, λ/α ∈ VS}.

2.3. Properties of Schur Functors

65

To ﬁnish the proof of (a) it is enough to construct a bijection h from Aα1 to Aα2 . Let (ν, i, j) ∈ Aα1 . We construct λ ∈ Aα2 as follows. Let us deﬁne the numbers a j (0 ≤ j). If α j is even, then α j − ν j = 2a j . If α j is odd, then α j − ν j = 2a j + 1. We also deﬁne a−1 = 0. Let us construct the partition β by adding to the j−th column of α 2a j−1 boxes if α j is even and 1 + 2a j−1 boxes if α j is odd. Then β is a partition of 2i + 2 j (we added j boxes to α) with each β j even. We construct λ = h(ν, i, j) by adding to the ﬁrst column of β 2(m − i − j) boxes. The reader will check easily that the map h deﬁnes a bijection from Aα1 to Aα2 . This proves the statement (a). We prove (b) using the same technique. We identify S = Sym( 2 E) with the polynomial ring in the variables X i, j = ei ∧ e j . Then we consider the n × n skew symmetric generic matrix X = (X i, j )1≤i, j≤n over S. For each even number 2r, 0 ≤ 2r ≤ n, we deﬁne the element g2r to be the Pfafﬁan of the skew symmetric 2r × 2r matrix obtained from X by deleting rows and columns with numbers > 2r . We see easily that for each 2r the element g2r is a U-invariant of the weight (12r , 0n−2r ). Now for a partition λ with all λi even we ﬁnd that gλ = i gλi is a nonzero U-invariant of weight λ . This shows that the right hand side of (b) is contained in the left hand side. Then we can ﬁnish the proof of (b) with a similar (but easier) reasoning to the one in the proof of (a). It turns out that the companion formulas for m (S2 E) and m ( 2 E) also can be easily obtained. Let us recall that every partition λ can be written in the hook notation as λ = (a1 , . . . ar |b1 , . . . , br ) (cf. section 1.1.2). Let us denote by Q 1 (m) the set of partitions λ of m for which ai = bi + 1 for each i. Similarly we denote by Q −1 (m) the set of partitions λ of m for which bi = ai + 1 for each i. Then we have (2.3.9) Proposition. Let K be a commutative ring of characteristic 0. (a)

m

(S2 E) =

L λ E.

λ∈Q −1 (2m)

(b) m 2 E =

L λ E.

λ∈Q 1 (2m)

Proof. The corresponding formulas for characters are given in [MD, chapter I]. Let us just indicate the nonzero U-invariants in m (S2 E) and in m ( 2 E).

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The proof of (2.3.9) can be ﬁnished by an argument similar to the one in the proof of (2.3.8). Let λ be a partition from Q −1 (2m) which can be written in the hook notation as λ = (a1 , . . . ar |a1 + 1, . . . , ar + 1). Let us denote by X i, j the element ei e j of S2 E. Then we deﬁne g j = X j, j ∧ X j, j+1 ∧ . . . ∧ X j, j+ai −1 and gλ = g1 ∧ g2 ∧ . . . ∧ gr . Then one checks easily that gλ is a U-invariant of weight λ . Similarly for m ( 2 E). We ﬁx λ ∈ Q 1 (2m) such that λ can be written in the hook notation λ = (a1 + 1, . . . ar + 1|a1 , . . . , ar ). We denote by X i, j the element ei ∧ e j from 2 E. Then we deﬁne g j = X j, j+1 ∧ X j, j+2 ∧ . . . ∧ X j, j+ai and gλ = g1 ∧ g2 ∧ . . . ∧ gr . It is easy to check that gλ is a nonzero U-invariant of weight λ . Finally let us mention that all the formulas proven in this section are functorial, so they extend to vector bundles. 2.4. The Schur Complexes In this section we review the theory of Schur complexes. First we deal with the case of a general Z2 -graded module and deﬁne Z2 -graded Schur modules, which are common generalizations of Schur and Weyl modules. We apply this deﬁnition to complexes. Over a ﬁeld of characteristic zero the Schur complexes obtained in this way have many nice acyclicity properties. We review the main properties of these complexes. Then we discuss the special case of complexes of length 1. It turns out that in this case the acyclicity properties are true in characteristic free settings. The Schur complexes will be used in section 6.2 when proving the properties of determinantal ideals in positive characteristic and in the discussion of the differentials in resolutions of determinantal ideals. We work in the category of Z2 -graded free modules over a commutative ring K. The objects of our category are Z2 -graded modules = F0 ⊕ F1 where both F0 , F1 are free K-modules. The maps are all K-linear maps of degree 0. Our theory associates to the Z2 -graded module the family of Z2 -graded modules L λ . They are a common generalization of Schur and Weyl modules. For F1 = 0 we have L λ = L λ F0 , and for F0 = 0 we have L λ = K λ F1 [|λ|], where the bracket denotes shift in homological degree. The strategy of our approach is similar when deﬁning Schur functors. First we deﬁne the exterior and symmetric powers of and then we imitate the deﬁnition from section 2.1. The i-th exterior power i is a Z2 -graded module deﬁned in the following way.

2.4. The Schur Complexes

67

Consider the i-fold tensor product ⊗i . The permutation σ ∈ i acts of ⊗i in the following way: σ (v1 ⊗ . . . ⊗ vi ) = ±vσ −1 (1) ⊗ . . . ⊗ vσ −1 (i) , where vi are homogeneous elements from and the sign ± is determined by the rule that exchanging the elements v and w contributes the sign

(−1)deg(v)deg(w) . This means that ± = (−1) N , where N = (i, j)∈Inv(σ ) deg(vi )deg(v j ), where we sum over inversions of σ . This formula deﬁnes a (Z2 -graded) action of i on ⊗i . We deﬁne the i-th exterior power of as the subset of antisymmetric elements in ⊗i with respect to this action of i . For elements of degree 0 this means antisymmetry of elements, but for degree 1 elements this means symmetry. More precisely, i is a Z-graded vector space whose t-th graded piece is i i−t = Dt F1 ⊗ F0 . t

The i-th symmetric power Si is a Z-graded module deﬁned as a factor of ⊗i by the span of all elements v − σ (v) (v ∈ ⊗i , σ ∈ i ). The t-th graded piece of St is (Si )t =

t

F1 ⊗ Si−t F0 .

(2.4.1) Proposition. (a) There exist natural maps of Z-graded modules :

i+ j

→

i

⊗

j

whose components are given by the products of exterior and divided diagonals. (b) There exist natural maps of Z-graded modules m : Si ⊗ S j → Si+ j whose components are given by the products of exterior and symmetric multiplications. (c) There exist natural maps of Z-graded modules m:

i

⊗

j

→

i+ j

whose components are given by the products of exterior and divided multiplications.

68

Schur Functors and Schur Complexes

In the case when K is a ﬁeld, all above maps are GL(F0 ) × GL(F1 )equivariant. Proof. We will just deﬁne the maps from parts (a), (b), (c) of the proposition. We start with part (a). Choose the pair of indices a, b such that 0 ≤ a ≤ i, 0 ≤ b ≤ j. We deﬁne the component i+ j j i ⊗ a,b : → a+b

as the following map: Da+b F1 ⊗

a

i+ j−a−b

b

F0

↓ ⊗ i−a F0 ⊗ j−b F0 ↓ t23 F0 ⊗ Db F1 ⊗ j−b F0 ,

Da F1 ⊗ Db F1 ⊗ Da F1 ⊗

i−a

where t23 is the map exchanging the second and third positions in the tensor product. Now we proceed with part (b) of the proposition. The component m a,b : (Si )a ⊗ (S j )b → (Si+ j )a+b is deﬁned as a composition a F1 ⊗ Si−a F0 ⊗ b F1 ⊗ S j−b F0 ↓ t23 b a F1 ⊗ F1 ⊗ Si−a F0 ⊗ S j−b F0 ↓ m⊗m a+b F1 ⊗ Si+ j−a−b F0 . Finally, the component j i+ j i ⊗ → m a,b : a

is deﬁned as a composition

b

F0 ⊗ Db F1 ⊗ j−b F0 ↓ t23 i−a Da F1 ⊗ Db F1 ⊗ F0 ⊗ j−b F0 ↓ m⊗m Da+b F1 ⊗ i+ j−a−a F0 . Da F1 ⊗

i−a

This completes the proof of the proposition.

a+b

2.4. The Schur Complexes

69

We call the maps and m from Proposition (2.4.1) the diagonal and multiplication maps. We could also deﬁne the diagonal maps on symmetric powers, and the multiplication maps on exterior powers, but they will not be needed in our application. Now we are ready to deﬁne the Z2 -graded Schur modules. Let = F0 ⊕ F1 be as above, and let λ be a partition. For two partitions µ ⊂ λ we deﬁne L λ/µ =

λ1 −µ1

⊗

λ2 −µ2

⊗ ... ⊗

λ s −µs

/R(λ/µ, ),

where R(λ/µ, ) is the sum of submodules: λ1 −µ1

⊗ ... ⊗

⊗ ... ⊗

λ s −µs

λa−1 −µa−1

⊗ Ra,a+1 () ⊗

λa+2 −µa+2

for 1 ≤ a ≤ s − 1, where Ra,a+1 () is the submodule spanned by the images of the following maps (a, u, v, ): u ⊗ λa −µa −u+λa+1 −µa+1 −v ⊗ v ↓ 1⊗⊗1 u λa −µa −u ⊗ ⊗ λa+1 −µa+1 −v ⊗ v ↓ m 12 ⊗m 34 λa −µa ⊗ λa+1 −µa+1 for u + v < λa+1 − µa . The next step is the description of the standard basis in L λ/µ . Let f 1 , . . . , f m and g1 , . . . , gn be ﬁxed bases in F0 and F1 respectively. We consider the Z2 -graded set A = (A0 , A1 ) where A0 = [1, m], A1 = [1, n]. Let us recall from section 1.1.2 that a Z2 -graded tableau of shape λ/µ with values in A is a map T : D(λ/µ) → A. We deﬁne the map ϕ : A → F0 ⊕ F1 by setting ϕ(i) = f i for i ∈ A0 and ϕ( j) = g j for j ∈ A1 . Let T be a Z2 -graded tableau of shape λ with the values in A. We associate to T the element in L λ which is a coset of the tensor ϕ(T (1, µ1 + 1)) ∧ . . . ∧ ϕ(T (1, λ1 )) ⊗ ϕ(T (2, µ2 + 1)) ∧ . . . ∧ϕ(T (2, λ2 )) ⊗ . . . ⊗ ϕ(T (s, µs + 1)) ∧ . . . ∧ ϕ(T (s, λs )). In the sequel we will identify these two objects and we will call both of them the (Z2 -graded) tableaux of shape λ/µ.

70

Schur Functors and Schur Complexes

Let us order the set A by an arbitrary order . Let us recall that in section 1.1 we deﬁned a standard Z2 -graded tableau relative to the order as a tableau satisfying the conditions (1) T (u, v) T (u, v + 1) with equality possible when T (u, v) ∈ A1 , (2) T (u, v) T (u + 1, v) with equality possible when T (u, v) ∈ A0 . We have the following generalization of Proposition (2.1.9) (b). (2.4.2) Proposition. Let us ﬁx the order on A. The standard Z2 -graded tableaux of shape λ/µ with values in A form a basis of the module L λ/µ . Proof of proposition (2.4.2). First we prove that the standard Z2 -graded tableaux generate L λ/µ . It is clear that the row standard Z2 -graded tableaux generate L λ/µ . Let us order the set of such tableaux by the order deﬁned in section 1.1.1. We will prove that if the Z2 -graded tableau T is not standard, then we can express it modulo R(λ/µ, ) as a combination of earlier Z2 -graded tableaux. Since T is not standard, we can ﬁnd a and w for which T (a, w) ( T (a + 1, w) with possible equality if T (a, w) ∈ A1 . Let w be such an index that T (a + 1, w) = T (a + 1, w + 1) = · · · = T (a + 1, w ) T (a + 1, w + 1). We consider the map (a, u, v; ) for u = w − µa − 1 and v = λa+1 − w . The key observation is that the image of the tensor U1 ⊗ . . . ⊗ Ua−1 ⊗ V1 ⊗ V2 ⊗ V3 ⊗ Ua+2 ⊗ . . . ⊗ Us , where U j = ϕ(T ( j, µ j + 1)) ∧ . . . ∧ ϕ(T ( j, λ j − µ j )) for j = a, a + 1, and where V1 = ϕ(T (a, µa + 1)) ∧ ϕ(T (a, µa + 2)) ∧ . . . ∧ ϕ(T (a, w − 1)), V2 = ϕ(T (a, w)) ∧ . . . ∧ ϕ(T (a, λa − µa )) ∧ ϕ(T (a + 1, µa+1 + 1)) ∧ . . . ∧ ϕ(T (a + 1, w )), V3 = ϕ(T (a + 1, w + 1)) ∧ . . . ∧ ϕ(T (a + 1, λa+1 − µa+1 )), contains the tableau T with the coefﬁcient 1, and all the other tableaux occurring in this image are earlier than T . It remains to prove that the standard tableaux are linearly independent in L λ/µ . Let us consider a map Hλ/µ :

λ1 −µ1

⊗

λ2 −µ2

⊗ ... ⊗ β

λ s −µs

α

−→ ⊗(i, j)∈D(λ/µ) (i, j)

−→ Sλ1 −µ1 ⊗ Sλ2 −µ2 ⊗ . . . ⊗ Sλt −µt ,

2.4. The Schur Complexes

71

where α is the tensor product of exterior diagonals δ:

λ j −µ j

→ ( j, µ j + 1) ⊗ ( j, µ j + 2) ⊗ . . . ⊗ ( j, λ j − µ j )

and β is the tensor product of multiplications m : (µi + 1, i) ⊗ (µi + 2, i) ⊗ . . . ⊗ (λi , i) → Sλi −µ1 . The map Hλ/µ is called the Z2 -graded Schur map associated to the partition λ/µ. The straightforward calculation (compare [ ABW2, II.2]) shows that (2.4.3) Proposition. The image Hλ (R(λ/µ, )) equals 0. This means that Hλ/µ induces a surjective map from L λ/µ to Im Hλ/µ . Now we will show that the map Hλ/µ maps standard tableaux to linearly independent elements of Sλ1 −µ1 ⊗ Sλ2 −µ2 ⊗ . . . ⊗ Sλt −µt . This will show (2.4.2), and at the same time it will prove that L λ/µ = Im Hλ/µ . A typical basis element of St is ϕ(s1 ) . . . ϕ(st ) where ϕ(s1 ) ϕ(s2 ) . . . ϕ(st ) with equality ϕ(si ) = ϕ(si+1 ) allowed only when si ∈ A0 . We order these elements lexicographically with respect to the sequence (s1 , . . . , st ). We denote this order by &. If w = w1 ⊗ w2 ⊗ . . . ⊗ wt and w = w1 ⊗ w2 ⊗ . . . ⊗ wt are two tensor products of such elements in Sλ1 −µ1 E ⊗ Sλ2 µ2 E ⊗ . . . ⊗ Sλt −µt E, we say that w & w iff w j & wj for the smallest j for which w j = wj . If T is a standard Z2 -graded tableau of shape λ/µ, then the smallest element (with respect to the order we just deﬁned) occurring in Hλ/µ (T ) is ϕ(T (µ1 + 1, 1)) . . . ϕ(T (λ1 , 1)) ⊗ . . . ⊗ ϕ(T (µt + 1, t)) . . . ϕ(T (λt , t)). Indeed, if, in applying the map α, we make the exchange of elements in some row, then we put bigger elements in earlier columns, so we get the earlier (with respect to &) elements. Moreover, it follows easily from the deﬁnitions that ϕ(T (µ1 + 1, 1)) . . . ϕ(T (λ1 , 1)) ⊗ . . . ⊗ ϕ(T (µt + 1, t)) . . . ϕ(T (λt , t)) occurs in α(T ) with coefﬁcient 1. It is also obvious that the elements ϕ(T (µ1 + 1, 1)) . . . ϕ(T (λ1 , 1)) ⊗ . . . ⊗ ϕ(T (µt + 1, t)) . . . ϕ(T (λt , t))

72

Schur Functors and Schur Complexes

are different for different standard tableaux T . This proves that the images Hλ/µ T of standard Z2 -graded tableaux T are linearly independent.• (2.4.4) Example. (a) The module L 2,1 is by deﬁnition the cokernel of the diagonal map : 3 → 2 ⊗ . It is a direct sum (L 2,1 )3 ⊕ (L 2,1 )2 ⊕ (L 2,1 )1 ⊕ (L 2,1 )0 . There are two basic descriptions of the graded components (L 2,1 )i . If we choose the order so F0 F1 , then (L 2,1 )3 = K 2,1 F1 , (L 2,1 )2 = F1 ⊗ F1 ⊗ F0 , (L 2,1 )1 has a ﬁltration with associated graded object F1 ⊗ 2 F0 ⊕ F1 ⊗ S2 F0 , and (L 2,1 )0 = L 2,1 F0 . If we set F1 F0 , we get (L 2,1 )3 = K 2,1 F1 , (L 2,1 )2 has a ﬁltration with associated graded object D2 F1 ⊗ F0 ⊕ 2 F1 ⊗ F0 , (L 2,1 )1 = F1 ⊗ F0 ⊗ F0 , and (L 2,1 )0 = L 2,1 F0 . (b) The module L 2,2 is by deﬁnition the factor of 2 ⊗ 2 di3 ⊗→ vided by the following images of maps (1, u, v; ): 2 ⊗ 2 (corresponding to u = 0, v = 1) and 4 → 2 ⊗ 2 (corresponding to u = v = 0). It is a direct sum (L 2,2 )4 ⊕ (L 2,2 )3 ⊕ (L 2,2 )2 ⊕ (L 2,2 )1 ⊕ (L 2,2 )0 . The graded components have the following descriptions, the same for both possible orders: (L 2,2 )4 = K 2,2 F1 , (L 2,2 )3 = K 2,1 F1 ⊗ F0 , (L 2,2 )1 = L 2,1 F0 ⊗ F1 (L 2,2 )0 = L 2,2 F0 . Here we use the isomorphisms K 2,2/1 F1 ∼ = K 2,1 F1 , L 2,2/1 F0 ∼ = L 2,1 F0 . The middle component has similar description whether we use the order F0 F1 or F1 F0 . The module (L 2,2 )2 has a ﬁltration with the associated graded object D2 F1 ⊗ 2 F0 ⊕ 2 F1 ⊗ S2 F0 . As a consequence of (2.4.2) we prove the following properties of Z2 -graded Schur modules. (2.4.5) Theorem. The Schur modules L λ/µ have the following properties: (a) The t-th term (L λ/µ )t has a natural ﬁltration with the associated graded object K λ/ν F1 ⊗ L ν/µ F0 . |ν|=|λ|−t

2.4. The Schur Complexes

73

(b) The t-th term (L λ/µ )t has a natural ﬁltration with the associated graded object K ν/µ F1 ⊗ L λ/ν F0 . |ν|=t

Proof. We start with (a). Let us choose the order by setting A0 A1 and i j if and only if i < j for i, j ∈ As for s = 0, 1. Let us order all sequences (u 1 , . . . , u s ) by saying that (u 1 , . . . , u s ) (v1 , . . . , vs ) if u j > v j for the smallest j for which u j = v j . For each ν we deﬁne (F≤ν )s as the span of the images of λ λ1 −µ1 s −µs , ⊗ ... ⊗ u1

us

where the sequence (u 1 , . . . , u s ) is than (ν1 , . . . , νs ). We notice that the proof of (2.4.2) implies that if we take the tableau T from (F≤ν )s and we standardize it, we express it as a linear combination of earlier standard tableaux from (F≤ν )s . Let us order all partitions ν by the order , and let us consider the factor (F≤η )s . (F≤ν )s / η λ2 , µ1 > µ2 . Deﬁne λ(1) = (λ1 − 1, λ2 ), µ(1) = (µ1 − 1, µ2 ). The shape λ(1)/µ(1) has rows of the same length as λ/µ, but the ﬁrst row is shifted by one place to the left, so we have t+1 overlaps. Show that there is a natural epimorphism π (λ, µ) : L λ/µ E → L λ(1)/µ(1) E. Show that the kernel of π (λ, µ) is isomorphic to L λ1 −µ2 ,λ2 −µ1 E (with the convention that Ker π (λ, µ) = 0 if λ2 < µ1 ). Formulate and prove the analogous result for skew Weyl modules.

Exercises for Chapter 2

79

Schur and Weyl Modules in Positive Characteristic 4. Deﬁne two morphisms jd : Sd E → Dd E, i d : Dd E → Sd E by formulas i d (e1a1 . . . enan ) =

n! e(a1 ) . . . en(an ) , (a1 )! . . . (an )! 1

jd (e1(a1 ) . . . en(an ) ) = (a1 )! . . . (an )!e1a1 . . . enan . Prove that i d , jd deﬁne GL(E)-equivariant maps and that the compositions i d jd = n!(id Sd E ), jd i d = n!(id Dd E ). 5. Let λ = (λ1 , . . . , λs ) be a partition. (a) Deﬁne a morphism ˜jλ :

λ1

E ⊗ ... ⊗

λs

φλ

E −→ Sλ1 E ⊗ . . . ⊗ Sλt E

jλ ⊗...⊗ jλs 1

−→

Dλ1 E ⊗ . . . ⊗ Dλt E.

Prove that ˜jλ factors to give an equivariant map jλ : L λ E → K λ E. (b) Deﬁne a morphism

ψλ i˜λ : Dλ1 E ⊗ . . . ⊗ Dλs E −→

λ1

E ⊗ ... ⊗

λt

E.

Prove that i˜λ factors to give an equivariant map i λ : K λ E → L λ E. (c) Prove that i λ jλ = h λ (id L λ E ), jλ i λ = h λ (id K λ E ), where h λ = (x,y)∈λ h λ (x, y), where h λ (x, y) = λ y − x + λx − y + 1 is the hook length of a hook in λ with the corner at (x, y). (d) Deduce that over a ﬁeld of characteristic p > 0 the module L λ E is irreducible as long as λ does not contain a box (x, y) such that h λ (x, y) is divisible by p. 6. We call a partition λ p-regular if it does not contain p rows of the same length. Otherwise we call λ p-singular. (a) For a partition λ deﬁne pi λ = ( pi λ1 , . . . , pi λs ). Prove that every

partition can be written uniquely as λ = j≥0 p j λ( j) where λ( j) are p-regular partitions. We call this decomposition a p-adic decomposition of λ.

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(b) Let |λ| = n = dim E. Prove that the module Mλ E contains a nonzero element of weight (1n ) if λ is p-regular. (c) Let K be an inﬁnite ﬁeld of characteristic p > 0. We deﬁne the Frobenius functor Fr : V ectK → V ectK by setting Fr (E) = E and p Fr (φ) = φ ( p) , where φ ( p) is a matrix (φi, j )1≤i≤m,1≤ j≤n . For a functor (i) H : V ectK → V ectK we denote H = H ◦ Fr i . This means that we take “the same” functor as H on object, but when evaluating H ◦ Fr i on a linear map η we raise every entry of the matrix H (η) to the power pi . (d) Deﬁne a U − p-invariant in L λ E to be a vector v ∈ L λ E such that

for the generic matrix t: = id + i< j ti, j E i, j from U tv = v +

α

pα

ti, j i, j vα .

i< j

Prove that if v is a U − p-invariant, then all vectors vα are also U − pinvariants. Prove that if λ is p-regular, then the only U − p-invariant in L λ E is the canonical tableau.

(e) (Steinberg theorem) Let λ be a partition, λ = j≥0 p j λ( j) its p-adic decomposition. Prove that ( j) Mλ( j) E. Mλ E = j≥0

Deduce that the reverse implication in (b) is also true. 7. Let K be an inﬁnite ﬁeld of characteristic p > 0. (a) Prove that the exterior power i E is an irreducible representation of GL(E). (b) Consider the symmetric power Sd E. Assume pi ≤ d < pi+1 . The sequence d = (d0 , . . . , di ) is a p-adic representation of d if d = d0 + d1 p + . . . + di pi . Assume that n = dim E ≥ d. For any p-adic representation d of d we denote by Nd the GL(E)-submodule of Sd E generated by the weight vector of weight (1d0 , p d1 , . . . , ( pi )di ). Describe Nd , and prove that these are the only equivariant subspaces in Sd E. (c) Let d, e be two p-adic representations of the same number d. We say that e is a reﬁnement of d if it can be obtained from d by several steps, each of which involves decreasing d j by 1 and simultaneously increasing d j−1 by p (for some j = 1, . . . , i). This deﬁnes a partial order on the set of p-adic representations of d, denoted d ⊂ e. Prove that d ⊂ e if and only if Nd ⊂ Ne .

Exercises for Chapter 2

81

(d) Let d be a p-adic representation of d. We deﬁne the partition λ( p, d)

where λv = ij=0 m v (d j ) p j , where the numbers m v (d) are deﬁned as follows p−1 if d ≥ (v + 1)( p − 1), m v (d) = d − v( p − 1) if v( p − 1) < d < (v + 1)( p − 1), 0 otherwise.

Prove that for each p-adic representation d of d the module Nd / e⊂d Ne is an irreducible representation of GL(E) of highest weight λ( p, d). 8. Let K be a ﬁeld. Consider a vector space E of dimension n. For a partition λ of m we denote by S λ the weight space of L λ E of weight (1m ). We can think of S λ as a span of tableaux of shape λ of weight (1m ) modulo the usual standard relations. The module S λ is called a Specht module corresponding to the partition λ. (a) Prove that S λ has the natural structure of a m -module. (b) Let K be a ﬁeld of characteristic 0. Prove that the modules S λ give a complete set of nonisomorphic irreducible m -modules. (c) Let K be a ﬁeld of characteristic p > 0. Let λ be a partition of m. Let M λ be the weight space of Mλ E of weight (1m ). The module M λ is = 0 if and only if λ is p-regular. It is proven in [ jm] that the representations M λ give a complete set of isomorphism classes of irreducible representations of Sm . 9. Let K be a ﬁeld of characteristic 3. Let E be a vector space of dimension n. Consider the Schur functors of degree 5. (a) Prove that L (3,1,1) E, L (4,1) E, L (5) E are irreducible, (b) Prove the following exact sequences, which imply the composition series of the remaining Schur functors: 0 → M(5) E → L (15 ) E → M(22 ,1) E → 0, 0 → M(4,1) E → L (2,13 ) E → M(3,2) E → 0, 0 → M(3,2) E → L (2,2,1) E → M(15 ) E → 0, 0 → M(2,2,1) E → L (3,2) E → M(2,13 ) E → 0.

Littlewood–Richardson Rule 10. Use the Littlewood–Richardson rule to ﬁnd the multiplicities of irreducible representations L λ E in the tensor products L 2,1 E ⊗ L 2,1 E, L 3,1 E ⊗ L 2,1 E .

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11. Let λ, µ be two rectangular partitions, i.e. λ = (l s ), µ = (m t ). Prove that the tensor product L λ E ⊗ L µ E is multiplicity free, i.e. that for each ν the multiplicity u(λ, µ; ν) equals 0 or 1. Characterize the partitions ν such that L ν E occurs in L λ E ⊗ L µ E. 12. Let ν = (m t ) be a rectangular partition. Show that L ν E occurs in the tensor product L λ E ⊗ L µ E if and only if λ and µ can be ﬁtted together to ﬁll the rectangle ν, i.e. when for λ = (λ1 , . . . , λt ), µ = (µ1 , . . . , µt ) we have λi + µt+1−i = m for 1 ≤ i ≤ t. Show that if λ and µ can be ﬁtted together to ﬁll the rectangle ν, then the multiplicity of L ν E in L λ E ⊗ L µ E is equal to 1. 13. Let us ﬁx ﬁve numbers a, b, c, d, e with a ≥ b ≥ c, d ≥ e, a + b + c = d + e. Denote by t(m) the multiplicity of L d m ,em E in L a m E ⊗ L bm E ⊗ t(1)+m−2 m L c E. Prove that t(m) = m−1 . 14. Let us ﬁx ﬁve numbers a, b, c, d, e with a ≥ b ≥ c ≥ d, a + b + c + d = 2e. Denote by t(m) the multiplicity of L e2m E in L a m E ⊗ L bm E ⊗ . L cm E ⊗ L d m E. Prove that t(m) = t(1)+m−2 m−1 15. Let us ﬁx seven numbers : a1 , a2 , b1 , b2 , c1 , c2 , d. Assume a1 + b1 + c1 + a2 + b2 + c2 = 3d. Denote by t(m) the multiplicity L d 3m E in t(1)+m−2of m m m m m m L a1 ,a2 E ⊗ L b1 ,b2 E ⊗ L c1 ,c2 E. Prove that t(m) = m−1 . 16. Let λ = (λ1 , . . . , λs ), µ = (µ1 , . . . , µt ) be two partitions. Let ν = (ν1 , . . . , νs+t ) be a partition resulting from permuting the sequence (λ1 , . . . , λs , µ1 , . . . , µt ) to be nonincreasing. Let π ∈ s+t be the resulting permutation, i.e. λi = νπ(i) for 1 ≤ i ≤ s, µ j = νs+ j for 1 ≤ j ≤ t. Prove that the morphism λ1

ˆ m(λ, µ) : →

ν1

E ⊗ ... ⊗

E ⊗ ... ⊗

νs+t

λs

E⊗

µ1

E ⊗ ... ⊗

µt

E

E

given by permuting the factors according to the permutation π factors to give an equivariant epimorphism m(λ, µ) : L λ E ⊗ L µ E → L ν E. Use the Littlewood–Richardson rule to show that the multiplicity of L ν E in L λ E ⊗ L µ E is equal to 1. Let us order the partitions lexicographically. Prove that for all partitions η such that L η E occurs in L λ E ⊗ L µ E we have η ≥ ν. Sometimes the factor L ν E is called the Cartan piece of the tensor product L λ E ⊗ L µ E.

Exercises for Chapter 2

83

17. Let λ = (λ1 , . . . , λs ) and µ = (µ1 , . . . , µs ) be two partitions. Let ν be a partition ν = (λ1 + µ1 , . . . , λs + µs ). Deﬁne the map ˆ (λ, µ) :

λ1 +µ1

⊗ ... ⊗

µs

E ⊗ ... ⊗

λ s +µs

E→

λ1

E ⊗ ... ⊗

λs

E⊗

µ1

E

E

to be the tensor product of the diagonals followed by a permutation of ˆ factors. Prove that (λ, µ) factors to give an equivariant map (λ, µ) : L ν E → L λ E ⊗ L µ E. Use the Littlewood–Richardson rule to show that the multiplicity of L ν E in L λ E ⊗ L µ E is equal to 1. Let us order the partitions lexicographically. Prove that for all partitions η such that L η E occurs in L λ E ⊗ L µ E we have η ≤ ν.

Schur Functors and Duality 18. Let E be a vector space of dimension n. (a) Prove the canonical isomorphisms ∗

L λ1 ,...,λs E = L n−λs ,...,n−λ1 E ⊗

n

E

∗

⊗s

,

(b) Prove the canonical isomorphism K λ1 ,...,λn E ∗ = K −λn ,...,−λ1 E. 19. Let E be a vector space of dimension n. Use duality and the Littlewood– Richardson rule to decompose i E ⊗ j E ∗ to the irreducible highest weight representations as a GL(E)-module.

Acyclicity Properties of Schur Complexes 20. Let R be a commutative ring, let M be an R-module, and let F1 → F0 → M → 0 be a presentation of M. Let be a complex F1 → F0 . We deﬁne the module L λ M to be L λ M = H0 (L λ ).

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Schur Functors and Schur Complexes

Prove that this deﬁnition does not depend on the choice of the presentation . 21. Let M be an R-module of projective dimension 1 with a free resolution 0 → F1 → F0 → M → 0. Let be the complex F1 → F0 . Prove that if λ is a partition of d and M is a (d–1)st syzygy, then L λ () gives a free resolution of L λ M. 22. Let : F1 → F0 be a linear map of free K-modules. For each partition ν the complex L λ has a subcomplex X u 1 , vt > . . . > v1 , u s− j ≥ vt− j for 1 ≤ j ≤ t. This means that when we reverse the order of numbers in this tableau, we will get the standardeness condition for both tableaux. Recalling the deﬁnition of the standard tableau from section 1.A. we see that it is natural to call the double tableau us ... . . . u 1 i1 i2 . . . it . . . is vt . . . v1 j1 j2 . . . jt a standard double tableau. (3.2.2) Proposition. The standard double tableaux form a basis of the coordinate ring K[U I0 ] (which by (3.2.1) is the polynomial ring in the variables z i j for 1 ≤ i ≤ r, 1 ≤ j ≤ n − r ). Proof. The coordinate ring K[U I0 ] is generated by functions pˆ (i 1 , . . . , ir ). The standard products of Pl¨ucker coordinates restrict to the standard double tableaux. If a product of functions pˆ (i 1 , . . . , ir ) is a double tableau which is not standard, we can express as a linear combinations of standard products of Pl¨ucker coordinates, which in turn will express our double tableau as a linear combination of standard double tableaux. This shows that standard double tableaux span K[U I0 ]. To show that the standard double tableaux are linearly independent on U I0 , we notice that if some linear combination of standard double tableaux (say in degree s) is zero, then by multiplying by some power of p(n − r + 1, . . . , n) we can extend each summand to Grass(r, E). Each summand extends to a standard tableau, since adding at the end the row (n − r + 1, . . . , n) does not affect standardness. Thus we get a nontrivial linear combination of standard tableaux on Grass(r, E) which vanishes on U I0 . This is a contradiction. Let us denote by F, G respectively the space of rows and columns of the matrix (∗). The polynomial ring K[U I0 ] is identiﬁed with Sym(F ⊗ G). We want to consider more closely the meaning of Pl¨ucker relations in terms of

94

Grassmannians and Flag Varieties

double tableaux. Let us consider the nonstandard tableau T =

i 1 i 2 . . . i a . . . ir . j1 j2 . . . ja . . . jr

More precisely, let us assume that i 1 < . . . < ir , j1 < . . . < jr , i a > ja , and i a ≤ n − r . To express T as a sum of earlier tableaux we use the relation R(i 1 , . . . , i a−1 ; ja+1 , . . . , jr ; j1 , . . . , ja , i a , . . . , ir ) from (3.1.2). Let us suppose that after the restriction to U I0 , p(i 1 , . . . , ir ) becomes an s × s minor of (∗) and p( j1 , . . . , jr ) becomes a t × t minor of (∗). This means that 1 ≤ i 1 < . . . < i s ≤ n − r < i s+1 < . . . < ir ≤ n, 1 ≤ j1 < . . . < jt ≤ n − r < jt+1 < . . . < jr ≤ n. We also assume that s ≤ t. We notice that in the summands of R(i 1 , . . . , i a−1 ; ja+1 , . . . , jr ; j1 , . . . , ja , i a , . . . , ir ) the number of entries that are ≤ n − r in the ﬁrst row is ≥ s. The summands in R(i 1 , . . . , i a−1 ; ja+1 , . . . , jr ; j1 , . . . , ja , i a , . . . , ir ) where the number of entries that are ≤ n − r in the ﬁrst row is equal to s correspond to shufﬂing j1 , . . . , ja with i a , . . . , i s . In all those summands the numbers bigger than n − r stay ﬁxed. This combination of products of minors of size s and t (s ≥ t) is, in terms of double tableaux, the relation of the type θ(a, u, v; F) on the left side of the double tableau with its right side ﬁxed. Our relation says that this combination belongs to the linear span of products of minors of size s + i multiplied by minors of size t − i for various i > 0. This establishes the following proposition. (3.2.3) Proposition. Let us consider the composition given by u F ⊗ s+t−u−v F ⊗ v F ⊗ s G ⊗ t G ↓ 1⊗⊗1⊗1⊗1 u F ⊗ s−u F ⊗ t−v F ⊗ v F ⊗ s G ⊗ t G ↓ m 12 ⊗m 34 ⊗1⊗1 s t F⊗ F⊗ sG⊗ tG ↓ 1⊗t23 ⊗1 s F⊗ sG⊗ tF⊗ tG ↓ ζs ⊗ζt Ss (F ⊗ G) ⊗ St (F ⊗ G) ↓m Ss+t (F ⊗ G), where ζs : s F ⊗ s G −→ Ss (F ⊗ G) sends f i1 ∧ f i2 ∧ . . . ∧ f is ⊗ g j1 ∧ g j2 ∧ . . . ∧ g js to the minor of the matrix Z = (z i, j )i, j ( f i ⊗ g j )i, j corresponding to rows i 1 , . . . , i s and columns j1 , . . . , js . Then each element in

3.2. The Standard Open Coverings of Flag Manifolds

95

Im is a linear combination of products of (s + i) × (s + i) minors and (t − i) × (t − i) minors for i > 0. (3.2.4) Example. 3 2 1 4 + =

5 4 3 2 1

1 2 3 4

4 3 1 2

4 2 1 3

1 2 3 4

1 2 3 4 5

−

5 4 3 2 1

1 2 3 4

−

3 2 1 4

−

−

=

1 2 3 4 5

4 3 2 5 1

1 2 3 4 5

−

4 3 2 1

1 2 3 4

,

5 4 2 3 1

1 2 3 4 5

1 2 3 4

5 3 2 4 1

1 2 3 4 5

5 4 3 2 1

1 2 3 5 4

.

For each partition λ = (λ1 , . . . , λt ) let us consider the maps ζλ :

λ1

F⊗

λ1

G⊗

λ2

F⊗

λ2

G ⊗ ... ⊗

λt

F⊗

λt

G

−→ Sym|λ| (F ⊗ G), where ζλ = ζλ1 ζλ2 . . . ζλt . Now we are ready to restate the characteristic free version of the Cauchy formula. (3.2.5) Theorem. The symmetric power Sm (F ⊗ G) has a natural GL(F) × GL(G)-invariant ﬁltration whose associated graded object equals |λ|=m L λ F ⊗ L λ G. Proof. Let us order all the partitions λ of weight m by the order ≤ (cf. section 1.1). We deﬁne Im ζµ , %λ = Im ζµ %λ = µ≤λ

µ dim F/ X they are zero by the relative version of the Grothendieck theorem ([H1, chapter III, Corollary 11.2]). Let us assume that j > i + 1 and that theorem is proven for smaller j − i. We take β = σi. (α), so β = (α1 , . . . , αi−1 , αi+1 − 1, αi + 1, αi+2 , . . . , αn ). Notice that β still satisﬁes our property, but now the pair (i, j) is changed to (i + 1, j). This means by induction that Ru h ∗ L(β) = 0 for all u ≥ 0. Now we again use the map h i . We notice that L(β) = L(α) ⊗ 'idi (L(α))+1 . and vice d (L(β))+1 versa L(α) = L(β) ⊗ 'i i . Since one of the line bundles L(α), L(β) has degree di which is ≥ −1, we can apply Proposition (4.2.2) to this bundle.

4.2. The Proof of Bott’s Theorem for the General Linear Group

121

We get either Ru h ∗ L(α) = Ru+1 h ∗ L(β) for all u or Ru h ∗ L(α) = Ru−1 h ∗ L(β) for all u. In both cases all higher direct images of Ru h ∗ L(α) are 0. Let us assume now that possibility (2) of (4.1.4) occurs. This means that the weight α + ρ has no repeated entries. Therefore there exists a unique permutation σ such that σ (α + ρ) is a strictly decreasing sequence, so σ . α is a partition β. Let us write a reduced expression σ = σv1 . . . σvl , where l = l(σ ). We deﬁne the weights β s = σvl−s+1 . . . σv.l (α) for s = 1, . . . , l. Obviously β l = β. We also set β 0 = α. We apply Proposition (4.2.2) to pvl−s and to L(β s+1 ). Since σv1 . . . σvl is a reduced expression, we have dls (L(β s+1 )) ≥ 0, so the proposition applies. We get Ru+1 h ∗ L(β s ) = Ru h ∗ L(β s+1 ) for all u and s. Putting these equalities together, we get Ru+l h ∗ L(α) = Ru h ∗ L(β) for all u. Therefore we are reduced to calculating the cohomology of the bundles L(β) where β is a partition. First we show that for such bundles Ru h ∗ L(β) = 0 for u > 0. In order to do this we use the permutation τ (i) = (n + 1 − i). Clearly l(τ ) = n2 . Let us choose the reduced expression τ = σ1 (σ2 σ1 ) . . . (σn−2 . . . σ1 )(σn−1 . . . σ1 ). Using Proposition (4.2.2) repeatedly, as above, we get n

Ru h ∗ L(β) = Ru+(2) h ∗ L(τ . (β))

for each u. Since n2 = dim F/ X , we get Ru h ∗ L(β) = 0 for u > 0. It remains to identify the direct images h ∗ L(β) for every partition β. To do this we ﬁrst notice that the question is local in X . Therefore we can assume that the bundle E is trivial, so E = X × E for some vector space E of dimension n over k. This means that we can identify Flag(E) with X × Flag(E). The map h is just the ﬁrst projection. The direct image h ∗ L(β) becomes O X ⊗ H 0 (Flag(E), L(β)). To conclude the proof we have to show that for every nonincreasing sequence β = (β1 , . . . , βn ) the space of sections

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Bott’s Theorem

H 0 (Flag(E), L(β)) is isomorphic to K (β1 ,...,βn ) E ∗ . Notice that we can also assume that βn = 0, because adding 1 to each coordinate corresponds to ten soring our bundle by n E ∗ , and the same is true for the functors K (β1 ,...,βn ) E ∗ . Pick β = (β1 , . . . , βn ). Let {b1 , . . . , bt } = {1 ≤ i ≤ n − 1 | βi > βi+1 }. Consider the graded ring S(β) = K nβ1 ,...,nβn E ∗ . n≥0

This is a homogeneous coordinate ring of the ﬂag variety Flag(b1 , . . . , bt ; E) embedded in a projective space P(L β E ∗ ) by using the line bundle L(β) on Flag(b1 , . . . , bt ; E). (4.2.4) Lemma. The ring S(β) is a domain. Proof. Assume S(β) is not a domain. Then the zero divisors of S(β) are the union of ﬁnitely many prime ideals P1 , . . . , Pm in S(β). The group GL(E) acts on S(β). Since GL(E) is connected, it follows that all ideals Pi are equivariant. This means P1 has to contain a U-invariant. But P1 is prime, so it has to contain a U-invariant in degree 1 – the canonical tableau. Since the representation L β E ∗ is irreducible, P1 contains all elements of degree 1. This is a contradiction proving that S(β) is a domain. By ([H1, chapter II, Exercise 5.14) we see that the normalization of S(β) S(β) = H 0 (Flag(b1 , . . . , bt ; E), L(nβ)). n≥0

Moreover, the same exercise shows that for n >> 0 we have H 0 (Flag(b1 , . . . , bt ; E), L(nβ)) = K nβ1 ,...,nβn E ∗ .

(∗∗)

Now it is easy to ﬁnish the proof. We will show in fact that S(β) is integrally closed. Without loss of generality we can assume that β is not a multiple of another weight. For such β we will show that (∗∗) holds for every n > 0. By (2.2.3) it is enough to show that every U-invariant in S(β) is a power of a canonical tableau cβ . Let x be a U-invariant in S(β). Then a high enough power x m is the power of the canonical tableau cβ , because S(β)/S(β) has to have ﬁnite length since it is supported at the origin. Let x m = cβl . If m divides l, we are done, because S(β) is a domain, so x has to be the power of canonical tableau. If not, then the weight of x is not an integral weight. The proof of Theorem (4.1.4) is complete.•

4.3. Bott’s Theorem for General Reductive Groups

123

4.3. Bott’s Theorem for General Reductive Groups In this section we assume that the reader is familiar with the basic notions concerning reductive groups and root systems. We state here Bott’s theorem for reductive groups. The results of this section will be used only in Chapter 8. We start with recalling some standard notation. Let K be an algebraically closed ﬁeld, and let G be a reductive linear group over K. Let T be a maximal torus, and B a Borel subgroup containing T. We denote by the root system associated to the pair G, T. This is by deﬁnition a ﬁnite set of vectors in Hom K (Lie(T), K), where Lie(T) is the Lie algebra of the torus T. The choice of B determines the subset + of positive roots in . The space HomK (Lie(T), K) is equipped with a nondegenerate scalar product ( , ). We deﬁne, for α, β ∈ HomK (Lie(T), K), (β, α) = 2(β, α)/(α, α). We denote by ( the lattice of integral weights in HomK (Lie(T), K): ( = {γ ∈ HomK (Lie(T), K) | ∀α ∈ , (γ , α) ∈ Z}. The lattice ( contains the cone (+ of dominant integral weights, (+ = {γ ∈ ( | ∀α ∈ + , (γ , α) ∈ Z+ }. Bott’s theorem gives a rule for calculating cohomology groups of the line bundles on the homogeneous space G/B. Such bundles are described by weights. Indeed, for each character γ of T we deﬁne a one dimensional rational B-module V (γ ) by letting the unipotent radical U of B act trivially on V (γ ) and the torus T act by the character γ , and by letting L(γ ) = G ×B V (γ ), where for any rational B-module V we denote by G ×B V the product G × V divided by the equivalence relation (g, v) ∼ (gb, b−1 v) for b ∈ B. We can identify the group of characters of T with the additive subgroup in ( by associating to each character its derivative at identity.

The Weyl group W of G acts naturally on weights. Let ρ = 12 α-0 α be half of the sum of positive roots. We deﬁne the dotted action of W on weights σ . (γ ) = σ (γ + ρ) − ρ. Let us recall that the irreducible representations of G correspond to the dominant integral weights. For a dominant integral weight β we denote by Vβ the irreducible G-module of highest weight β. (4.3.1) Theorem (Bott). Let G, T, B, W , be as above. Let γ be an integral weight, and let L(γ ) be the corresponding line bundle over G/B. Then one

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of two mutually exclusive possibilities occurs: (1) There exists σ ∈ W , σ = 1, such that σ . (γ ) = γ . Then the cohomology groups H i (G/B, L(γ )) are zero for i ≥ 0. (2) There exists a unique σ ∈ W such that σ . (γ ) := (α) is a dominant integral weight. In this case all cohomology groups H i (G/B, L(γ )) are zero for i = l(σ ), and H l(σ ) (G/B, L(γ )) = Vα . (4.3.2) Remark. The proof in the general case follows the same scheme as in the case of the general linear group. The role of the ﬂag varieties Flag (1, 2, . . . , i − 1, i + 1, . . . , n − 1; E) is played by the homogeneous space G/Pα , where Pα is a parabolic subgroup corresponding to a simple root α. (4.3.3) Examples. (a) Let us ﬁx a vector space E of dimension n, and let us consider the general linear group G = GL(E). Then we can choose the maximal torus T to be the subgroup of diagonal matrices, and the Borel subgroup B to be the subgroup of upper triangular matrices. The homogeneous space G/B can be identiﬁed with Flag(E), the set ( = Zn , and the Weyl group is isomorphic to n . The statement of Bott’s theorem reduces to Corollary (4.1.2). (b) Let F be a vector space of dimension 2n + 1 with a nondegenerate symmetric bilinear form ( , ). Let us take G = SO(F) to be the special orthogonal group. The lattice ( = Zn . The Weyl group W is a hyperoctahedral group acting on ( by signed permutations. The half sum of the positive roots is ρ = ( 2n−1 , 2n−3 , . . . , 12 ) 2 2 (c) Let F be a vector space of dimension 2n with a nondegenerate skew symmetric bilinear form ( , ) on F. Let us take G = Sp(F) to be the symplectic group associated to F. The lattice ( = Zn . The Weyl group W is a hyperoctahedral group acting on ( by signed permutations. The half sum of the positive roots is ρ = (n, n − 1, . . . , 1) (d) Let F be a vector space of dimension 2n with a nondegenerate symmetric bilinear form ( , ). Let us take G = SO(F) to be the special orthogonal group. The lattice ( = Zn . The Weyl group W is a subgroup of hyperoctahedral group acting on ( by signed permutations with even number of sign changes. The half sum of the positive roots is ρ = (n − 1, n − 2, . . . , 0).

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We proceed with some explicit calculations on homogeneous spaces G/P where P is a maximal parabolic subgroup in G. We will need these kinds of calculations in chapter 8. We limit ourselves to some examples related to classical groups. They can be viewed as analogues of (4.1.9). We recall that by the general theory ([Hu2], [Bou]) there are (up to conjugation) ﬁnitely many types of parabolic subgroups in G, and they correspond to subsets of simple roots. We start with the symplectic group. Let F be a vector space of even dimension 2n with a nondegenerate skew symmetric form ( , ). The simple roots of are α j = j − j+1 for j = 1, . . . , n − 1 and αn = 2n . Next we give a concrete description of the homogeneous spaces G/B. Let us recall that a subspace R ⊂ F is isotropic if the restriction of ( , ) to R is zero. We consider the set IFlag(F) = {(R1 , . . . , Rn ) ∈ Flag(1, 2, . . . , n; F) | Rn is isotropic}. The group Sp(F) acts on the set IFlag(F) transitively. To see this, observe that for a ﬂag (R1 . . . , Rn ) ∈ IFlag(F) we can choose a symplectic basis e1 , . . . , en , e¯ n , . . . , e¯ 1 of F so e1 , . . . , ei is a basis of Ri (1 ≤ i ≤ n). Since the symplectic group operates transitively on symplectic bases, we are done. The space IFlag(F) can be identiﬁed with the homogeneous space G/H, where H is a subgroup of elements in Sp(F) stabilizing the ﬂag (R1 , . . . , Rn ). By Borel’s theorem there exists a Borel subgroup B contained in H. Since every parabolic subgroup of a connected reductive group is connected ([Hu2]), we have H = B. This realization of the homogeneous space G/B allows us to develop relative theory in the same spirit as in section 3.3. We just give the deﬁnitions, leaving the details to the reader. Let F be a symplectic vector bundle over a scheme X , i.e. a vector bundle F equipped with a map ( , ) : 2 F → O X for which the restriction to each ﬁber gives a nondegenerate skew symmetric form on it. We can construct a relative isotropic ﬂag variety IFlag(F) with the structure map p : IFlag(F) → X . The statement of Bott’s theorem is true in a relative version with higher direct images replacing cohomology groups. We leave the formulation of this result to the reader. For each j = 1, . . . , n we consider the maximal parabolic subgoup P j in G = Sp(2n) which corresponds to the subset of all simple roots except α j . The space G/P j can be identiﬁed with the isotropic Grassmannian IGrass( j, F) of isotropic subspaces of dimension j in F. This is a closed subset in Grass(F), so we can talk about the tautological subbundle R j on IGrass( j, F). For any

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isotropic subspace from IGrass( j, F) we deﬁne the orthogonal complement R ∨ = {x ∈ F | ∀y ∈ R (x, y) = 0}. The space R ∨ contains R and has dimension 2n − j. The correspondence R → R ∨ deﬁnes a tautological bundle R∨j of dimension 2n − j on IGrass ( j, F). We have the inclusions of bundles on IGrass( j, F) R j ⊂ R∨j ⊂ F × IGrass( j, F). The bundle R∨j /R j is a symplectic bundle. Indeed, we have a map of vector bundles 2

R∨j /R j → OIGrass( j,F) ,

which on each ﬁber is induced by the form ( , ), so it is nondegenerate. This means that for each dominant weight µ = (µ1 , . . . , µn− j ) for the root system of type Cn− j we can talk about the bundle Vµ (R∨j /R j ). Its ﬁber over a point corresponding to the isotropic space R is Vµ (R ∨ /R). (4.3.4) Corollary. Let us consider the vector bundle Vβ,µ = K β R j ⊗ Vµ (R∨j /R j ) over IGrass(r, F), where β = (β1 , . . . , β j ) is a dominant integral weight for the root system of type A j−1 and µ = (µ1 , . . . , µn− j ) is the integral dominant weight for the root system of type C n− j . Let us consider the weight γ = (−β j , . . . , −β1 , µ1 , . . . , µn− j ). Then one of the mutually exclusive possibilities occurs: (1) There exists σ ∈ W , σ = 1, such that σ (γ ) = γ . Then all cohomology groups H i (IGrass( j, F), Vβ,µ ) are 0 for i ≥ 0, (2) There exists unique σ ∈ W such that σ . (γ ) := α is a dominant integral weight for the root system of type C n . Then all cohomology groups H i (IGrass( j, F), Vβ,µ ) are 0 for i = l(σ ), and H l(σ ) (IGrass( j, F), Vβ,µ ) = Vα (F). Proof. Let us consider the projection p : G/B → G/P j . Identifying, as above, G/B with the space of isotropic ﬂags and G/P with the isotropic Grassmannian, we see that the ﬁber over a point corresponding to a subspace R is p −1 (R) = Flag(R) × IFlag(R ∨ /R). Therefore we can identify G/B with the relative variety Flag(R j ) × IFlag(R∨j /R j ). Consider the line bundle L(γ ) on G/B. Using Bott’s theorem in relative situation for the types A j−1 and Cn− j

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we see that Ri p∗ (L(γ ) = 0 for i > 0 and R0 L(γ ) = Vβ,µ . Now our statement follows from Bott’s theorem (4.3.1) and the spectral sequence of the composition. (4.3.5) Remark. In the above corollary and in the following calculations we adopt the convention that for a vector space E of dimension m, K (β1 ,...,βm ) E ∗ ∼ = K (−βm ,...,−β1 ) E. Thus the above proposition also allows the calculation of the cohomology groups H i (IGrass( j, F), K β R∗j ⊗ Vα (R∨j /R j )). Let us look more closely at the isotropic Grassmannian IGrass( j, F) as a subset of the Grassmannian Grass ( j, F). (4.3.6) Proposition. The isotropic Grassmannian IGrass( j, F) is locally a complete intersection in Grass( j, F). The structure sheaf of IGrass( j, F) can be resolved by locally free sheaves over Grass( j, F) by means if the Koszul complex 0→

(2j ) 2

Rj

→ ... →

2

ψ

R j → OGrass( j,F) .

Proof. Since F is a symplectic space, we have the following map of locally free sheaves over Grass( j, F): 2

Rj →

2

F × OGrass( j,F) → OGrass( j,F)

with the left map coming from tautological inclusion and the right one induced by the form ( , ). The composition gives us the cosection ψ of 2 R j which deﬁnes our Koszul complex. It is clear by deﬁnition that IGrass( j, F) is equal to the set of zeros of this cosection. Moreover, an easy calculation shows that locally these equations deﬁne a reduced subscheme of Grass( j, F). The dimension count shows that locally the equations give a regular sequence, so the Koszul complex is acyclic. The situation for the orthogonal group is very similar, but there is one difference. The special orthogonal group does not act transitively on the isotropic ﬂags; only the orthogonal group does. This leads to some minor differences, which we highlight below. We just formulate the results we need, as the proofs are the same as in the case of a symplectic group. We ﬁrst consider the group of type Bn . Let F be a vector space of odd dimension 2n + 1 with a nondegenerate symmetric form ( , ). The simple

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roots of are α j = j − j+1 for j = 1, . . . n − 1 and αn = n . We take G = SO(F). This group is not simply connected. Since we have in mind some applications to nilpotent orbits in the corresponding Lie algebra, we do not need to discuss spinor groups. We start with a description of the homogeneous spaces G/B in terms of ﬂags. Let us recall that a subspace R ⊂ F is isotropic if the restriction of ( , ) to R is zero. We consider the set IFlag(F) = {(R1 , . . . , Rn ) ∈ Flag(1, 2, . . . , n; F) | Rn is isotropic}. The orthogonal group O(F) acts transitively on the set IFlag(F). Indeed, for each ﬂag (R1 . . . , Rn ) ∈ IFlag(F) we can choose a hyperbolic basis e1 , . . . , en , e, e¯ n , . . . , e¯ 1 of F so e1 , . . . , ei is a basis of Ri (1 ≤ i ≤ n) and e1 , . . . , en , e is a basis of Rn∨ . Now the orthogonal group O(F) operates transitively on hyperbolic bases. We will show that even SO(F) does. Indeed, let g ∈ O(F) be an element from O(F) which in given hyperbolic basis sends for each i ei to ei , e¯ i to e¯ i and e to −e. The element g ﬁxes he ﬂag (R1 , . . . , Rn ) where Ri is spanned by e1 , . . . , ei for i = 1, . . . , n. This means that for every h ∈ O(F) we have h(R1 , . . . , Rn ) = hg(R1 , . . . , Rn ). One of these elements has to lie in SO(F). As for symplectic group we now identify G/B with IFlag(F). Moreover, we can again give the relative version of the whole setup. Let F be an orthogonal vector bundle of dimension 2n + 1 over a scheme X , i.e. a vector bundle F equipped with a map ( , ) : S2 F → O X for which the restriction to each ﬁber gives a nondegenerate symmetric form on it. We can construct a relative isotropic ﬂag variety IFlag(F) with the structure map p : IFlag(F) → X . The relative version of Bott’s Theorem (4.3.1) is true if we replace cohomology groups with higher direct images. We leave the formulation of this result to the reader. For each j = 1, . . . , n we consider the maximal parabolic subgoup P j in G = SO(2n + 1) which corresponds to the subset of all simple roots except α j . The space G/P j can be identiﬁed with the isotropic Grassmannian IGrass( j, F) of isotropic subspaces of dimension j in F. This is a closed subset in Grass(F), so we can talk about the tautological subbundle R j on IGrass( j, F). For any isotropic subspace from IGrass( j, F) we deﬁne the orthogonal complement R ∨ = {x ∈ F | ∀y ∈ R (x, y) = 0}. The space R ∨ contains R and has dimension 2n + 1 − j. The correspondence R → R ∨ deﬁnes a tautological bundle R∨j of dimension 2n + 1 − j

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on IGrass( j, F). We have the inclusions of bundles on IGrass( j, F) R j ⊂ R∨j ⊂ F × IGrass( j, F). The bundle R∨j /R j is an orthogonal bundle of dimension 2(n − j) + 1. Indeed, we have a map of vector bundles S2 (R∨j /R j ) → OIGrass( j,F) which on each ﬁber is induced by the form ( , ), so it is nondegenerate. This means that for each dominant weight µ = (µ1 , . . . , µn− j ) for the root system of type Bn− j we can talk about the bundle Vµ (R∨j /R j ). Its ﬁber over a point corresponding to to the isotropic space R is Vµ (R ∨ /R). (4.3.7) Corollary. Let us consider the vector bundle Vβ,µ = K β R j ⊗ Vµ (R∨j /R j ) over IGrass(r, F), where β = (β1 , . . . , β j ) is a dominant integral weight for the root system of type A j−1 and µ = (µ1 , . . . , µn− j ) is the integral dominant weight for the root system of type Bn− j . Let us consider the weight γ = (−β j , . . . , −β1 , µ1 , . . . , µn− j ). Then one of the mutually exclusive possibilities occurs: (1) There exists σ ∈ W , σ = 1 such that σ (γ ) = γ . Then all cohomology groups H i (IGrass( j, F), Vβ,µ ) are 0 for i ≥ 0, (2) There exists unique σ ∈ W such that σ . (γ ) := α is a dominant integral weight for the root system of type Bn . Then all cohomology groups H i (IGrass( j, F), Vβ,µ ) are 0 for i = l(σ ), and H l(σ ) (IGrass( j, F), Vβ,µ ) = Vα (F). The proof of (4.3.7) is identical to that of (4.3.4). Let us look more closely at the isotropic Grassmannian IGrass( j, F) as a subset of the Grassmannian Grass( j, F). (4.3.8) Proposition. The isotropic Grassmannian IGrass( j, F) is locally a complete intersection in Grass( j, F). The structure sheaf of IGrass( j, F) can be resolved by locally free sheaves over Grass( j, F) by means if the Koszul complex 0→

j+1 ( 2 )

ψ

(S2 R j ) → . . . → S2 R j → OGrass( j,F) .

The proof is identical to that of (4.3.6).

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Finally we consider the group of type Dn . Let F be a vector space of even dimension 2n with a nondegenerate symmetric form ( , ). The simple roots of are α j = j − j+1 for j = 1, . . . n − 1 and αn = n−1 + n . We take G = SO(F). Again we are not interested in the spinor group. The description of the homogeneous spaces G/B in terms of ﬂags is now different. We again start with the set IFlag(F) of isotropic ﬂags: IFlag(F) = {(R1 , . . . , Rn ) ∈ Flag(1, 2, . . . , n; F) | Rn is isotropic}. As above, the orthogonal group O(F) acts transitively on the set IFlag(F). However, here IFlag(F) has two connected components and SO(F) operates transitively on each of them. In order to see this, let us ﬁx a hyperbolic basis e1 , . . . , en , e¯ n , . . . , e¯ 1 . We associate to it a ﬂag (R10 , . . . , Rn0 ) where Ri0 is spanned by e1 , . . . , ei . For a given ﬂag (R1 , . . . , Rn ) there exists h ∈ O(F) such that h Ri0 = Ri for i = 1, . . . , n. Both ﬂags are in the same component if h ∈ SO(F). Both components are homogeneous spaces for SO(F), so they are connected. The only thing to show is that they do not coincide. Let (R1 , . . . , Rn ) be in both components. Then there exist elements h 1 ∈ SO(F) and h 2 ∈ O(F) \ SO(F) such that h j Ri0 = Ri for i = 1, . . . , n, j = 1, 2. This 0 means that h −1 2 h 1 ﬁxes Ri for i = 1, . . . , n. Now simple linear algebra shows −1 that det(h 2 h 1 ) = 1, which is a contradiction. We will denote two components of IFlag(F) by IFlag+ (F) and IFlag− (F). We identify G/B with Flag+ (F). Moreover, we can again give the relative version of the whole setup. Let F be an orthogonal vector bundle of dimension 2n over a scheme X , i.e. a vector bundle F equipped with a map ( , ) : S2 F → O X for which the restriction to each ﬁber gives a nondegenerate symmetric form on it. We can construct a relative isotropic ﬂag variety IFlag(F) with the structure map p : IFlag(F) → X . The relative version of Bott’s theorem (4.3.1) is true if we replace cohomology groups with higher direct images. We leave the formulation of this result to the reader. For each j = 1, . . . , n we consider the maximal parabolic subgoup P j in G = SO(2n) which corresponds to the subset of all simple roots except α j . Similar arguments to those for type Bn show that for j = 1, . . . , n − 2 the space G/P j can be identiﬁed with the isotropic Grassmannian IGrass( j, F) of isotropic subspaces of dimension j in F. For j = n the isotropic Grassmannian IGrass(n, F) has two connected components IGrass+ (F) and IGrass− (F). They can be identiﬁed with the homogeneous spaces SO(F)/P j for j = n − 1, n. For the remainder of this section we assume that 1 ≤ j ≤ n − 2. Each SO(F)/P j is a closed subset in Grass( j, F), so we can talk about the

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131

tautological subbundle R j on IGrass( j, F). For any isotropic subspace from IGrass( j, F) we deﬁne the orthogonal complement R ∨ = {x ∈ F | ∀y ∈ R (x, y) = 0}. The space R ∨ contains R and has dimension 2n − j. The correspondence R → R ∨ deﬁnes a tautological bundle R∨j of dimension 2n − j on IGrass( j, F). We have the inclusions of bundles on IGrass( j, F) R j ⊂ R∨j ⊂ F × IGrass( j, F). The bundle R∨j /R j is an orthogonal bundle of dimension 2(n − j) + 1. Indeed, we have a map of vector bundles S2 (R∨j /R j ) → OIGrass( j,F) which on each ﬁber is induced by the form ( , ), so it is nondegenerate. This means that for each dominant weight µ = (µ1 , . . . , µn− j ) for the root system of type Dn− j we can talk about the bundle Vµ (R∨j /R j ). Its ﬁber over a point corresponding to to the isotropic space R is Vµ (R ∨ /R). (4.3.9) Corollary. Let us consider the vector bundle Vβ,µ = K β R j ⊗ Vµ (R∨j /R j ) over IGrass( j, F), where β = (β1 , . . . , β j ) is a dominant integral weight for the root system of type A j−1 and µ = (µ1 , . . . , µn− j ) is the integral dominant weight for the root system of type Dn− j . Let us consider the weight γ = (−β j , . . . , −β1 , µ1 , . . . , µn− j ). Then one of the mutually exclusive possibilities occurs: (1) There exists σ ∈ W , σ = 1 such that σ (γ ) = γ . Then all cohomology groups H i (IGrass( j, F), Vβ,µ ) are 0 for i ≥ 0. (2) There exists unique σ ∈ W such that σ . (γ ) := α is a dominant integral weght for the root system of type Dn . Then all cohomology groups H i (IGrass( j, F), Vβ,µ ) are 0 for i = l(σ ), and H l(σ ) (IGrass( j, F), Vβ,µ ) = Vα (F). The proof of (4.3.8) is identical to that of (4.3.4). Let us look more closely at the isotropic Grassmannian IGrass( j, F) as a subset of the Grassmannian Grass( j, F).

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Bott’s Theorem

(4.3.10) Proposition. The isotropic Grassmannian IGrass( j, F) is locally a complete intersection in Grass( j, F). The structure sheaf of IGrass( j, F) can be resolved by locally free sheaves over Grass( j, F) by means if the Koszul complex 0→

j+1 ( 2 )

ψ

(S2 R j ) → . . . → S2 R j → OGrass( j,F) .

The proof is identical to that of (4.3.5).

Exercises for Chapter 4 The General Linear Group 1. (a) Calculate the cohomology groups of bundles L(1, 4, 7, 5), L(3, 2, 1, 5) on G/B for G = GL(4, C). (b) Calculate the cohomology of vector bundles K (3,2,1) Q ⊗ K (7,6,1) R and of K (7,6,6) R on Grass(3, E) with dim E = 6. 2. Calculate the cohomology groups of bundles K λ Q∗ on the Grassmannian with tautological sequence 0 → R → F → Q → 0, where dim R = r , dim Q = q. 3. Let E be an n-dimensional space. Consider the Grassmannian Grass(r ; E) with the tautological sequence 0 → R → E × Grass(r ; E) → Q → 0. Let ξ be a subbundle of S2 E × Grass(r ; E) ﬁtting into the exact sequence 0 → ξ → S2 E × Grass(r ; E) → S2 Q → 0. Notice that ξ also ﬁts into an exact sequence 0 → S2 R → ξ → R ⊗ Q → 0. Calculate the cohomology groups of 2 ξ and 3 ξ using the information from the Schur complexes associated to the map S2 E × Grass(r ; E) → S2 Q. Calculate this cohomology using the ﬁltration induced by the second sequence on 2 ξ , 3 ξ . 4. We recall from Proposition (3.3.5) that the tangent bundle TGrass(r ;E) can be identiﬁed with R∗ ⊗ Q.

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133

(a) Calculate the cohomology groups of the exterior algebra on the tangent bundle TGrass(r ;E) . Prove that the higher cohomology groups vanish. (b) Calculate the cohomology groups of the exterior algebra on the cotan∗ i gent bundle TGrass(r ;E) . Prove that only the group H (Grass(r ; E), j ∗ (TGrass(r ;E) )) is nonzero if and only if when i = j. Prove that ∗ H i (Grass(r ; E), i (TGrass(r ;E) )) consists of P(i, r, n − r ) copies of trivial representation, where P(i, r, n − r ) is the number of partitions of i contained in the r × (n − r ) rectangle. 5. Some characteristic free cases of Bott’s theorem: (a) Let λ = (λ1 , . . . , λn ) be such that for some s < t we have λ1 ≥ . . . ≥ λs > λs+1 = . . . = λt−1 ≥ λt+1 − 1 ≥ . . . ≥ λn − 1 λt = λs+1 + t − s. Then H i (G/B, L(λ)) = 0 for i = t − s − 1 and H t−s (G/B, L(λ)) = L ν E where ν = (λ1 , . . . , λs , λt − (t − s) + 1, λs+1 + 1, . . . , λt−1 + 1, λt+1 , . . . , λn ). (b) Let λ = (λ1 , . . . , λn ) be such that for some s < t we have λ1 ≥ . . . ≥ λs > λs+1 = . . . = λt−1 , λs+1 < λt < λs+1 + t − s. Then H i (G/B, L(λ)) = 0 for all i.

Other Classical Groups 6. Let F be a symplectic space of dimension 2n. Consider the isotropic Grassmannian IGrass( j, F). Let λ = (λ1 , . . . , λ j ) be a partition. Prove that if λ1 ≤ 2n − j + 1, then the cohomology of K λ R j can be zero or can contain only a trivial representation of Sp(F). More precisely, the cohomology is nonzero precisely when λ is one of the partitions occurring in Proposition (6.4.3). 7. Let IGrass(r ; F) be the isotropic Grassmannian of r -dimensional isotropic subspaces in a symplectic space (F, (−, −)) of dimension 2n. Calculate the cohomology groups of the exterior powers of the vector bundle R∨ . 8. Formulate and prove the analogues of exercises 6 and 7 for the even and odd orthogonal groups. 9. Let (F, ( , )) be a symplectic space of dimension 2n. For 1 ≤ r ≤ n, let IGrass(r ; F) be the isotropic Grassmannian of r -dimensional isotropic spaces in F with tautological subbundle R. Consider the ﬁltration 0 ⊂ R ⊂ R∨ ⊂ F × IGrass(r ; F) of the vector bundles on IGrass(r ; F). The factor (F × IGrass(r ; F))/R∨ can be identiﬁed with R∗ . Therefore we

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Bott’s Theorem

have an epimorphism π of vector bundles on IGrass(r ; F) which is a composition π : (TGrass(r ;F) )|IGrass(r ;F) = R∗ ⊗ ((F × IGrass(r ; F))/R) 2 → R∗ ⊗ R∗ → R∗ . (a) Prove that the embedding IGrass(r ; F) ⊂ Grass(r ; F) allows one to identify the vector bundle TIGrass(r ;F) with the kernel of π. (b) Prove the exact sequence 0 → R∗ ⊗ (R∨ /R) → TIGrass(r ;F) → D2 R∗ → 0. 10. Let (F, (−, −)) be an orthogonal space of dimension n. For 1 ≤ r ≤ n2 , let IGrass(r ; F) be the isotropic Grassmannian of r -dimensional isotropic spaces in F with tautological subbundle R. Consider the ﬁltration 0 ⊂ R ⊂ R∨ ⊂ F × IGrass(r ; F) of the vector bundles on IGrass(r ; F). The factor (F × IGrass(r ; F))/R∨ can be identiﬁed with R∗ . Therefore we have an epimorphism π of vector bundles on IGrass(r ; F) which is a composition π : (TGrass(r ;F) )|IGrass(r ;F) = R∗ ⊗ ((F × IGrass(r ; F))/R) → R∗ ⊗ R∗ → S2 R∗ . (a) Prove that the embedding IGrass(r ; F) ⊂ Grass(r ; F) allows to identify the vector bundle TIGrass(r ;F) with the kernel of π. (b) Prove the exact sequences 0 → R∗ ⊗ (R∨ /R) → TIGrass(r ;F) →

2

R∗ → 0,

0 → TIGrass(r ;F) → R∗ ⊗ F/R → S2 R∗ → 0. 11. Use the results of exercises 9 and 10 to calculate cohomology groups of exterior powers of tangent and cotangent bundles on isotropic Grassmannians of isotropic subspaces of maximal dimension. Compare to the results of exercise 4.

Tensor Product Multiplicities 12. (Brauer and Klimyk’s formula.) Let λ, µ be two dominant weights for a reductive group G. Prove that the decomposition of the tensor product

Exercises for Chapter 4

135

Vλ ⊗ Vµ can be calculated as follows. Consider the set of weights (µ) = {ν1 , . . . , ν N } occurring in Vµ (with N = dim Vµ ). For each weight in λ + (µ) = {λ + ν1 , . . . , λ + ν N } calculate the Euler characteristic of the line bundle L(λ + νi ) on G/B. Then make cancellations if the same representation occurs with a positive and a negative sign. The remaining representations occur with positive sign and give the irreducible representations occurring in Vλ ⊗ Vµ .

5 The Geometric Technique

In this chapter we develop the basic technique for calculating syzygies. It applies to the subvarieties Y in an afﬁne space X with a desingularization Z which is a total space of a vector bundle over some projective variety V , which is a subbundle of the trivial bundle X × V over V . In such situation the Koszul complex of sheaves on X × V resolving the structure sheaf of Z has terms that are pullbacks of vector bundles over V . Taking the direct image of this Koszul complex by the projection p : X × V → V , one gets the formula expressing terms on the free resolution of the coordinate ring of Y in terms of cohomology of bundles on V . One also gets interesting complexes by taking direct images of the Koszul complex twisted by a pullback of a vector bundle on V . In this chapter we discuss the general construction and properties of direct images of Koszul complexes. The examples will be given in following chapters. The chapter is organized as follows. In section 5.1 we state the properties of the twisted direct images F(V)• of Koszul complexes. In particular we give the expressions for their terms and homology. We also state the criteria for F(V)• to be acyclic, the duality theorem for such complexes, and the result expressing the codimension and degree of Y in terms of the complex F(V)• . In section 5.2 we give the actual construction of complexes F(V)• . It involves constructing certain double complexes of sheaves on X × V resolving the Koszul complex. In section 5.3 we prove the other statements announced in section 5.1. In some of the proofs in sections 5.2 and 5.3 we rely on the machinery of derived categories. The necessary information is collected in section 1.2.5. Section 5.4 contains the equivariant setup. We prove that if a reductive group G acts on X and the action stabilizes Y , the variety V is a homogeneous space G/P, and the bundle V is a homogeneous bundle, then the terms and homology of the complexes F(V)• also carry an action of G. We also discuss the results of Kempf on rational singularities of the subvarieties Y and on the geometry of the desingularization Z in the case when V is a homogeneous space. 136

5.1. The Formulation of the Basic Theorem

137

In section 5.5 we give more explicit description of the differentials of F(V)• . Section 5.6 describes the technique of degeneration sequences which allows us to compare complexes F(V)• supported in different subvarieties.

5.1. The Formulation of the Basic Theorem Throughout this chapter we work over the algebraically closed ﬁeld K of arbitrary characteristic. Let us consider the projective variety V of dimension m. Let X = AKN be the afﬁne space. The space X × V can be viewed as a total space of trivial vector bundle E of dimension N over V . Let us consider the subvariety Z in X × V which is the total space of a subbundle S in E. We denote by q the projection q : X × V −→ X and by q the restriction of q to Z . Let Y = q(Z ). We get the basic diagram Z ⊂ ↓ q Y ⊂

X×V ↓q X

The projection from X × V onto V is denoted by p, and the quotient bundle E/S by T . Thus we have the exact sequence of vector bundles on V , 0 −→ S −→ E −→ T −→ 0. The dimensions of S and T will be denoted by s, t respectively. The coordinate ring of X will be denoted by A. It is a polynomial ring in N variables over K. We will identify the sheaves on X with A-modules. (5.1.1) Proposition. (a) The locally free resolution of the sheaf O Z as an O X ×V -module is given by the Koszul complex K(ξ )• : 0 →

t

( p∗ ξ ) → . . . →

2

( p ∗ ξ ) → p ∗ (ξ ) → O X ×V

where ξ = T ∗ . The differentials in this complex are homogeneous of degree 1 in the coordinate functions on X . (b) The direct image p∗ (O Z ) can be identiﬁed with the the sheaf of algebras Sym(η) where η = S ∗ . Proof. Let us identify X with the vector space E of dimension N over K. The bundle S is the s-dimensional subbundle of the N -dimensional trivial bundle

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over V . By the universal property (3.3.2) of the Grassmannian, there exists a map f : V −→ Grass(s, E) such that S = f ∗ (R). Let us consider the complex K• (Q∗ ) from (3.3.3). We set K(ξ )• := f ∗ K• (Q∗ ). The proposition follows by the same arguments as in the proof of (3.3.3). The idea of the geometric technique is to use the Koszul complex K(ξ )• to construct for each vector bundle V on V the free complex F(V)• of A-modules with the homology supported in Y . These complexes are the main subject of this book. In many cases the complex F(OV )• gives the free resolution of the deﬁning ideal of Y . In this section we state the theorems establishing the existence and basic properties of complexes F(V)• . The most important is the basic theorem (5.1.2) below, which gives the terms and the precise description of homology of complexes F(V)• . The next two sections will be devoted to the proofs of all the results that follow. Before we state the basic theorem, let us introduce the twisted Koszul complex. For every vector bundle V on V we introduce the complex K(ξ, V)• := K(ξ )• ⊗O X ×V p ∗ V. This complex is a locally free resolution of the O X ×V -module M(V) := O Z ⊗ p ∗ V. Now we are ready to state the basic theorem. (5.1.2) Basic Theorem. For a vector bundle V on V we deﬁne free graded A-modules i+ j H j V, ξ ⊗ V ⊗k A(−i − j). F(V)i = j≥0

(a) There exist minimal differentials di (V) : F(V)i → F(V)i−1 of degree 0 such that F(V)• is a complex of free graded A-modules with H−i (F(V)• ) = Ri q∗ M(V). In particular the complex F(V)• is exact in positive degrees.

5.1. The Formulation of the Basic Theorem

139

(b) The sheaf Ri q∗ M(V) is equal to H i (Z , M(V)) and it can be also identiﬁed with the graded A-module H i (V, Sym(η) ⊗ V). (c) If φ : M(V) → M(V )(n) is a morphism of graded sheaves then there exists a morphism of complexes f • (φ) : F(V)• → F(V )• (n) Its induced map H−i ( f • (φ)) can be identiﬁed with the induced map H i (Z , M(V)) → H i (Z , M(V ))(n). This theorem will be proven in section 5.2. If V is a trivial bundle of rank one on V , then the complex F(V)• is denoted simply by F• . The next theorem gives the criterion for the complex F• to be the free resolution of the coordinate ring of Y . (5.1.3) Theorem. Let us assume that the map q : Z −→ Y is a birational isomorphism. Then the following properties hold: (a) The module q∗ O Z is the normalization of K[Y ]. (b) If Ri q∗ O Z = 0 for i > 0, then F• is a ﬁnite free resolution of the normalization of K[Y ] treated as an A-module. (c) If Ri q∗ O Z = 0 for i > 0 and F0 = H 0 (V, 0 ξ ) ⊗ A = A, then Y is normal and it has rational singularities. The complexes F(V)• satisfy a Grothendieck type duality. Let ωV denote the canonical divisor on V . (5.1.4) Theorem. Let V be a vector bundle on V . Let us introduce the dual bundle t V ∨ = ωV ⊗ ξ ∗ ⊗ V ∗. Then F(V ∨ )• = F(V)∗• [m − t]. This result can be applied to give a criterion for the twisted module to be Cohen–Macaulay. (5.1.5) Corollary. Let us assume that dim Z = dim Y . Assume that for some vector bundle V on V we have Ri q∗ (O Z ⊗ p ∗ V) = 0 for i > 0. Then the

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module R0 q∗ (O Z ⊗ p ∗ V) is a maximal Cohen–Macaulay module supported in Y if and only if Ri q∗ (O Z ⊗ p ∗ V ∨ ) = 0 for i > 0. In that case the module R0 q∗ (O Z ⊗ p ∗ V ∨ ) is also a maximal Cohen–Macaulay module, dual to R0 q∗ (O Z ⊗ p ∗ V) in the sense of (1.2.26). Proof. We apply (5.1.4) to the complexes F(V)• and F(V ∨ )• . Our assumption implies that codim Y = dim X − dim Y = dim X − dim Z = dim X −(dim X + m − t) = t − m. Now (5.1.4) implies that the length of F(V)• equals t − m if and only if F(V ∨ )• has all the terms in nonnegative degrees. This establishes the ﬁrst claim. The duality statement follows because the two complexes are dual to each other. If the complex F(V)• satisﬁes the conditions of Corollary (5.1.5), we say that it has the Cohen–Macaulay property. In particular, when codim Y = 1 the complexes with Cohen–Macaulay property have length one, so they are just matrices. The determinant of such a matrix equals g rank V , where g is an irreducible equation of Y . In that case the complex F(V)• is called a determinantal complex. We will analyze such complexes for the case of discriminants and resultants in chapter 9. We conclude this section by showing that the complex F• contains the information about the codimension and the degree of Y . This fact will be useful in the cases when q is not necessarily a birational map. (5.1.6) Theorem. (a) codim X Y = max { i | Fi = 0 }. (b) Let us assume that dim Z = s + m < dim X = N . Let r = N − m − s. Then we have deg(q ) deg Y =

i+ j (i + j)r j ξ (−1)i+r h V, r! i, j

where by deﬁnition deg(q ) is 0 when dim Y < dim Z . Theorems (5.1.3), (5.1.4), and (5.1.6) as well as Corollary (5.1.5) will be proved in section 5.3.

5.2. The Proof of the Basic Theorem

141

5.2. The Proof of the Basic Theorem Before we prove Theorem (5.1.2) we recall several facts we will need. The ﬁrst one is the result on an equivalence of categories of graded modules and sheaves. Let S be a graded ring with S0 = A a ﬁnitely generated K-algebra and S1 a ﬁnitely generated A-module. For a graded S-module M we denote by M the corresponding sheaf on Proj S. For a sheaf F on X we deﬁne *(Proj S, F(n)). *∗ (F ) = n∈Z

We deﬁne an equivalence relation ≈ on graded S-modules by saying M ≈ M if there exists an integer d such that M≥d / M≥d . Here M≥d = n≥d Mn . We say that a graded S-module M is quasiﬁnitely generated if M is equivalent to a ﬁnitely generated module. In this setting we have (5.2.1) Proposition. The functors and *∗ induce an equivalence of categories between the category of quasiﬁnitely generated graded S-modules modulo the equivalence ≈ and the category of coherent OProj S -modules. Proof. This is exercise 5.9 in section II.5 of [H1]. We proceed with the discussion of some general facts concerning free complexes. Let A be a graded ring with A0 = K. (5.2.2) Proposition. (a) Let G • be a complex of ﬁnitely generated graded free A-modules. The complex G • decomposes into a direct sum G • G • = G • where G • is a minimal complex and G • is exact. The terms of the complex G • are G i = Hi (G • ⊗ A K) ⊗k A. (b) Let M• be a complex of graded A-modules with Mi = 0 for i < i 0 . Then there exists a minimal complex G • of free graded A-modules and a map φ : G • → M• which is a quasiisomorphism. Proof. We will start with the proof of (a). Let G i = 1≤ j≤gi A(−e j,i ), where gi = dimG − i. Let us consider the differential di : G i → G i−1 . We can

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The Geometric Technique

identify di with the matrix over A where the (k, j)th entry is homogeneous of degree e j,i − ek,i−1 . We will prove that we can change the basis in G i in a homogeneous way so the module G i will decompose as follows: G i = Bi ⊕ Ui ⊕ Bi ,

(∗)

so the differential di has a block decomposition 0 0 I di = 0 di 0 . 0 0 0 If this is done, then obviously we can set G i = Ui and G i = Bi ⊕ Bi , and the proposition follows. To get our decomposition we ﬁx an index m, choose a basis in G m in the appropriate way, and then spread this choice to the left and to the right by induction. We start by changing the basis in G m and G m−1 in a homogeneous way to bring dm to the canonical form dm 0 dm = 0 dm where dm is a minimal matrix with homogeneous entries and dm is an identity matrix. This means we can write G m = Wm ⊕ Vm , G m−1 = Wm−1 ⊕ Vm−1 , with dm corresponding to the map Wm → Wm−1 and dm corresponding to the map Vm → Vm−1 . Now we notice that the rows of dm+1 corresponding to Vm are zero because G • is a complex. Therefore it is really a map from G m+1 to Um . Bringing this map to the canonical form as above, we see that we can decompose Wm = Um ⊕ Bm and G m+1 = Wm+1 ⊕ Vm+1 in such way that dm+1 is a direct sum of the minimal map from Wm+1 to Um and the identity map from Vm+1 to Bm . We get the required choice of basis in G m by setting Bm = Vm . In fact we have also chosen the direct summand Bm−1 = Vm−1 in = Vm+1 in G m+1 . G m−1 and the direct summand Bm+1 Next we show how to extend our choice of basis to the right. Assume that we have the block decomposition (∗) for i > j together with the decomposition G j = B j ⊕ W j . We notice that B j ⊂ Ker d j . Therefore we treat d j as a map from W j to G j−1 . We bring it to the canonical form, which means we can write W j = U j ⊕ B j and G j−1 = B j−1 ⊕ W j−1 , so d j is a direct sum of the minimal map from U j to W j−1 and the identity from B j to B j−1 . Similarly, let us assume that the decomposition (∗) is achieved for i < j together with the decomposition G j = W j ⊕ B j . We notice that the image of

5.2. The Proof of the Basic Theorem

143

d j+1 is contained in W j , so we can treat is as a map from G j+1 to W j . Reducing this map to the canonical form, we get the decompositions W j = B j ⊕ U j and G j+1 = W j+1 ⊕ B j+1 such that d j+1 is a direct sum of a minimal map from W j+1 to U j and the identity map from B j+1 to B j . This completes the proof of (a). Let us prove (b). It is enough to construct the quasiisomorphism φ : G • → M• where G • is a complex of free graded A-modules. We can achieve minimality by applying (a) and taking G • as our complex. Let us denote the submodule Ker d in Mi by Z i and the submodule d(Mi+1 ) by Bi . Then we have exact sequences 0 → Bi → Z i → Hi → 0, 0 → Z i → Mi → Bi−1 → 0. Let 0 → Bi → Zi → Hi → 0, 0 → Zi → Mi → Bi−1 → 0 be the short exact sequences of free complexes covering the above maps. For each i we consider the map of complexes ηi given by the composition Mi → Bi−1 → Zi−1 → Mi−1 . It is clear that ηi−1 ηi = 0. We consider the double complex ηi

ηi−1

→ . . . Mi −→Mi−1 −→Mi−2 → . . . . We deﬁne G • to be the total complex of this double complex. By construction it is equipped with the natural map φ to M• . It is clear from the spectral sequence associated to our double complex that φ is a quasiisomorphism. After these preparations we can proceed with the proof of (5.1.2). Let us embed V in a projective space. Let OV (1) be the ample line bundle corresponding to this embedding. We can assume that the higher cohomology of the sheaves OV (n) vanishes for n > 0. This can be achieved by Serre’s theorem ([H1, Proposition III.5.3]). Let us denote by R the homogeneous coordinate ring of V in this embedding. The key step in the construction of complexes F(V)• is the existence of a certain right resolution of the twisted Koszul complex K(ξ, V)• .

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(5.2.3) Lemma. There exists a right resolution 0 → K(ξ, V)• → P(V)•• such that the following properties hold: (a) Each module P(V)i j is a direct sum of sheaves A(i) ⊗ OV (n), where n > 0, and therefore is q∗ -acyclic. (b) Each column 0 → K(ξ, V) j → P(V) r j is a q∗ -acyclic resolution of K(ξ, V) j by coherent O X ×V -modules which is the tensor product of A(− j) with the *-acyclic resolution of j (ξ ) ⊗ V. Proof. Let us start with the dual complex K(ξ, V)∗• . Its j-th term equals K(ξ, V)∗j =

−j ( p∗ ξ ∗ ) ⊗ V ∗ .

This is a complex of sheaves over X × V whose differential is homogeneous of degree 1 with respect to the generators of A. Let us apply to this complex the functor *∗ from (5.2.1). We get a complex of bigraded A ⊗ R-modules. The generators of the j-th term here have A-degree − j. Now we replace this complex by the complex of equivalent modules by cutting out in each module the components in nonpositive R-degrees. We get a complex C(V)• of bigraded A ⊗ R-modules where the j-th term has generators in A-degree − j and in positive R-degree. •• of the complex C(V)• . Next we consider the minimal free resolution C(V) Each module Ci j has generators in positive R-degree. •• , Now we construct P(V)•• by applying the functorto the complex C(V) and dualizing. By (5.2.1) it is the right resolution of the Koszul complex. Properties (a) and (b) are obviously satisﬁed. Consider the double complex q∗ (P(V)•• ). By Lemma (5.2.3) (a) it is a double complex of free graded A-modules. Let us consider the total complex associated to q∗ (P(V)•• ), G(V)• := T ot• (q∗ (P(V)•• )). Since the resolution P(V)•• is q∗ -acyclic, we get H−i G(V)• = Ri q∗ M(V).

5.2. The Proof of the Basic Theorem

145

Now we apply Proposition (5.2.2) to the complex G(V)• . We want to calculate the components of the minimal part of this complex. It is enough to calculate the homology of the complex G(V)• ⊗ K. We consider the double complex of vector spaces q∗ (P•• ) ⊗ K. The horizontal differentials in this double complex are 0 because the horizontal maps in P•• are by Lemma (5.2.3) (a) the matrices with entries of degree 1 in A. By Lemma (5.2.3) (b) the homology of each column q∗ (P• j ) consists of cohomology groups H . (V, j (ξ ) ⊗ V). Therefore H l (G(V)• ⊗ K) =

l+ j ξ ⊗V . H j V,

j≥0

Proposition (5.2.2) applied to G(V)• gives G(V)• = F(V)• L(V)• for some exact complex L(V)• . This proves part (a) of theorem (5.1.2). Let us prove part (b). The ﬁrst part of the statement follows from the fact that the module on X is determined by its global sections. The second part follows from the spectral sequence of the composition of maps X × V → V → ∗, from the fact that p is afﬁne, and from (5.1.1) (b). Before we turn to part (c), let us state some facts about the complexes F(V)• that follow easily from the proof of part (a) of Theorem (5.1.2). (5.2.4) Proposition. (a) The component i+ j (di )( j, j ) : H j V, ξ ⊗ V ⊗k A(−i − j) → Hj

V,

i−1+ j

ξ ⊗ V ⊗k A(−i + 1 − j )

of the differential di (V) : F(V)i → F(V)i−1 is of homogeneous degree j − j + 1. (b) The component (di )( j, j ) is zero if j < j . Proof. Part (a) follows from the fact that di is a homogeneous map of degree 0. Part (b) follows from minimality of the complex F(V)• . We need a result characterizing the complex F(V)• .

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(5.2.5) Proposition. The complex F(V)• is the unique minimal free complex quasiisomorphic to Rq∗ (O Z ⊗ p ∗ V). In particular, the complex F(V)• does not depend on the choice of the sheaf OV (1) and of the resolution P(V)•• . Proof. We constructed the complex F(V)• as a minimal free complex quasiisomorphic to the total complex of the double complex q∗ (P(V)•• ). However, the total complex of P(V)•• is a q∗ -acyclic complex quasiisomorphic to O Z ⊗ p ∗ (V). The statement now follows from (5.2.2) (b) and the fact implicitly contained in (5.2.2) that every quasiisomorphism between minimal free complexes has to be an isomorphism. Now we conclude the proof of part (c) of Theorem (5.1.2). A morphism φ : M(V) → M(V )(n) induces the map Rq∗ (φ) : Rq∗ (O Z ⊗ p ∗ V) → Rq∗ (O Z ⊗ p ∗ V )(n) in the derived category of bounded complexes of A-modules. However, every map between free complexes in this derived category is represented by a genuine map ψ of complexes ([H2], the dual version of Proposition I.4.7, or [GM], the dual version of Theorem 21, chapter III, section 5). The map ψ does not need to be homogeneous. However, since both complexes are graded, the homogeneous component ψ0 of ψ of degree zero will also be a map of complexes. Moreover, since the induced map H (ψ)∗ has degree zero, the map ψ − ψ0 is a map of free complexes inducing the trivial map on homology. Such a map is homotopic to zero, and thus ψ is homotopic to the map ψ0 of degree zero. •

5.3. The Proof of Properties of Complexes F(V)• In this section we prove Theorems (5.1.3), (5.1.4), and (5.1.6). Proof of Theorem (5.1.3). First of all, we notice that parts (b) and (c) follow from Theorem (5.1.2) and part (a) of (5.1.3). Thus it is enough to prove part (a). This statement follows from the following elementary lemma applied to the normalization of Y . (5.3.1) Lemma. Let q : Z → Y be a desingularization of Y . Let us assume that Y is normal. Then q∗ O Z = OY .

5.3. The Proof of Properties of Complexes F(V)•

147

Proof. The question is local on Y , so we can assume that Y = Spec A where A is a normal domain. The sheaf q∗ O Z is the sheaf associated to the ring *(Z , O Z ). Therefore it is enough to show that *(Z , O Z ) = A. Since q is birational, it is clear that *(Z , O Z ) is contained in the ﬁeld of fractions of A. It is also a ﬁnitely generated A-module, because q is proper. This proves the lemma. Proof of Theorem (5.1.4). We use the duality theorem for proper morphisms (Theorem (1.2.22)) for the map f = q : X × V → X , for F • = • ( p ∗ ξ ) ⊗ p ∗ V, and for G • = O X . The complex F(V)• is, by its construction, a free graded minimal representative of the object R f ∗ (F • ). Therefore the right side of the theorem gives R Hom•X (R f ∗ (F • ), G • ) = R Hom•X (F(V)• , O X ). Now R can be dropped because F(V)• is its own projective resolution (we calculate R Hom as R I I R I : compare Lemma 6.3, p. 66 in [H2]), and we are left with the complex F(V)∗• . To identify the right side we notice that by (1.2.21) (c) we have f ! (G • ) = f ∗ (G • ) ⊗ ω X ×V /V [n] = O X ×V ⊗ p ∗ (ωV )[n], because f is smooth. Therefore the inside term on the left side in (1.2.22) can be written as • R Hom•X ×V (F • , f ! (G • )) = R Hom•X ×V ( p ∗ ξ ) ⊗ p ∗ V, p ∗ (ωV )[n] . By proposition 5.16 (p. 113) of [H2], with L = identify the right hand side with R Hom•X ×V (O X ×V , O X ×V ) ⊗

•

( p ∗ ξ ) ⊗ p ∗ V) we can

•

( p ∗ ξ )∗ ⊗ p ∗ V ∗ ⊗ p ∗ (ωV )

which can be written as R Hom•X ×V (O X ×V , O X ×V ) ⊗

•

( p∗ ξ ) ⊗

t

(ξ ∗ ) ⊗ p ∗ V ∗ ⊗ p ∗ (ωV )

where t = rank ξ . We can drop R in the above expression, because in the left place we have a locally free complex (again we calculate R Hom as R I I R I , using Lemma 6.3, p. 66 in [H2]). This means the complex above is quasiisomorphic to • ( p ∗ ξ ) ⊗ t ( p ∗ ξ )∗ ⊗ p ∗ V ∗ ⊗ p ∗ (ωV ). Therefore the left hand side in

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(1.2.22) can be identiﬁed with • t R f∗ ( p ∗ ξ ) ⊗ ( p ∗ ξ )∗ ⊗ p ∗ V ∗ ⊗ p ∗ (ωV ) =F

t

∗

∗

∗

∗

∗

( p ξ ) ⊗ p V ⊗ p (ωV ) •,

as claimed in (5.1.4). Proof of Theorem (5.1.6). We start with part (a). Let us consider the canonical sheaf ω Z . By the adjunction formula ([H1, Proposition II.8.20]), ω Z = ω X ×V | Z ⊗ t ξ ∗ . Since X is just an afﬁne space, ω X ×V = p ∗ K . Therefore ω Z = O Z ⊗ K ⊗ t ξ ∗ . By the Grauert–Riemenschneider theorem (1.2.28) we know that Ri q∗ (ω Z ) = 0 for i > dim Z − dim Y . Therefore the terms F(K ⊗ t ξ ∗ )i are zero for i < dim Y − dim Z . By the duality (5.1.4) this means that Fi = 0 for i > codim X Y . It remains to show that for i = dim Z − dim Y we have Ri q∗ (ω Z ) = 0. After shrinking Y we can assume that q is smooth and projective. Then the last claim follows from the uppersemicontinuity theorem ([H1, III.12.11]) and the adjunction formula ([H1, II.8.20]), since each ﬁber Z y is smooth of dimension i, so H i (Z y , ω Z y ) is one dimensional (hence nonzero) by Serre duality. This proves (a). To prove (b) we consider the graded Hilbert function P(F• , t) =

i i+ j

(−1) t

h

j

i+ j V, ξ (1 − t)−N .

i, j≥0

Writing P(F• , t) = a≥0 P(a)t a , we know that for big a the function P(a) is polynomial in a. We also know that P(F• , t) is the alternating sum of graded Hilbert functions of the homology modules Ri O Z of F• . The homology modules Ri O Z are supported in Y . Moreover, the modules Ri O Z for i > 0 are supported in the locus of points in Y where the ﬁbers of q have dimension at least 1, which is a proper subvariety of Y . The sheaf q∗ O Z is generically of rank deg q . This means that P(a) is a polynomial of degree ≤ N − r and that the highest coefﬁcient of P(a) equals (N − r )! deg q deg Y in the case dim Y = dim Z and is zero otherwise. Statement (b) of (5.1.6) now follows by standard calculation. (5.3.2) Remarks. The geometric method was ﬁrst applied to determinantal varieties ([Ke 0],[L2], [JPW]). The general forms of statements (5.1.2), (5.1.3), (5.1.4), related to derived categories, were ﬁrst used to deal with

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149

examples related to nilpotent orbit closures and discriminants ([W2], [W3], [Br5]). We follow the approach from [Br5] to prove the ﬁrst part of (5.1.2) without derived categories.

5.4. The G-Equivariant Setup In this section we consider the special case of the construction from section 5.1 related to the situation when the variety V is a homogeneous space. This is the most important class of known examples where the geometric method applies. In fact, all examples considered in the following chapters are of this kind. Let G be a linearly reductive group, and let P be a parabolic subgroup in G. We assume that the variety V is the homogeneous space G/P. We also assume that the group G acts linearly on the afﬁne space X , so X can be identiﬁed with a representation of G. Let U be a P-submodule of X . We associate to U the vector bundle Z = G ×P U , which is by deﬁnition the orbit space G × U/P with P acting by p . (g, y) = (gp −1 , py). The projection G × U → G induces the G-equivariant morphism p : Z = G ×P U → V = G/P. Since U is a submodule of a G-module X , we have the embedding Z = G ×P U → G ×P X = G/P × X The identiﬁcation on the right hand side is made by the morphism (g, x) → (gP, gx). We denote by q the projection G/P × X → X , and by q its restriction to Z . As before, we denote Y = q (Z ). This places us in the situation of section 5.1. Let W be another P-module. We associate to W the vector bundle V(W ) := G ×P W . We can apply the construction from section 5.1 to get the complex F(V(W ))• of free graded modules over the ring A = K[X ] = Sym(X ∗ ). Let us recall that F(V(W ))i is given by the formula F(V(W ))i =

H

j

V,

i+ j

ξ ⊗ V(W ) ⊗k A(−i − j).

j≥0

Since the group G acts naturally on the bundles V(W ) and ξ , it acts ratio nally on the cohomology groups H j (V, i+ j ξ ⊗ V(W )) . Therefore G acts rationally on free modules F(V(W ))• via the diagonal action.

150

The Geometric Technique

(5.4.1) Theorem. Let G, P, V , X , Z , and V(W ) be as above. Then the complex F(V(W ))• can be constructed in such way that all the differentials di (V(W )) : F(V(W ))i → F(V(W ))i−1 are G-equivariant. Proof. We just have to follow the proof of Theorem (5.1.2) to assure that each step can be made G-equivariant. Before we do that, we need a G-equivariant analogue of (5.2.2). Let A be a graded ring over K with A0 = K. Let us assume that G acts rationally on A, i.e., G acts as a group of automorphisms of the graded ring A, so that each graded component Ai is a representation of G and that the multi

plication maps are G-equivariant. Let M = i≥i0 Mi be a graded A-module. We say that the group G acts rationally on M (compatibly with the action on A ) if each graded component Mi is a G-representation and the structure maps for the A-module M are G-equivariant. We will call a complex M• of ﬁnitely generated modules over A G-equivariant if G acts rationally on each module G i and the differentials are G-equivariant. Let M be a graded A-module on which G acts rationally. Then we can choose a minimal set of generators for M which forms a G submodule in M. Indeed, the projection M → M/A+ M splits as a map of G-modules. This means that every projective graded ﬁnitely generated A-module P on

which G acts rationally is of the form P = j P j ⊗ A(− j) for some ﬁnite dimensional G-representations P j . Therefore a complex G • of ﬁnitely generated graded free A-modules is G-equivariant if each G i is of the form

G i = j G i, j ⊗ A(− j) for some ﬁnite dimensional representations G i, j of G, and the differentials di : G i → G i−1 are G-equivariant. We can now recover all the standard results on minimal free resolutions of graded modules and complexes in G-equivariant form. In particular we have (5.4.2) Proposition. (a) Let G • be a G-equivariant complex of ﬁnitely generated graded free A-modules. The complex G • decomposes into a direct sum G • = G • ⊕ G • , where G • is a minimal complex, G • is exact, and both G • and G • are G-equivariant. The terms of the complex G • are G i = Hi (G • ⊗ A K) ⊗K A.

5.4. The G-Equivariant Setup

151

(b) Let M• be a G-equivariant complex of graded A-modules with Mi = 0 for i < i 0 . Then there exists a minimal G-equivariant complex G • of free graded A-modules and a map φ : G • → M• which is a quasiisomorphism. Proof. To prove both statements we just repeat the proof of (5.2.2). For (a) we notice that the decompositions G i = Bi ⊕ Wi ⊕ Bi

(∗)

can be chosen in G-equivariant way. In the proof of (b) all modules Z i , Bi , etc. are the modules with the rational G actions, so all the exact sequences are G-equivariant. Therefore the covering complexes of free modules can be also choosen in a G-equivariant way. Now we go through all the steps of the proof of (5.1.2). (1) The embedding of V in the projective space can be chosen in a Gequivariant way. Indeed, we can use any positive line bundle L(α) (cf. section 4.3). Then the sheaf OV (1) corresponds to the ample line bundle whose total space admits an action of G. Therefore the homogeneous coordinate ring R of V in this embedding admits a rational G-action. The key step in the construction of complexes F(V)• is the existence of a certain right resolution of the twisted Koszul complex K(ξ, V)• . (2) The twisted Koszul complex K(ξ, V(W ))• of sheaves on X × V consists of vector bundles admitting G-action. Therefore, after applying the functor *∗ to its dual, we get a G-equivariant complex of bigraded A ⊗ R-modules. Therefore its minimal resolution C(V(W ))•• is a Gequivariant complex, and thus the double complex P(V(W ))•• is a G-equivariant complex of sheaves. (3) It follows that the double complex q∗ (P(V)•• ) is a G-equivariant double complex of graded free A-modules. The rest of the proof follows by applying (5.4.2). This concludes the proof of Theorem (5.4.1). (5.4.3) Remark. The construction we applied in this section is also possible when the group G is only assumed to be reductive. In such case we cannot claim that the complex F(V(W ))• is G-equivariant. We get the G-action on the terms of F(V(W ))i , and we can claim the G-equivariance of linear strands of F(V(W ))• . However, the higher degree maps come from the spectral sequence, so some lifting is required. Therefore the higher degree maps need not be G-equivariant.

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The Geometric Technique

5.5. The Differentials in Complexes F(V)• . In this section we discuss the description of differentials in the complexes of type F(V)• . The point is that if one follows through the proof of Proposition (5.2.2), one gets an inductive procedure for calculating the differential in F(V)• which is not convenient to use. The following result, due to Eisenbud and Schreyer, allows us to describe the differential in a closed form. (5.5.1) Theorem ([ES]). Let F be a double complex ↑ ...

→

...

→

↑ dh

F ji+1

→

F ji ↑

→

dv ↑

dh

i+1 F j+1 ↑dv

→ ...

i F j+1 ↑

→ ...

in some abelian category. Assume that F ji = 0 for i & 0. Suppose that the vertical differential of F splits, so that for each i, j there is a decomposition i−1 i i i F ji = G ij ⊕ dv (G i−1 j ) ⊕ H j such that the kernel of dv in F j is H j ⊕ dv (G j ), isomorphically to dv (G ij ). Let us write s : F ji → and such that dv maps G i−1 j H ji for the projection corresponding to this decomposition and p : F ji → i−1 for the composition of the projection with the inverse of dv dv (G i−1 j ) → Gj i−1 restricted to G j . Then the total complex of F is homotopic to the complex ... →

d

H ji →

i+ j=k

with differential d=

H ji → . . .

i+ j=k−1

s(dh p)+ dh .

+≥0

Proof. We write dt = dv ± dh for the differential of the total complex. We i−+ i−+ note ﬁrst that s(dh p) j dh takes H ji to Hi+++1 . Since F j+++1 = 0 for + 1 0, the sum in the deﬁnition of d is ﬁnite. Let F denote F without a differential, i.e. viewed as a bigraded module. We will ﬁrst show that F is the direct sum of three components G ij , dt (G), and H = H ji G= i, j

and that dt is a monomorphism on G.

i, j

5.5. The Differentials in Complexes F(V)• .

153

The same statements with dv replacing dt are true by hypothesis. In particular, any element of F can be written in the form g + dv (G) + h with i g ∈ G ij , g ∈ G i−1 j , h ∈ H j for some i, j. Modulo G + dt (G) + H , such an element can be written as dh (G) ∈ F i−1 + j + 1. Since Fts = 0 for s & 0, we can use induction on i and assume dh (g) ∈ G + dt (G) + H , so we see that F = G + dt (G) + H . Suppose G ij , g∈G= G ij , and h ∈ H = H ji g ∈ G = i, j

i, j

i, j

be such that g + dt (g) + h = 0. We need to show that g = g = h = 0. Write

k−1 with gts ∈ G st . If b − a = −1, then dt = 0 and the desired g = bk=a g+−k result is a special case of the hypothesis. In any case, there is no componnt b−1 . From of g in G b+−b−1 , so the component of dt (g) in G b+−b is equal to dv g+−b b−1 b−1 the hypothesis we see that dv g+−b = 0, so g+−b = 0, and we are done by induction on b − a. This shows that F = G ⊕ dt (G) ⊕ H and that dt is an isomorphism from G to dt (G). The modules G ⊕ dt (G) form a double complex contained in F that we will call G. Since dt : G → dt (G) is an isomorphism, the total complex of G is split exact. It follows that the total complex tot(F) is homotopic to F/tot(G), and the modules in the last complex are isomorphic to i+ j=k H ji . We will complete the proof by showing that the induced differential on tot(F)/tot(G) is the differential d deﬁned in the statement of the theorem. Choose h ∈ H ji . The image of h under the induced differential is the unique element h ∈ H such that dt (h) ≡ h ((mod G) + dt (G)). Now dt h = dh h ≡ sdh h + (dv p)dh h

(mod G).

However, dv p ≡ dh p ≡ s(dh p) + dv p(dh p)

(mod G + dt (G)).

Continuing this way, and using again the fact that F ji = 0 for i & 0, we obtain s(dh p)+ dh h (mod G + dt (G)), dt h ≡ +

as required. Let us specialize to the situation where our abelian category is the category of graded A-modules for some graded ring A = d≥0 Ad , with A0 = K. Assume that the modules F ji above are free A-modules.

154

The Geometric Technique

(5.5.2) Corollary. Let F be a double complex of graded free A-modules. Assume that the differential dv is of degree 0 and that dh is minimal. Then the assumptions of Theorem (5.5.1) are satisﬁed, and H is a minimal complex homotopically equivalent to tot(F). Proof. The only claim needing veriﬁcation is that the vertical differential splits. Since A0 = K and the modules F ji are free, each column of F is obtained from some complex of vector spaces over K by tensoring with A. The splitting can also be chosen over K. The most efﬁcient general procedure to calculate the differential on the complexes of type F• (V) consists in applying Corollary (5.5.2) to the complex q∗ (P(V)•• ) constructed in the course of the proof of Theorem (5.1.2) in section 5.2. Still, that procedure cannot be carried to its completion for large complexes. We will see in the following chapters that for equivariant complexes representation theory is the best tool for identifying the differentials. 5.6. Degeneration Sequences So far we discussed the complexes F(V)• and their properties. They often give the terms of the minimal resolution of the module q∗ (O Z ⊗ p ∗ (V)). Sometimes it is useful to consider the exact sequences formed by such modules. This is especially useful in the “equivariant” situations, i.e. when our projective variety V is a homogeneous space. Such analysis allows sometimes to compare the resolutions of two orbit closures Y and Y1 such that Y1 ⊂ Y , i.e., Y1 is a degeneration of Y . Let us consider the basic diagram Z ⊂ ↓ q Y ⊂

X×V ↓q . X

We assume that V = G/P for some reductive algebraic group G and a parabolic subgroup P. The variety Z is a total space of a vector subbundle of X × V which can be identiﬁed with η∗ . Assume that η is a homogeneous bundle, i.e., it is of the form η∗ = G ×P U for some rational P-module U . We denote B := Sym(U ∗ ). This is a polynomial ring with a rational P-action. Let I ⊂ B be a P-equivariant ideal. We have a corresponding G-equivariant sheaf of ideals I ⊂ O Z . The degeneration technique comes from trying to exploit the resolution of B/I as a B-module.

5.6. Degeneration Sequences

155

(5.6.1) Proposition. Assume that we can ﬁnd a P-equivariant resolution 0 → G m → G m−1 → . . . → G 1 → G 0 of B/I with G i = Wi ⊗ B. Then we have an induced exact sequence 0 → Gm → Gm−1 → . . . → G1 → G0

(∗)

of vector bundles G j = (G ×P G j ) ⊗ O Z which is a resolution of O Z /I. Assume that higher cohomology groups H i (G/P, G j ) = 0 for i ≥ 1, 0 ≤ j ≤ m. Then we have a G-equivariant acyclic sequence of A-modules 0 → Mm → Mm−1 → . . . → M1 → M0 where M j = H 0 (G/P, G j ). Proof. Decompose the exact sequence (∗) into short exact sequences, and use long cohomology sequences. The existence of such a P-equivariant resolution is in general a rather subtle question. There are, however, two cases when such a resolution exists. The ﬁrst case occurs when the unipotent radical N of P acts on B trivially. Denote by L a Levi factor of P. (5.6.2) Proposition. Assume that N acts trivially on B and that L is linearly reductive. Then the resolution (∗) of O Z /I exists. Proof. Since N acts on B trivially, B is really an L-module. Since L is linearly reductive, we can construct an L-equivariant resolution of B/I by the arguments used in section 5.4. The other case occurs when the ideal I is a complete intersection deﬁned by some P-semiinvariants. Let’s recall that if an algebraic group acts rationally on a vector space U , then the ring of semiinvariants S I (H, U ) = S I (H, U )χ , χ∈char(H)

where S I (H, U )χ = { f ∈ Sym(U ∗ ) | h ◦ f = χ (h) f ∀h ∈ H }, is the ring of functions that transform according to a certain character of H.

156

The Geometric Technique

There is an important special case when one can predict which semiinvariants one should look at. It occurs when the rational H-module U has an open H-orbit. In such case one can classify the H-semiinvariants. This is due to the following result of Sato and Kimura. (5.6.3) Lemma (Sato–Kimura [SK]). Let H be a linear algebraic group acting rationally on a vector space U . Assume that this action has an open orbit. Then the ring S I (H, U ) is a polynomial ring. Moreover, the characters of the generators are linearly independent. The generators of the ring of semiinvariants can be described as follows. Assume O x is the open orbit of H in U . Let U \ O x = D1 ∪ . . . ∪ Dt be a decomposition into irreducible components. Assume that the ﬁrst s components have codimension 1 in U , while the other components have codimension bigger than 1. Then the generators of S I (H, U ) are the irreducible equations v1 , . . . , vs of D1 , . . . , Ds . For the proof of this lemma the reader should consult [Kr1, Theorem 2, section 3.6]. Coming back to our basic situation, i.e. H = P, U = η∗ , and using the Koszul complex, we have (5.6.4) Proposition. Assume that the ideal I is a complete intersection deﬁned by P-semiinvariants v1 , . . . , vs of weights λ1 , . . . , λs . Then the Koszul complex gives a resolution (∗) of length s, with G j = 1≤t1 0. (c) The Hilbert function of the normalization of K[Y ] does not depend on the characteristic of K. Proof. To prove (a) we consider the open subset U = {(φ, R) ∈ Z | rank φ = r }. By the deﬁnition of Z we see that if (φ, R) ∈ U then R = Im φ. Therefore the algebraic map φ → (φ, Im φ) is the inverse of the map q |U . This means q |U is an isomorphism, so q is a birational isomorphism. In order to establish (b) we apply Proposition (5.1.1)(b). We see that q∗ O Z = Sym(F ⊗ R∗ ). Using Theorem (5.1.2)(b) (applied with V = OV ), we see that it is enough to show that H i (V, Sym(F ⊗ R∗ )) = 0

for i > 0.

By the Cauchy formula (3.2.5), every symmetric power St (F ⊗ R∗ ) has a

ﬁltration with associated graded object |λ|=t L λ F ⊗ L λ R∗ . By the version (4.1.12) of Kempf’s theorem for Grassmannians, the higher cohomology of each L λ F ⊗ L λ R∗ vanishes. This proves part (b). Notice that if the characteristic of K is zero, then we can use the version (4.1.9) of the Bott’s theorem for Grassmannians to establish the vanishing of higher cohomology groups

162

The Determinantal Varieties

of Sym(F ⊗ R∗ ). Since the dimension of H 0 (V, L λ R∗ ) does not depend on the characteristic of K by Kempf’s theorem, (c) follows from (5.1.2)(b) and part (b). It follows from (6.1.1) and (5.1.3) that the complex F• gives a minimal resolution of the normalization of the coordinate ring K[Y ] as an A-module. We want to calculate the terms of the complex F• . Since the cohomology depends on the characteristic of K, we have to treat the cases of characteristic 0 and p separately. Thus for the remainder of this section we assume that char K = 0. By Theorem (5.1.2) the terms of the complex F• are given by the cohomo logy groups H i (V, i+ j (F ⊗ Q∗ )). Applying the Cauchy formula (2.3.3), we know that t L λ F ⊗ K λ Q∗ . (F ⊗ Q∗ ) = |λ|=t

Therefore we have t L λ F ⊗ H i (V, K λ Q∗ ). H i V, (F ⊗ Q∗ ) = |λ|=t

We calculate the groups H i (V, K λ Q∗ ). Using the version of Bott’s theorem for Grassmannians (Corollary (4.1.9)), we see that we have to apply Bott’s algorithm (4.1.5) to the sequence (0, λ) + ρ = (n − 1, . . . , n − r, λ1 + n − r − 1, . . . , λq ). The ﬁrst r numbers in this sequence are consecutive, and the last q numbers form a decreasing sequence. Let us assume that (0, λ) is regular. Let s be the biggest number such that λs + n − r − s > n − 1. Then λs+1 + n − r − s − 1 < n − r . In terms of λ this means that λs ≥ r + s,

λs+1 < s + 1.

We denote the set of partitions satisfying the above inequalities by P(r, s). Reordering (0, λ) + ρ means that the numbers λ1 + n − r − 1, . . . , λs + n − r − s go in front of n − 1, . . . , n − r . Let us denote the corresponding permutation by w(s). Clearly +(w(s)) = r s. The conditions for λ ∈ P(r, s) mean that if the partition λ has the Durfee square size s (cf. section 1.1.2), then the sequence (0, λ) is regular if and only if λ contains an additional s × r rectangle to the right of the Durfee square. In that case the partition w(s) · (0, λ) = (λ1 − r, . . . , λs − r, s r , λs+1 , . . . , λq ). This means we have proved

6.1. The Lascoux Resolution

163

(6.1.2) Proposition. The i-th term of the complex F• is given by L λ F ⊗ K w(s). (0,λ) G ∗ ⊗K A. Fi = s≥0

λ∈P(r,s), |λ|−r s=i

Let us try to rewrite this result in a more symmetric way. Every partition λ from P(r, s) can be written as λ = (r + s + α1 , . . . , r + s + αs , β1 , . . . , βn−s ) where α and β are two partitions. In this setup we have w(s) · (0, λ) = (s + α1 , . . . , s + αs , s r , β1 , . . . , βq−s ). The dual partition (w(s) · (0, λ)) can in this notation be expressed as ). (w(s) · (0, λ)) = (r + s + β1 , . . . r + s + βs , α1 , . . . , αm−s

The term corresponding to λ appears in Fi with i = s 2 + |α| + |β|. We can rewrite (6.1.2) in terms of partitions α and β. For any s we consider the set Q(s) = {(α, β)|α ⊂ (m − r − s)s , β ⊂ (s)(n−r −s) }. For (α, β) ∈ Q(s) we denote P1 (α, β) = (r + s + α1 , . . . , r + s + αs , β1 , . . . , βn−s ), ). P2 (α, β) = (r + s + β1 , . . . , r + s + βs , α1 , . . . , αm−s

Then (6.1.2) can be rewritten in following way: (6.1.3) Proposition. Fi = s≥0

(α,β)∈Q(s),

L P1 (α,β) F ⊗ L P2 (α,β) G ∗ ⊗K A.

i=s 2 +|α|+|β|

This way of writing the term is symmetric in F and G ∗ . Now we can state the basic result on the syzygies of determinantal varieties. (6.1.4) Theorem (Lascoux [L2]). Assume that char K = 0. The complex F• is a minimal resolution of the coordinate ring K[Yr ]. Therefore the i-th module in this minimal resolution is given by the formula (6.1.3). Proof. The only thing that we have to prove is that K[Yr ] is normal. However, it is clear from (6.1.3) that F0 = A. By (5.1.3)(c) the normality follows. Let us state some other consequences of Theorem (6.1.4).

164

The Determinantal Varieties

(6.1.5) Corollary. Assume that char K = 0. (a) The determinantal ideal Ir +1 is a prime ideal. The quotient ring A/Ir +1 is a normal domain. (b) The quotient ring A/Ir +1 is a Cohen–Macaulay ring with rational singularities. (c) Let us assume that m ≥ n. Then the type of the module A/Ir +1 is equal to the dimension of the module L (n−r )(m−n) G ∗ . Therefore A/Ir +1 is a Gorenstein ring if and only if m = n. (d) The t-th homogeneous component of the ring A/Ir +1 decomposes as a GL(F) × GL(G)-module as follows: Lλ F ⊗ LλG∗ (A/Ir +1 )t = |λ|=t, λ1 ≤r

Proof. To prove (a) let us notice that from (6.1.3) it follows at once that F1 = r +1 F ⊗ r +1 G ∗ ⊗K A(−r − 1). The map from F1 to F0 has to be GL(F) × GL(G)-invariant. By the Cauchy formula (3.2.5) it has to be given by (r + 1) × (r + 1) minors of the matrix . Since the variety Y is irreducible, its deﬁning ideal has to be prime. Part (b) follows from (5.1.3)(c). To see that (c) is true, let us assume that m ≥ n. We look at the last module in the resolution. It is clear from (6.1.3) that it is F(m−r )(n−r ) = L (m)(n−r ) F ⊗ L (n)(n−r ) ,(n−r )(m−n) G ∗ ⊗K A(−m(n − r )). The result follows, since the codimension of Yr equals (m − r )(n − r ). This is also a way to see that A/Ir +1 is Cohen–Macaulay without using rational singularities. Finally, (d) follows from (5.1.2)(b) and from Corollary (4.1.9). The remainder of this section is devoted to a more explicit description of the resolution of determinantal ideals in some important special cases. Historically these cases preceeded Lascoux’s paper. We preserve all our notation, assuming again that m ≥ n. (6.1.6) The Eagon–Northcott Complex. Let us consider the case r = n − 1. The complex F• is the resolution of the ideal In of maximal minors of the matrix . We notice that all modules corresponding to elements of Q(s) are zero for s ≥ 2. Obviously the only contribution from the set Q(0) is F0 = L (0) F ⊗ L (0) G ∗ ⊗K A = A.

6.1. The Lascoux Resolution

165

The elements from the set Q(1) give the contribution to the terms Fi for i > 0. In fact Fi = L (n+i−1) F ⊗ L (n,1i−1 ) G ∗ ⊗K A(−n − i + 1) for i ≥ 1, and identifying these Schur functors, we get Fi =

n+i−1

n

F⊗

G ∗ ⊗ Di−1 G ∗ ⊗K A(−n − i + 1).

We used the divided power because in such form the description of our complex will be characteristic free. Since our complex is the minimal resolution of In and the differentials are GL(m) × GL(n)-equivariant (section 5.4), we can use the representation theory to identify completely the differentials in F• . The differential d1 :

n

F⊗

n

G ∗ ⊗K A(−n) −→ A

has to be the composition n

F⊗

n

G ∗ ⊗K A(−n) → Sn (F ⊗ G ∗ ) ⊗K A(−n) → A,

where the left map is the embedding via n × n minors (cf. (3.2.5)) and the right map is the multiplication in A. Similarly, for i ≥ 1 the map di+1 :

n+i

F⊗

n

n+i−1

→

G ∗ ⊗ Di G ∗ ⊗K A(−n − i)

F⊗

n

G ∗ ⊗ Di−1 G ∗ ⊗K A(−n − i + 1)

is determined by its homogeneous component n+i

F⊗

=

n

n+i−1

G ∗ ⊗ Di G ∗ →

F⊗

n

n+i−1

F⊗

n

G ∗ ⊗ Di−1 G ∗ ⊗ A1

G ∗ ⊗ Di−1 G ∗ ⊗ F ⊗ G ∗

and thus has to be (up to a scalar we choose to be equal to 1) the tensor product of diagonalizations n+i

F→

n+i−1

F ⊗ F, Di G ∗ → Di−1 G ∗ ⊗ G ∗

tensored with n G ∗ . One can describe di+1 by an explicit formula. If { f 1 , . . . , f m } is a basis of F, {g1 , . . . , gn } is a basis of G, and i 1 , . . . , i n are nonnegative integers such

166

The Determinantal Varieties

that i 1 + . . . + i n = i, then the image di+1 ( f u 1 ∧ . . . ∧ f u n+i ⊗ g1∗ (i1 ) . . . gn∗ (in ) ) equals (−1)s+1 φu s ,t f u 1 ∧ . . . ∧ fˆu s ∧ . . . ∧ f u n+i ⊗ g1∗ (i1 ) . . . gt∗ (it −1) . . . gn∗ (in ) s,t

One can check easily that for i ≥ 1 one has di di+1 = 0. Our result generalizes to arbitrary characteristic. (6.1.7) Proposition. The complex F• deﬁned above is a minimal free resolution of the A-module A/In over a ﬁeld K of arbitrary characteristic, and thus when K is replaced by an arbitrary commutative ring. Proof. The proof is a repetition of proof of Lascoux’s theorem in a characteristic free setting. We notice that in the case under consideration the bundle Q∗ used in our Koszul complex is a line bundle. Therefore i (F ⊗ Q∗ ) = i F ⊗ Si Q∗ . The cohomology of bundles Si Q∗ has a characteristic free description by Serre’s theorem ([H1, chapter 3, section 5]). This proves that the terms of the complex F• are given by the formulas above. Then the reasoning we have just given in characteristic 0 case allows us to identify the differentials. Of course there exists an alternative proof of the general version based on Buchsbaum–Eisenbud acyclicity criterion (1.2.12) (see for example [BV, section 2C]). (6.1.8) The Gulliksen–Negard Complex ([GN]). This is the case m = n and r = n − 2. The nonzero terms of F• are F0 = L (0) F ⊗ L (0) G ∗ ⊗K A(0) = A(0), F1 = L (n−1) F ⊗ L (n−1) G ∗ ⊗K A(−n + 1), F2 = (L (n) F ⊗ L (n−1,1) G ∗ ⊗K A(−n)) ⊕ (L (n−1,1) F ⊗ L (n) G ∗ ⊗K A(−n)), F3 = L (n,1) F ⊗ L (n,1) G ∗ ⊗K A(−n − 1), F4 = L (n,n) F ⊗ L (n,n) G ∗ ⊗K A(−2n). We give an explicit description of the differentials. The idea is to construct the middle strand of F• (consisting of F3 , F2 , and F1 ) as a complex ﬁrst. We treat the map : F ⊗K A(−1) → G ⊗K A

6.1. The Lascoux Resolution

167

as a complex with nonzero terms appearing in homological degrees 1 and 0. The complex ∗ [−1] : G ∗ ⊗K A → F ∗ ⊗K A(1) has nonzero terms in degrees 0 and −1. Let f 1 , . . . , f m and g1 , . . . gn be the bases of F and G respectively. We have two equivariant maps of complexes Ev : ⊗ ∗ → A, Tr : A → ⊗ ∗ given by formulas Ev( f i ⊗ f j∗ ) = δi, j ,

Ev(gi ⊗ g ∗j ) = δi, j ,

Ev( f i ⊗ g ∗j ) = 0,

Ev(gi ⊗ f j∗ ) = 0,

and Tr(1) =

n i=0

gi ⊗ gi∗ −

m

f j ⊗ f j∗ .

j=0

This implies that the composition Ev Tr = 0. We consider the complex of complexes A[−1] → ⊗ ∗ [−1] → A[−1], and let H• be the complex which is the homology in the middle term. The nonzero terms of the complex H• are F ⊗ G ∗ ⊗K A(−1) → U ⊗K A → G ⊗ F ∗ ⊗K A(1), where U is the homology of the complex of vector spaces K → (F ⊗ F ∗ ) ⊕ (G ⊗ G ∗ ) → K with the maps coming from trace and evaluations according to the formulas given. Now we can see that after tensoring H• by n F ⊗ n G ∗ , shifting the grading by −n, and shifting the homological degree, we get the complex with the terms F3 , F2 , F1 . Reasoning as with the Eagon–Northcott com plex, we see that H• ⊗ n F ⊗ n G ∗ has to be a middle strand of F• . We

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augment our complex from both sides. We set F0 = F0 , F4 = F4 . We deﬁne the maps d1 : F1 → F0 by sending the generator f 1 ∧ f 2 ∧ . . . ∧ fˆi ∧ . . . ∧ f m ⊗ g1∗ ∧ . . . ∧ gˆ∗j ∧ . . . ∧ gn∗ ⊗ 1 to the determinant M(i, j) of the matrix with the i-th row and j-th column deleted. We also deﬁne the map d4 : F4 → F3

n

by sending the generator to i, j=1 (−1)i+ j M(i, j) f i ⊗ g ∗j ⊗ 1. Here we identify L (n,1) F with F, and L (n,1) G ∗ with G ∗ . Reasoning as with the Eagon– Northcott complex, we see that H• ⊗ n F ⊗ n G ∗ has to be a middle strand of F• . (6.1.9) Remarks. (a) The fact that determinantal ideals are perfect was ﬁrst proved by Eagon and Hochster in [HE]. The proof using the straightening law was given by DeConcini, Eisenbud, and Procesi in [DEP1]. (b) The idea of using higher direct images to calculate syzygies is due to Kempf. In his thesis ([Ke1]) he constructed the Eagon–Northcott complex in the way described above, using Serre’s theorem. Lascoux in his groundbreaking paper [L2] constructed the syzygies in the general case (in characteristic 0) using this method. The precise description of the differentials in Lascoux’s resolution was given by P. Roberts in his unpublished preprint [R1.]. (c) Several special cases of the resolutions of determinantal ideals were known before Lascoux’s proof. In addition to the Eagon–Northcott and Gulliksen–Negard complexes mentioned above, Poon ([Pn]) treated the case m = r + 3, n = r + 2. In all these papers the approach was algebraic. Various criteria for the exactness of the complex and localization were used. The resolutions in these cases were proven to be characteristic free.

6.2. The Resolutions of Determinantal Ideals in Positive Characteristic Throughout this section we assume that K is a ﬁeld of characteristic p. We retain the rest of the notation from the previous section. Let us recall that we

6.2. The Resolutions of Determinantal Ideals

169

constructed the free complex F• of A-modules with the i-th term i+ j j ∗ H V, (F ⊗ Q ) ⊗K A, Fi = j≥0

which is a minimal resolution of the normalization of the coordinate ring K[Yr ]. We will use the Schur complexes (section 2.3) to obtain some information about the terms of the complex F• in characteristic p. Our information is far from being complete, because we do not have the analogue of Bott’s theorem. Still, we can prove that K[Yr ] is normal and that the determinantal ideal Ir +1 is radical. This will establish properties (1)–(3) from the beginning of the previous section in characteristic p. We will also show that even though the Hilbert function of K[Yr ] does not depend on the characteristic of K, the resolution F• does. More precisely, for m = n = 5 and r = 1 the complex F• is different in characteristic 0 and 3. This is the example given by Hashimoto. Consider the natural epimorphism π : G ∗ × V → R∗ deﬁned over V = Grass(r, G). Applying (2.4.12)(c), we see that the complex L λ π gives a right resolution of K λ Q∗ . By (2.4.12)(a), the terms of L λ π have a ﬁltration whose associated graded object is µ K λ/µ G ∗ ⊗L µ R∗ . By Corollary (4.1.12) we see that each term of this ﬁltration is *-acyclic. This means that L λ π is a *acyclic resolution of K λ Q∗ , so the complex *(L λ π ) can be used to calculate the cohomology of K λ Q∗ . We want to identify *(L λ π) more precisely. Let us consider the Schur complex L λ (id), where id : G ∗ → G ∗ is the identity map. Applying (2.4.12)(a), we see that there exists a natural subcomplex (L λ (id))(r + 1) whose terms have a ﬁltration with the associated graded object K λ/µ G ∗ ⊗ L µ G ∗ . µ,µ1 >r

(6.2.1) Proposition. The complex *(L λ π ) is naturally isomorphic to L λ (id)/ L λ (id)(r + 1). Proof. We have a commutative diagram id

G ∗ −→ G ∗ ↓ id ↓π π ∗ G −→ R∗ This induces a map of complexes θ¯ : L λ (id) → L λ (π). Applying the functor * to this map, we get a map of complexes *(θ¯ ) : L λ (id) → *(L λ π). We

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The Determinantal Varieties

¯ is an epimorphism with the kernel L λ (id)(r + 1). First want to show that *(θ) of all it is clear that the image of L λ (id)(r + 1) is zero for dimension reasons. This means we get the induced map θ : L λ (id)/L λ (id)(r + 1) → *(L λ π). Let us deﬁne the subcomplexes X 0,

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The Determinantal Varieties

because Y is an afﬁne variety. Using Theorem (5.1.3)(a), we see that q∗ (O Z 1 ) is the normalization of K[Yra ] and this normalization has rational singularities. a ]. We just saw that q∗ (O Z 1 ) can Finally let us show that q∗ (O Z 1 ) = K[Y2u a 0 be identiﬁed with the A -module H (V, Sym( 2 Q)). Using (2.3.8)(b) and Corollary (4.1.9), we see that this module decomposes into the representations of GL(E) in the following way: 2 0 H V, Sym( Q) = L λ E. λ,λ1 ≤2u, λi even for all i a K[Y2u ]

To prove that the ring is normal it is therefore enough to show that each representation on the right hand side of the above formula occurs in a ]. K[Y2u We use the explicit description of U-invariants in Sym( 2 E) given in the proof of (2.3.8)(b). It is obvious that these functions do not vanish on Y2a u a ]. and therefore the corresponding representations occur in K[Y2u 2 The description of U-invariants in Sym( E) has another application. a consists It is clear from what we just proved that the deﬁning ideal of Y2u of all representations L λ E occurring in (2.3.8)(b) (for arbitrary t) for which λ1 > 2u. Also, by (2.3.8)(b) we see that the U-invariant corresponding to a a such representation is contained in I2u+2 . Since I2u+2 is GL(E)-equivariant, a . This shows that the the whole representation L λ E is contained in I2u+2 a a is equal to I2u+2 . deﬁning ideal of Y2u This completes the proof of ﬁrst two parts of Theorem (6.4.1). 2. The second incidence variety. Let us ﬁx n and r = 2u. We consider V = Grass(n − u, E) with the tautological sequence 0 → R → E × V → Q → 0, where dim R = n − u and dim Q = q = u. For a subspace R in E we denote by i the embedding of R into E. We consider the variety Z 2 = {(φ, R) ∈ X × V | i ∗ φ i = 0 }. The variety Z 2 is the total space of the bundle S2 deﬁned by the following sequence: 0 → S2 →

2

E∗ × V →

2

R∗ → 0.

This means that we are in the setting of the section 5.1 with E = 2 E ∗ × V . 2 ∗ R . It follows that T2 = Note that the variety Y , which is by deﬁnition q(Z 2 ), is not a priori equal a to Y2u .

6.4. The Determinantal Ideals for Skew Symmetric Matrices

191

We consider the basic diagram Z2 ⊂ ↓ q Y ⊂

X×V ↓q X

Applying Proposition (5.1.1), we see that the resolution of O Z as an O X ×V module is given by the Koszul complex 2 K R •:0→ 1 2 (n−u)(n−u−1)

p

∗

2

R

→...→ p

∗

2

R → O X ×V .

We calculate the terms of the complex F• . By the formula (2.3.9)(b) we see that t 2 K λ R. R = λ∈Q −1 (2t)

Therefore we have to calculate the cohomology groups H i (V, K λ R) for λ ∈ Q −1 . Using Corollary (4.1.9), we see that we have to consider a sequence (0, λ) + ρ = (n − 1, . . . , n − u, λ1 + n − u − 1, . . . , λn−u ). The ﬁrst u numbers in this sequence are consecutive, and the last n − u numbers form a decreasing sequence. Let us assume that (0, λ) is regular. Let s be the biggest number such that λs + n − u − s > n − 1. Then λs+1 + n − u − s − 1 < n − u. In terms of λ this means that λs ≥ u + s,

λs+1 < s + 1.

The only nonzero cohomology group of K λ R will be H us (V, K λ R) = K µ E, where µ = (λ1 − u, . . . , λs − u, s u , λs+1 , . . . , λn−u ). Now we use the fact that λ ∈ Q −1 . This means that there exists a partition α (with α1 ≤ n − 2u − s + 1) such that ). λ = λ(α, s) = (α1 + s + u, . . . , αs + s + u, s u+1 , α1 , . . . , αn−2u−s+1

In terms of α the partition µ equals ). µ = µ(α, s) = (α1 + s, . . . , αs + s, s 2u+1 , α1 , . . . , αn−2u−s+1

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The Determinantal Varieties

The term corresponding to µ(α, s) occurs in F• in the place |α| + 12 s(s + 1). We have proved the following proposition. (6.4.3) Proposition. The terms of the complex F• are given by the formula Fi = K µ(α,s) E ⊗K Aa . (s,α), α1 ≤n−2u−s−1, i=|α|+ 12 s(s+1)

We proceed with the analysis of the complex F• . First of all, it is clear that Fi = 0 for i < 0. Therefore F• is acyclic. Let us analyze the terms F0 and F1 . Clearly F0 = K 0 E and F1 = K (12u+2 ) E. This means that the complex F• a is the resolution of K[Y2u ], and the last part of Theorem (6.4.1) follows.• We ﬁnish this section by giving two examples in low codimension. (6.4.4) The Buchsbaum–Eisenbud complex ([BE3]). This is the case n = 2t + 1, r = 2t − 2. The complex G • gives the resolution of the module Aa /I2ta . The calculation of the terms of the complex G • from Theorem (6.4.1) gives G 3 = L (2t+1,2t+1) E ⊗ Aa (−2t − 1), G 2 = L (2t+1,1) E ⊗ Aa (−t − 1), G 1 = L (2t) E ⊗ Aa (−t), G 0 = Aa (0). The middle strand G 2 −→ G 1 is just ⊗

n

E. where ∗

: E ⊗K A (−1) → E ⊗K Aa a

is the generic map. We augment the complex G 2 −→ G 1 by two maps d1 : G 1 −→ G 0 and d3 : G 3 −→ G 2 given by the formulas ˆ d1 (e1 ∧ . . . ∧ eˆ j ∧ . . . ∧ e2t+1 ) = Pf( j) d3 (1) =

2t+1

ˆ j (−1) j Pf( j)e

j=1

ˆ is the Pfafﬁan of the 2t × 2t skew symmetric matrix obtained by where Pf( j) deleting in the j-th row and the j-th column. In the second formula we identify L (2t+1,1) E with E. One can check directly that these formulas deﬁne the GL(E)-equivariant complex with the same terms as G • . Then the argument we used in (6.1.6) for the Eagon–Northcott complex applies, and we see that G • is isomorphic to G • and thus it is a resolution of Aa /I2ta . One can use the Buchsbaum–Eisenbud acyclicity criterion (1.2.12) to

6.4. The Determinantal Ideals for Skew Symmetric Matrices

193

show that the complex deﬁned in this way gives a resolution of Aa /I2ta when K is replaced by an arbitrary commutative ring. (6.4.5) Remarks. Buchsbaum and Eisenbud proved in [BE3] that the complex G • deﬁned above is the universal resolution of Gorenstein ideals of codimension 3. This means that if S is a commutative ring and J is a Gorenstein ideal of codimension 3 in S, then there exists a homomorphism of rings ψ : Aa → S such that J = ψ(I2ta ), i.e., J is the ideal of 2t × 2t Pfafﬁans of the (2t + 1) × (2t + 1) matrix ψ(). The resolution of S/J as an S-module is given by G • ⊗ Aa S. (6.4.6) The J´ozeﬁak–Pragacz complex ([JP2]). This is the complex of length 6 which arises when n = 2t + 2, r = 2t − 2. The complex G • gives the resolution of Aa /I2ta . The nonzero terms are G 0 = Aa (0), G 1 = L (2t) E ⊗K Aa (−t), G 2 = L (2t+1,1) E ⊗K Aa (−t − 1), G 3 = (L (2t+2,1,1) E ⊗K Aa (−t − 2) ⊕ (L (2t+1,2t+1) E ⊗K Aa (−2t − 1), G 4 = L (2t+2,2t+1,1) E ⊗K Aa (−2t − 2), G 5 = L (2t+2,2t+2,2) E ⊗K Aa (−2t − 3), G 6 = L (2t+2,2t+2,2t+2) E ⊗K Aa (−3t − 3). As before, we identify the linear strands of G • using Schur complexes and trace and evaluation maps. The complex G • has two linear strands (apart from the trivial ones occurring at both ends). The ﬁrst one is the complex H•1 with the terms L (2t+2,1,1) E ⊗K Aa (−t − 2), L (2t+1,1) E ⊗K Aa (−t − 1), L (2t) E ⊗K Aa (−t). Since is a skew symmetric matrix, we can identify and ∗ [−1]. We will denote the identifying isomorphism by τ . We have a map EV of complexes 2

1⊗τ

EV

() −→ ⊗ −→ ⊗ ∗ [−1]−→Aa [−1].

Deﬁne B(1,1) () := Ker EV. We deﬁne H•1 = B(1,1) () ⊗ This is the ﬁrst linear strand of G • .

2t+2

E(−t)[−1].

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The Determinantal Varieties

The second linear strand, H•2 , is the complex with the terms L (2t+2,2t+2,2) E ⊗K Aa (−2t − 3), L (2t+2,2t+1,1) E ⊗K Aa (−2t − 2), L (2t+1,2t+1) E ⊗K Aa (−2t − 1). We have a map TR of complexes TR

1⊗τ

Aa [−1]−→ ⊗ ∗ [−1]−→ ⊗ −→ S2 (). Deﬁne A2 () = Coker TR. Finally, H•2 = A2 () ⊗

2t+2

E ⊗2 (−2t − 1)[−3].

Of course we can deﬁne the 0th linear strand to be H•0 Aa (0), and the strand H•3 = L (2t+2,2t+2,2t+2) E ⊗ Aa (−2t − 3). Notice that up to grading and homological shifts, H•i is dual to H•3−i . It remains to deﬁne the maps di from H•i to H•i−1 The map d1 is nonzero on the term L (2t) E ⊗ Aa (−t), and it sends the generator e1 ∧ . . . ∧ eˆi ∧ . . . ∧ ˆ j), ˆ where the last symbol denotes the Pfafﬁan of the eˆ j ∧ . . . ∧ en to Pf(i, 2t × 2t skew symmetric matrix we get from by deleting the i-th and j-th rows and the i-th and j-th columns. The map d3 can be deﬁned as the dual of d1 (up to shifts in grading and homological degree). The most difﬁcult part is the deﬁnition of d2 . We just sketch its construction. We start with the map of complexes: ⊗TR

1⊗1⊗

S2 ()[−1] −→ ⊗ ⊗ S2 () −→ ⊗ ⊗ ⊗ 2 2 m 1,3 ⊗m 2,4 −→ () ⊗ (). Here m 1,3 ⊗ m 2,4 denotes the map which multiplies the ﬁrst factor by the third one, and the second factor by the fourth one. We can check easily that this map induces the map of complexes A2 ()[−1] −→ B(1,1) () ⊗ B(1,1) () However, up to grading and homological shifts, B(1,1) () is the same as H•1 . Therefore we have a map d1 : B(1,1) () −→ Aa . We deﬁne d2 as a composition 1⊗d1

A2 ()[−1] −→ B(1,1) () ⊗ B(1,1) ()−→B(1,1) ().

6.5. Modules Supported in Determinantal Varieties

195

Now it is easy to check that the maps di deﬁne a GL(E)-equivariant double complex H•3 −→ H•2 −→ H•1 −→ H•0 . In order to check that this complex is isomorphic to G • it is enough to show that the components of the differentials are nonzero when restricted to every summand L λ E ⊗ Aa . Indeed, inducting on the homological and the homogeneous degree, we see by exactness of G • that in given homological and homogeneous degrees the module of cycles in G • has only one irreducible representation of the required kind, so it has to he the one covered by the differential we just constructed. Checking that the differentials deﬁned above have the required properties can be carried out using the U-invariant elements in each representation. (6.4.7) Remarks. The minimal resolutions of ideals of Pfafﬁans are not, in general, characteristic free. In fact, Kurano showed in [K3], [K4] that the relations between 4 × 4 Pfafﬁans of an 8 × 8 skew symmetric matrix are not spanned by linear relations when char K = 2. The more general family of similar examples is provided in [Ha5]. In the case of complex (6.4.6) Pragacz showed in [Pi ] that the ranks of syzygies do not depend on characteristic of K.

6.5. Modules Supported in Determinantal Varieties We preserve the notation from section 6.1. We will study the GL(F) × GL(G)equivariant modules supported in determinantal varieties. We are interested in the structure of such modules. An interesting family of modules supported in determinantal varieties are the direct images of equivariant sheaves on the desingularization we studied in section 6.1. Our constructions allow us to calculate the minimal resolutions of such modules, given by the appropriate complexes F(V)• . It turns out that in addition to the desingularization used in section 6.1, we have two other resolutions of singularities of the determinantal variety Yr , leading to three such families. The choice of desingularization does not make a difference when investigating the coordinate rings Ar := K[Yr ], but different desingularizations lead to different families of equivariant modules. It turns out that each of these families generates the Grothendieck group of graded GL(F) × GL(G)-equivariant modules supported in Yr . This means that, at least in principle, the resolution of an equivariant GL(F) × GL(G)module supported in Yr can be obtained from complexes F(V)• .

196

The Determinantal Varieties

We show that some of the terms of the complexes F(V)• depend on the characteristic of the ﬁeld K. When studying determinantal varieties Yr in section 6.1, we used a desingularization ¯ ∈ X × Grass(r, G) | Im φ ⊂ R}. Z r(2) = {(φ, R) In fact we could have made another choice: Z r(1) = {(φ, R) ∈ X × Grass(m − r, F) | φ |

R¯

= 0}.

The choice of desingularization was irrelevant when studying the coordinate rings of determinantal varieties, but it makes a difference when looking at twisted complexes F(V)• . In order to make the situation symmetric it is also necessary to study the ﬁbered product ¯ ∈ X × Grass(r, G) Z r = Z r(1) × X r Z r(2) = {(φ, R, R) × Grass(m − r, F) | Im φ ⊂ R, φ | R¯ = 0}. Throughout this section we denote by 0 → R → F × Grass(m − r, F) → ¯ → Q → 0 the tautological sequence on Grass(m − r, F), and by 0 → R ¯ G × Grass(r, G) → Q → 0 the tautological sequence on Grass(r, G). Let Cr (F, G) be the category of graded Ar -modules with rational GL(F) × GL(G)-action compatible with the module structure, and equivariant degree 0 maps. We denote by K 0 (Ar ) the Grothendieck group of the category Cr (F, G). For an equivariant graded module M ∈ Ob(Cr (F, G)) and for q ∈ Z, we denote by M(q) the module M with gradation shifted by q, i.e. M(q)n = Mq+n . For M ∈ Ob(Cr (F, G)) we deﬁne the graded character of M, char(Mn ) q n ∈ Rep(GL(F) × GL(G ∗ ))[[q]][q −1 ]. char(M) = n∈Z

where Rep(GL(F) × GL(G ∗ )) denotes the representation ring of GL(F) × GL(G ∗ ) and char is the product of character maps described in (2.2.10). We recall from section 2.2, that an integral weight for GL(m) is just an mtuple α = (α1 , . . . , αm ) of integers. The weight α is dominant if α1 ≥ . . . ≥ αm . Let α = (α1 , . . . , αm ) be an integral weight for GL(F). We set α (1) = (α1 , . . . , αr ), α (2) = (αr +1 , . . . , αm ). Let β = (β1 , . . . , βn ) be an integral weight for GL(G ∗ ). We deﬁne β (1) = (β1 , . . . , βr ), β (2) = (βr +1 , . . . , βn ). Let α = (α (1) , α (2) ). Assume that both α (1) and α (2) are dominant. Let β be a dominant weight for GL(G ∗ ). For each such pair (α, β) we deﬁne a sheaf M(α, β) = p (1)∗ (L α(1) Q ⊗ L α(2) R) ⊗ L β G ∗ ⊗ O Z (1)

6.5. Modules Supported in Determinantal Varieties

197

of graded modules on Z (1) . Symmetrically, let α be a dominant weight for GL(F), and let β be a weight for GL(G ∗ ) such that β (1) and β (2) are both dominant. We deﬁne a sheaf ¯ ⊗ O Z (2) N (α, β) = L α F ⊗ p (2)∗ (L β (1) Q¯ ⊗ L β (2) R) of graded modules on Z (2) . Finally, let α, β be weights such that α (1) , α (2) , β (1) , β (2) are dominant. We deﬁne a sheaf ¯ ⊗ OZ P(α, β) = p (1)∗ (L α(1) Q ⊗ L α(2) R) ⊗ p (2)∗ (L β (1) Q¯ ⊗ L β (2) R) of graded modules on Z . We deﬁne the equivariant graded modules M(α, β) = H 0 (Z (1) , M(α, β)), N (α, β) = H 0 (Z (2) , N (α, β)), P(α, β) = H 0 (Z , P(α, β)). Let us start with providing some examples of modules from three families. We look ﬁrst at the family M(α, β). Since in this case M(α, β) = M(α, (0)) ⊗K L β G ∗ , we can assume that β = (0). (6.5.1) Example. Let α (2) = (0). The sheaf M(α, β) equals L α(1) Q ⊗ O Z (1) . Therefore the direct image p∗(1) M(α, β) equals L α(1) Q ⊗ Sym(Q ⊗ G ∗ ). Using the straightening formula (2.3.2) and the characteristic free version [Bo2] of the Littlewood–Richardson rule, we see that the higher cohomology of M(α, β) vanishes. The j-th homogeneous component M j (α, (0)) has a ﬁltration with the associated graded object (L α(1) F ⊗ L µ F)≤r ⊗ L µ G ∗ , |µ|= j

where (L α(1) F ⊗ L µ F)≤r denotes a factor of (L α(1) F ⊗ L µ F) consisting of all L ν F in the tensor product with ν1 ≤ r . In characteristic zero we can write the direct sum instead of ﬁltration. (6.5.2) Example. The simplest examples of modules M(α, (0)) are provided by the ones corresponding to line bundles on Grass(m − r, F). The line bundles on Grass(m − r, F) correspond to tensor powers OGrass(m−r,F) (t) = ( r Q)⊗t for t ∈ Z. For t ≥ 0 this is a special case of the example (6.5.1). Let us describe the minimal free resolution of the module M((n r ), (0)). Assume that the characteristic of K is zero. The terms come from the cohomology of the vector bundles ( r Q)⊗t ⊗ • (R ⊗ G ∗ ). Decomposing by Cauchy’s

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The Determinantal Varieties

formula we see that one needs to apply Bott’s theorem (4.1.9) to the weights (t r , λ1 , . . . , λm−r ). The description is similar to the description of syzygies of determinantal varieties. The surviving terms decompose to families depending on a parameter s saying how many parts λ1 , . . . , λs have to move in front of r parts t. Notice that if λ1 ≤ t, there are no exchanges and we have s = 0. The condition for s terms being exchanged is λs ≥ s + t + r , λt+1 ≤ s + t. In such case we write λ := λ(t, s, α, β) = (α1 s + t + r, . . . , αt + s + t + r, β1 , . . . , βm−r −t ). The corresponding term is K µ F ⊗ L λ G ∗ where µ := µ(t, s, α, β) = ((α1 + s + t, . . . , αt + s + t, (s + t)r β1 , . . . , βm−r −t ). The term corresponding to the set of data t, s, α, β occurs in homological degree |α| + |β| + s 2 + st. In particular, take r = 3, t = 1, s = 2. Take α = (3, 1), β = (3, 2). The partitions λ, µ are X X λ= ◦ ◦

• • • •

X X X X ◦ ◦ ◦

,

X X X • • • X X X • .

µ= ◦ ◦ ◦ ◦ ◦

Here the boxes corresponding to α are ﬁlled by •, the boxes corresponding to β are ﬁlled by ◦, and the boxes corresponding to the (s + t) × s rectangle both partitions have to contain are ﬁlled by X . (6.5.3) Theorem. The group K 0 (Ar ) is generated by the classes of the modules of each of the families M(α, β)(q), N (α, β)(q), P(α, β)(q) where α and β are both dominant weights and q ∈ Z. (6.5.4) Theorem. (a) The group K 0 (Ar ) is isomorphic to the additive subgroup of the ring Rep(GL(F) × GL(G ∗ ))[[q]][q −1 ] generated by shifted graded characters of modules M(α, β) (or N (α, β), or P(α, β)(q)) for α, β dominant, (b) The group K 0 (Ar ) is isomorphic to the additive group of the ring Rep(GL(F) × GL(G ∗ ))[q][q −1 ].

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199

We provide the proofs only for the family M(α, β). The proofs for the family N (α, β) are symmetric. The proofs for the family P(α, β) and the transition formulas are given in [W6]. We start with basic observations about the cohomology groups of the sheaves M(α, β). Since p (1) is an afﬁne map, R i p∗(1) O Z (1) = 0 for i > 0. We also have (1) p∗ O Z (1) = Sym(Q ⊗ G ∗ ). Using the Leray spectral sequence and a projection formula (assuming α arbitrary, β dominant), we have H i (Z (1) , M(α, β)) = H i (Grass(m − r, F), K α(1) Q ⊗ K α(2) R ⊗ Sym(Q ⊗ G ∗ )). (6.5.5) Proposition. Let α, β be dominant weights for GL(F), GL(G ∗ ) respectively. Then H i (Z (1) , M(α, β)) = 0 for i > 0. Proof. By deﬁnition M(α, β) = p (1)∗ (L α(1) Q ⊗ L α(2) R) ⊗ L β G ∗ ⊗ O Z (1) . Using this, we see that p∗(1) M(α, β) = L α(1) Q ⊗ L α(2) R ⊗ L β G ∗ ⊗ Sym(Q ⊗ G ∗ ). Using the straightening law (3.2.5) and the fact that the tensor product of two Schur functors has a ﬁltration whose associated graded object is a direct sum of Schur functors ([Bo2]), we are reduced to proving that if α is dominant then H i (Grass(m − r, F), L α(1) Q ⊗ L α(2) R) = 0 for i > 0. In characteristic 0 this follows at once from (4.1.9). The characteristic free version follows from (4.1.12). We can get additional information about the cohomology of M(α, β) when K is a ﬁeld of characteristic zero. Let α be a weight with α (1) , α (2) dominant. We deﬁne the number l(α) as follows. Consider the weight α + ρ where ρ = (m − 1, m − 2, . . . , 1, 0) = (u 1 , . . . , u m ). Deﬁne, by reverse induction on s (from s = r to s = 1) the numbers δs = min{t | t ≥ δs+1 , t +m −s ∈ / {δs+1 + m − s − 1, . . . , δr + m − r, αr +1 + m − r −1, . . . , αm }}.

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By construction the weight (δ1 , . . . , δr , αr +1 , . . . , αm ) + ρ is not orthogonal to any root. By (4.1.9) there exists a unique l such that H l (Grass(m − r, F), K δ Q ⊗ K α(2) R) = 0. We deﬁne l(α) := l. (6.5.6) Proposition. Let K be a ﬁeld of characteristic zero. Let α = (α (1) , α (2) ) be a weight for GL(F) with α (1) , α (2) dominant. Let l(α) be deﬁned as above. Assume that β is a dominant weight for GL(G ∗ ). (a) H i (Z (1) , M(α, β)) = 0

for i > l(α).

(b) H l(α) (Z (1) , M(α, β)) = 0. Proof. We can assume that β = (0) because, by the projection formula, tensoring with K β G ∗ commutes with taking cohomology. This means we are reduced to calculating the cohomology H ∗ (Grass(m − r, F), L α(1) Q ⊗ L α(2) R ⊗ Sym(Q ⊗ G ∗ )). This can be rewritten as H ∗ (Grass(m − r, F), L δ Q ⊗ L α(2) R ⊗ L γ G ∗ ). δ∈α (1) ⊗γ

By the Littlewood–Richardson rule (2.3.4), every weight occurring in the tensor product α (1) ⊗ γ is bigger than or equal to α (1) termwise. Also, since dim Q = r ≤ dim G ∗ , all such weights δ will occur in α ⊗ γ for some γ . Consider α = (α (1) , α (2) ) satisfying the assumptions of the proposition. Let δ0 be the weight constructed in deﬁning l(α). This, by deﬁnition, is the termwise minimal weight for which L δ0 Q ⊗ L α(2) R has nonzero cohomology. This cohomology occurs in degree l(α). Also it is clear from (4.1.9) that for δ that is bigger termwise than δ0 the cohomology of L δ Q ⊗ L α(2) R, if nonzero, occurs in degree ≤ l(α). This proves both parts of the proposition. We also get information on the support of cohomology modules H i (Z (1) , M(α, β)). In oder to state the result we need to introduce one more notion. The permutation σ ∈ m is an r -Grassmannian permutation if σ (1) > . . . > σ (r ), σ (r + 1) > . . . > σ (m). For each r -Grassmannian permutation σ we denote by Cσ the Weyl chamber of all weights (γ1 , . . . , γm ) such that the entries in

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201

γ + ρ are pairwise different and ordered in the same way as the sequence σ (1), . . . , σ (m). Let σ be an r -Grassmannian permutation of length i. For α arbitrary and β dominant we deﬁne the Ar -module H i (Grass(m − r, F), M(α, β))σ to be the part of H i (Grass(m − r, F), M(α, β)) consisting of all cohomology modules of sheaves K δ Q ⊗ K α(2) R ⊗ K γ G ∗ for which the weight (δ, α (2) ) ∈ Cσ . It is clear that this is a direct summand of the Ar -module H i (Z (1) , M(α, β)). (6.5.7) Proposition. Let K be a ﬁeld of characteristic 0. Asume that α is arbitrary and β is dominant. (a) The module H i (Z (1) , M(α, β))σ is nonzero if and only if there exists δ = (δ1 , . . . , δr ) such that δ ≥ α (1) (termwise) and (δ, α (2) ) ∈ Cσ . (b) The support of H i (Z (1) , M(α, β))σ is the determinantal variety Ys−1 for s = σ (r + 1). Proof. As in the proof of (6.5.6), we can assume that β = (0). The ﬁrst part of the proposition follows as in the proof of (6.5.6). Let us choose an r Grassmannian permutation σ of length i. We are interested in the support of the cohomology modules of M(α, β)σ = K δ Q ⊗ K α(2) R ⊗ K γ G ∗ . γ

δ∈α (1) ⊗γ , δ∈Cσ

Let σ (r + 1) = s. Then we can increase δ1 , . . . , δs−1 as we please to still get the weights (δ, α (2) ) from Cσ . On the other hand, the indices δs , . . . , δr can increase only by a limited number if we are to get a weight from Cσ . Now the use of Littlewood–Richardson rule and (6.1.5)(d) shows that the support of the module H i (Z (1) , M(α, β))σ equals Ys−1 . We proceed to prove Theorems (6.5.3) and (6.5.4) for the family M(α, β). Each equivariant sheaf M on Z (1) has its Euler characteristic class χ(M) = (−1)i [H i (Z (1) , M)] ∈ K 0 (Ar ). i≥0

(6.5.8) Proposition. The group K 0 (Ar ) is generated by the Euler characteristic classes χ (M(α, β)) for (α (1) , α (2) , β dominant). Proof. Let M be a graded Ar -module with a rational GL(F) × GL(G ∗ ) action. Then the natural morphism M → (q∗(1) )(q (1)∗ M)

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has a kernel and cokernel supported in Yr −1 . It is therefore enough to show. (1) If M is a module supported in Yr −1 , then its class in K 0 (Ar ) is in the span of classes χ(M(α, β)) for (α (1) , α (2) ), β dominant. (2) The class of q (1)∗ M is contained in the span of sheaves M(α, β) in the Grothendieck goup of equivariant sheaves on Z (1) . We start with the proof of (1). Since in this argument all constructions will commute with tensoring by L β G ∗ , we will drop it from our notation, dealing with sheaves M(α (1) , α (2) ) := M(α (1) , α (2) , (0)). It is enough to show that the Euler characteristic of each sheaf of type ˆ is in the subgroup of K 0 (Ar ) generated by M(α), for Yr −1 denoted by M(α), Euler characteristics of sheaves M(α) for Yr . Consider the Grassmannian Grass(m − r + 1, F) with the tautological sequence ˆ → F × Grass(m − r + 1, F) → Qˆ → 0. 0→R Consider the ﬂag variety Flag(m − r, m − r + 1; F) with universal ﬂag ˆ ⊂ F. Let 0⊂R⊂R v1 : Flag(m − r, m − r + 1; F) → Grass(m − r, F), v2 : Flag(m − r, m − r + 1; F) → Grass(m − r + 1, F) denote the natural projections. We have by deﬁnition ˆ (1) , . . . , α (1) , α (2) , . . . , α (2) M(α 1 r −1 1 m−r +1 ) ˆ ⊗ L (2) = v2∗ (L α(1) Qˆ ⊗ L (2) (R/R)

(2) R (α2 ,...,αm−r +1 )

α1

⊗ Sym(Qˆ ⊗ G ∗ )).

The higher direct images of the tensor product sheaf on the right hand side vanish. This sheaf has a Koszul type resolution of locally free modules over Sym(Q ⊗ G ∗ ) on Flag(m − r, m − r + 1; F) with terms ˆ ⊗ L (α(2) ,...,α(2) L α(1) Qˆ ⊗ L α(2) (R/R) ⊗

•

1

2

m−r +1 )

R

ˆ ⊗ G ∗ ) ⊗ Sym(Qˆ ⊗ G ∗ ), (Ker(Q → Q)

which can be rewritten as ˆ ⊗ L (α(2) ,...,α(2) L α(1) Qˆ ⊗ L α(2) +• (R/R) 1

2

m−r +1

)R ⊗

•

(G ∗ ) ⊗ Sym(Qˆ ⊗ G ∗ ),

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203

ˆ is isomorphic to R/R, ˆ because Ker(Q → Q) as can be seen from the commutative diagram 0 0

→ R → ↓ ˆ → → R

F × Flag(m − r, m − r + 1; F) → Q → 0 ↓ ↓ F × Flag(m − r, m − r + 1; F) → Qˆ → 0

of vector bundles over Flag(m − r, m − r + 1; F). We push down the terms of the resolution by v1∗ . The bundle Q is induced from Grass(m − r, F). Thus each term ˆ L α(1) Qˆ ⊗ L α(2) +t (R/R) ⊗ L (α(2) ,...,α(2) 1

2

m−r +1

)R ⊗

t (G ∗ ) ⊗ Sym(Qˆ ⊗ G ∗ )

gives us a sheaf M(γ (1) , γ (2) ), possibly with sign. Taking Euler characteristics ˆ (1) , α (2) ) as a combination of terms of the type gives an expression of χ (M(α (1) (2) χ (M(γ , γ ). To prove statement (2) we notice that q1∗ is a sheaf of graded Sym(Q ⊗ G ∗ )-modules. We can take its free GL(Q) × GL(R) × GL(G ∗ )equivariant resolution. Its terms are, up to ﬁltration, direct sums of sheaves of type M(α (1) , α (2) , β). This completes the proof of Proposition (6.5.4). (6.5.9) Remarks. Notice that the proof of Proposition (6.5.8) shows that if ˆ (1) , α (2) , β) is (α (1) , α (2) ) is dominant, then the Euler characteristic of M(α in the subgroup of K 0 (Ar ) spanned by the Euler characteristics of sheaves M(α, β) with α, β dominant. Theorem (6.5.3) for modules of type M(α, β) follows from the following statement: (6.5.10) Proposition. The classes χ (M(α (1) , α (2) , β)) such that (α (1) , α (2) ) is dominant generate the group K 0 (Ar ). Proof. We ﬁrst reduce the proof to the case when the characteristic of K equals zero. First of all we notice that the representation ring of the general linear group GL(F) is spanned by the classes of Schur modules L α F. This follows from the description of irreducible representations of GL(F) given in Theorem (2.2.9). Moreover, the Littlewood–Richardson rule holds in the representation ring of GL(F) in a characteristic free way, by the remark following Theorem (2.3.4). Finally, even though Bott’s theorem is true only over a ﬁeld of characteristic zero, the Euler characteristic of any line bundle over a ﬂag variety is characteristic free. Our statement involves only Euler

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characteristics and the characters of modules M(α, β) for α, β dominant. Both these notions are independent of the characteristic and can be described using only Schur modules and the Littlewood–Richardson rule. We see that the proof in characteristic zero implies that the same statement holds over a ﬁeld of arbitrary characteristic. Let H be the subgroup of K 0 (Ar ) spanned by the Euler characteristic classes of sheaves M(α, β) with α and β dominant. Consider an arbitrary sheaf M(α (1) , α (2) , β). We use induction on s := s(α (1) , α (2) ) = α1(2) − αr(1) . If s ≤ 0, then (α (1) , α (2) ) is dominant and there is nothing to prove. Suppose that for (γ (1) , γ (2) ) with smaller s the corresponding sheaves are in H . We identify M(α (1) , α (2) , β) with its direct image p1∗ M(α (1) , α (2) , β) = L α(1) Qˆ ⊗ L α(2) (R) ⊗ Sym(Q ⊗ G ∗ ). Consider the subsheaf M<s (α (1) , α (2) , β) of M(α (1) , α (2) , β) consisting of all summands L γ (1) Q ⊗ L γ (2) R ⊗ Sδ G ∗ such that γ1(2) − γr(1) < s. It follows from the Littlewood–Richardson rule (2.3.4) that it is a Sym(Q ⊗ G ∗ ) submodule. Denote by Ms (α (1) , α (2) , β) the factor M(α (1) , α (2) , β)/ M<s (α (1) , α (2) , β). We have an exact sequence 0 → M<s (α (1) , α (2) , β) → M(α (1) , α (2) , β) → Ms (α (1) , α (2) , β) → 0. The support of all cohomology groups of the sheaf Ms (α (1) , α (2) , β) is contained in Yr −1 . Indeed, if we multiply the summand L γ (1) Q ⊗ L γ (2) R ⊗ Sδ G ∗ by r Q ⊗ r G ∗ corresponding to r × r minors, we add one to each entry of γ (1) , so s has to decrease. This means that the ideal of r × r minors annihilates all cohomology groups of Ms (α (1) , α (2) , β). Now Remark (6.5.9) and an induction on r imply that the class χ (Ms (α (1) , α (2) , β)) is in H . Consider the relative version of the GL(Q) × GL(R) × GL(G ∗ )equivariant free resolution of the sheaf M<s (α (1) , α (2) , β). Its i-th term Fi is a direct sum of sheaves M(γ (1) , γ (2) , β) and each term occurring in Fi has smaller s than (α (1) , α (2) ). Indeed, by induction on i and by the Littlewood– Richardson rule (2.3.4) it follows that on multiplying M(γ (1) , γ (2) , β) occurring in Fi by summands of Sym(Q ⊗ G ∗ ), the invariant s in all resulting summands can only decrease. We conclude that, by induction on s, the Euler characteristic χ(M<s (α (1) , (2) α , β)) is in H and Proposition (6.5.10) is proven. Proof of Theorem (6.5.4). Part (a) is a consequence of the fact that the characters of M(α, β)(−i) are linearly independent in Rep(GL(F) × GL(G ∗ ))

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205

[[q]][q −1 ]. To prove (b) we deﬁne the homomorphism of abelian groups : Rep(GL(F) × GL(G ∗ ))[q][q −1 ] → K 0 (Ar ) by sending the class [(L α F ⊗ L β G ∗ )q i ] to [M(α, β)(−i)]. By Theorem (6.5.3) the homomorphism is an epimorphism. It is also a monomorphism, because the characters of modules M(α, β)(−i) are linearly independent in Rep(GL(F) × GL(G ∗ ))[[q]][q −1 ] and therefore in K 0 (Ar ). Theorem (6.5.3) implies that for any graded GL(F) × GL(G ∗ )-equivariant Ar -module M the class [M] can be expressed as a linear combination of classes [M(α, β)(−i)]. In particular we can describe the class of Ar −1 . (6.5.11) Proposition. The class of Ar −1 in K 0 (Ar ) is given by the formula [Ar −1 ] = [Ar ] −

n−r

[M((i + 1, 1r −1 ), (1r +i , 0n−r −i ))].

i=0

Proof. Let A denote the sheaf of algebras Sym(Q ⊗ G ∗ ) over Grass(m − r, F). Consider the relative Eagon–Northcott complex over A, 0 → En−r → En−r −1 → . . . → E1 → E0 → A, where Ei = Di Q ⊗

r

Q⊗

n

G ∗ ⊗ A(−i − r ).

This is a sheaf resolution of a sheaf B of algebras. Using the relative version of the straightening law (3.2.5), we deduce that the sheaf B has a ﬁltration whose associated graded object is L λ Q ⊗ L λ G ∗ . [B] = λ=(λ1 ,...,λr −1 )

Kempf’s theorem (4.1.10) implies that the higher cohomology of B vanishes and that the sections of B are isomorphic to Ar −1 . We also have the identiﬁcation Ei = M((i + 1), 1r −1 ), (1r +i , 0n−r −i )). This concludes the proof. (6.5.12) Remarks. The relative Eagon–Northcott complex used in the proof of (6.5.11) and its sections is the simplest example of a degeneration sequence discussed in section 5.6.

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For the remainder of this section we assume that K is a ﬁeld of characteristic 0. We calculate the depth of modules M(α, β). Of course the depth of M(α, β) is independent of β, so we assume that β is zero and consider M(α) := M(α, (0)). We start with arbitrary (α (1) , α (2) ), not necessarily dominant. We assume only that the sheaf M(α (1) , α (2) ) has no higher cohomology. (6.5.13) Proposition. The sheaf M(α (1) , α (2) ) has no higher cohomology if (2) and only if αr(1) ≥ α1(2) − t where t is such that α1(2) = . . . = αt(2) > αt+1 . Proof. Let δ1 , . . . , δr be the numbers deﬁned before the statement of Proposition (6.5.6). The condition of Proposition (6.5.13) means that δr ≥ α1(2) , and that is equivalent to l(α) = 0. We want to ﬁnd the projective dimension of M(α (1) , α (2) ), because by the Auslander–Buchsbaum formula (1.2.7) we have depth( A, M) + pd A (M) = mn. The resolution of M(α (1) , α (2) ) is given by the complex F(L α(1) Q ⊗ L α(2) R)• , whose i-th term is given by i+ j j ∗ (1) (2) (R ⊗ G ) ⊗ A(−i − j). H Grass(m − r, F), L α Q ⊗ L α R ⊗ j≥0

In order to study the top of the resolution, we look ﬁrst at the last term of the Koszul complex. The corresponding GL(F)-weight is (2) + n). t(α) = (α1(1) , . . . , αr(1) , α1(2) + n, . . . , αm−r

Let’s look at the weight t(α) + ρ = (a1 , . . . , ar , b1 , . . . , bm−r ). For j = 1, . . . , m − r we deﬁne the sequences t j (α) inductively by setting t j (α) = (a1 , . . . , ar , d1 , . . . , d j , b j+1 , . . . , bm−r ), where d j+1 is deﬁned as d j+1 = max{t | t ≤ b j+1 , b < d j , t ∈ / {a1 , . . . , ar }}. Notice that in this deﬁnition the condition t < d j could be skipped, because each b j is essentially lowered to the ﬁrst possible number that is not one of

6.5. Modules Supported in Determinantal Varieties

207

the ai ’s or previous dK ’s. Let us also deﬁne numbers q j = n − (b j − dn ), p j = q j − #{i | ai < d j } for j = 1, . . . , m − r . (6.5.14) Lemma. For each j = 1, . . . , m − r we have p j ≥ n − r . Proof. Let us imagine that we construct the sequences t j (α) by the following process. We look at b j and start lowering it by 1 until we reach a number that is not equal to any ai or dK for k < j. Then every step of lowering by one accounts for some ai satisfying d j + 1 ≤ ai ≤ b j for some dK (which comes from a previous aK > b j ). This means that we have a set of b j − d j aK ’s which is disjoint from the set {i | ai < d j }. Therefore p j = n − (b j − d j ) − #{i | ai < d j } ≥ n − r. This concludes the proof of the lemma. (6.5.15) Theorem. Let K be a ﬁeld of characteristic zero. Let (α (1) , α (2) ) satisfy the condition of Proposition (6.5.13). Then the projective dimension

of M(α (1) , α (2) ) over A equals m−r j=1 p j . Proof. Let us decompose the terms of the complex F(L α(1) Q ⊗ L α(2) R)• using the Cauchy formula (2.3.3) and the Littlewood–Richardson rule (2.3.4). The GL(F) weights of all terms have the form (α (1) , δ (2) ) with δ (2) containing α (2) such that the difference between corresponding terms of δ (2) and α (2) does not exceed n. Also, all such weights occur. For all weights of this form we need to ﬁnd the supremum of the numbers |δ (2) | − |α (2) | − l(w) where w is a permutation ordering the weight (α (1) , δ (2) ) + ρ. It is clear tht the top number is obtained from the sequence tm−r (α). We need therefore to ﬁnd in which term F(L α(1) Q ⊗ L α(2) R)i the corresponding term occurs. The homo

geneous degree is m−r j=1 (n − (b j − d j )), and the length of the permutation w

m−r is j=1 #{i | ai < d j }. The statement of the theorem follows. (6.5.16) Corollary. Let K be a ﬁeld of characteristic zero. Assume that the weight (α (1) , α (2) ) satisﬁes the condition of Proposition (6.5.13). The depth of

(1) (2) M(α (1) , α (2) ) equals mn − m−r j=1 p j . The module M(α , α ) is a maximal Cohen–Macaulay module over Ar if and only if every number p j equals n − r.

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The Determinantal Varieties

Let us look more closely at the weights giving maximal Cohen–Macaulay modules over Ar . Let us look at the process of getting sequences t j (α) by lowering the numbers b j to d j . At each stage we modify the sequence (a1 , . . . , ar ) ( j) ( j) as follows. We deﬁne inductively the sequences (a1 , . . . , ar ) by setting (a1(0) , . . . , ar(0) ) = (a1 , . . . , ar ), ( j−1) ( j−1) ( j) ( j−1) (a1 , . . . , aˆ i , . . . ar , d j ) if b j = ai , ( j) (a1 , . . . , ar( j) ) = ( j−1) ( j−1) ) otherwise. (a1 , . . . ar Now p j = n − r for j = 1, . . . , m − r if and only if we have b j ≥ ( j−1) ( j−1) } for each j = 1, . . . , m − r . Indeed, each ai either inmax{a1 , . . . ar duces a number between b j and d j + 1 or is smaller than d j , so the overall number by which the projective dimension decreases at the j-th stage is n − r . Stating the last result, we have (6.5.17) Corollary. Let (α (1) , α (2) ) satisfy the condition of Proposition ( j) ( j) (6.5.13). Deﬁne the sets (a1 , . . . ar ) as above. Then M(α (1) , α (2) ) is a maximal Cohen–Macaulay Ar -module if and only if for every j = 1, . . . , m − r we have ( j−1)

b j ≥ max{a1

, . . . ar( j−1) }.

We conclude the discussion of modules M(α) by showing an example of a module of type M(α) whose Cohen–Macaulay property depends on the characteristic of the ﬁeld K. (6.5.18) Example. Let us set r = 2, m = 3, n = 4. Consider the module M((2, 0, 0)). It consists of sections of the sheaf M = S2 Q ⊗ Sym(Q ⊗ G ∗ ) over Grass(1, F). The module M((2, 0, 0)) is maximal Cohen–Macaulay over ﬁelds of characteristic = 2, but in characteristic 2 it is not maximal Cohen– Macaulay. Proof. The higher cohomology groups of M(2, 0, 0) vanish. This follows from the Cauchy formula (3.2.5), the Kempf vanishing theorem (4.1.10), and the fact that the tensor product of Schur functors has a ﬁltration with factors isomorphic to Schur modules. (In characteristic 0 one could just use Bott’s theorem (4.1.9).)

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209

A minimal free resolution of M(2, 0, 0) is given by the complex F((2, 0, 0))• = S2 Q ⊗

•

(R ⊗ G ∗ ).

In this case dim R = 1, so for each i, 0 ≤ i ≤ 4, we deal with the cohomology of S2 Q ⊗ Si R ⊗ i G ∗ . In characteristic 0 we get the resolution 0 → L 3,2 F ⊗

3

G ∗ ⊗ A(−3) → L 3,1 F ⊗

2

G ∗ ⊗ A(−2) → S2 F ⊗ A.

In fact one can see that this complex is acyclic over all ﬁelds K with char K = 2. If char K = 2, then the bundle S2 Q ⊗ S4 R has nonzero cohomology. In fact H 1 (Grass(1, F), S2 Q ⊗ S4 R) = H 2 (Grass(1, F), S2 Q ⊗ S4 R) = L 3,3 F. Therefore the resolution of M((2, 0, 0)) over a ﬁeld of characteristic 2 is 0 → L 3,3 F ⊗ ⊗

3

4

G ∗ ⊗ A(−4) → L 3,3 F ⊗

G ∗ ⊗ A(−3) → L 3,1 F ⊗

2

4

G ∗ ⊗ A(−4) ⊕ L 3,2 F

G ∗ ⊗ A(−2) → S2 F ⊗ A.

We see that M((2, 0, 0)) is a maximal Cohen–Macaulay module over Ar over ﬁelds of characteristic = 2, but in characteristic 2 its depth drops by one. 6.6. Modules Supported in Symmetric Determinantal Varieties We preserve the notation from section 6.3. To construct a family of modules supported in Yrs we use the ﬁrst incidence variety from section 6.3. Let us ﬁx n and r . We take Y = Yrs and V = Grass(n − r, E) with the tautological sequence 0 → R → E × V → Q → 0. Here dim R = n − r and dim Q = q = r . For a subspace R in E, we denote by i the embedding of R into E. We consider the variety Z 1 = {(φ, R) ∈ X × V | φ i = 0}. As usual, we have the diagram Z1 ⊂ ↓ q Y ⊂

X×V ↓q X

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The variety Z 1 is the total space of the bundle S1 = S2 Q∗ . The bundle T1 is deﬁned by the exact sequence 0 → S2 Q∗ → S2 E ∗ × V → T1 → 0. Let Crs (E) be the category of graded Ars : = K[Yrs ]-modules with rational GL(E)-action compatible with the module structure, and equivariant degree 0 maps. We denote K 0 (Ars ) the Grothendieck group of the category Crs (E). For an equivariant graded module M ∈ Ob(Crs (E)) and for q ∈ Z, we denote by M(q) the module M with gradation shifted by q, i.e. M(q)n = Mq+n . For M ∈ Ob(Crs (E)) we deﬁne the graded character of M, char(M) = char(Mn ) q n ∈ Rep(GL(E))[[q]][q −1 ], n∈Z

where Rep(GL(E)) denotes the representation ring of GL(E), and char denotes the character map described in (2.2.10). Let α = (α1 , . . . , αn ) be an integral weight for GL(E). We set α (1) = (α1 , . . . , αr ), α (2) = (αr +1 , . . . , αn ). Let α = (α (1) , α (2) ). Assume that both α (1) and α (2) are dominant. We deﬁne a sheaf Ms (α) = p ∗ (L α(1) Q ⊗ L α(2) R) ⊗ O Z 1 of graded modules on Z 1 . We deﬁne the equivariant graded Ars -modules M s (α) = H 0 (Z 1 , Ms (α)). (6.6.1) Example. The simplest examples of modules M s (α) are provided by the pushdowns of line bundles on Grass(n − r, E). The line bundles on Grass(n − r, E) correspond to tensor powers OGrass(n−r,E) (m) = ( r Q)⊗m for m ∈ Z. Notice that the formula (5.1.4) for the dual bundle implies that the canonical module K Ars = M((r + 1)r , 0n−r ) ⊗ n F ⊗−r −1 . For m ≥ 0 the corresponding modules occur as a subset of a family from the next example. (6.6.2) Example. Let α (2) = (0). The sheaf Ms (α) equals L α(1) Q ⊗ O Z 1 . Therefore the direct image p∗ Ms (α) equals L α(1) Q ⊗ Sym(S2 Q). The higher cohomology of Ms (α) vanishes by Bott’s theorem (4.1.9). The j-th

6.6. Modules Supported in Symmetric Determinantal Varieties

211

homogeneous component M sj (α) decomposes as follows: (L α(1) E ⊗ L µ E)≤r , |µ|= j, µi even for all i,µ1 ≤r

where (L α(1) E ⊗ L µ E)≤r denotes a factor of (L α(1) E ⊗ L µ E) consisting of all L ν E in the tensor product with ν1 ≤ r . The main results of this section are the analogues of the results from the previous section. (6.6.3) Theorem. The Grothendieck group K 0 (Ars ) is generated by the classes of the modules M s (α)(q), where α is a dominant weight and q ∈ Z. (6.6.4) Theorem. (a) The Grothendieck group K 0 (Ars ) is isomorphic to the additive subgroup of the ring Rep(GL(E))[[q]][q −1 ] generated by shifted graded characters of modules M s (α) for α dominant. (b) The group K 0 (Ars ) is isomorphic to the additive group of the ring Rep(GL(E))[q][q −1 ]. The proof of Theorems (6.6.3) and (6.6.4) follows the same steps as the proof of Theorems (6.5.3) and (6.5.4). We only give the statements, leaving the detais as an exercise to the reader. (6.6.5) Proposition. Let α be a dominant weight for GL(E). Then H i (Z 1 , Ms (α)) = 0 for i > 0. Proof. This is an analogue of (6.5.5). Let α be a weight with α (1) , α (2) dominant. We deﬁne the number l(α) as follows. Consider the weight α + ρ, where ρ = (n − 1, n − 2, . . . , 1, 0) = (u 1 , . . . , u n ). Deﬁne, by reverse induction on s (from s = r to s = 1), the numbers δs = min{t | t ≥ δs+1 , t +n−s ∈ / {δs+1 + n − s − 1, . . . , δr + n − r, αr +1 + n − r − 1, . . . , αn }}. By construction the weight (δ1 , . . . , δr , αr +1 , . . . , αn ) + ρ is not orthogonal to any root. By (4.1.9) there exists a unique l such that H l (Grass (n − r, E), K δ Q ⊗ K α(2) R) = 0. We deﬁne l(α) := l.

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(6.6.6) Proposition. Let K be a ﬁeld of characteristic zero. Let α = (α (1) , α (2) ) be a weight for GL(F) with α (1) , α (2) dominant. Let l(α) be deﬁned as above. Then H i (Z 1 , Ms (α)) = 0 for i > l(α). Proof. This is an analogue of the ﬁrst statement of (6.5.6). Notice that the analogue of the second statement need not be true. Each equivariant sheaf Ms on Z 1 has its Euler characteristic class (−1)i [H i (Z 1 , Ms )] ∈ K 0 (Ars ). χ (Ms ) = i≥0

(6.6.7) Proposition. The group K 0 (Ars ) is generated by the Euler characteristic classes χ(Ms (α)) for (α (1) , α (2) ) dominant. Proof. This is an analogue of (6.5.8). For the proof one has to use the same type of induction, using the ﬂag variety Flag(n − r, n − r + 1; E). (6.6.8) Remarks. Notice that, as in the previous section, we can also show ˆ s (α (1) , α (2) ) that if (α (1) , α (2) ) is dominant, then the Euler characteristic of M s is in the subgroup of K 0 (Ar ) spanned by the Euler characteristics of sheaves Ms (α, β) with α dominant. Theorem (6.6.3) follows from the following statement: (6.6.9) Proposition. The classes χ (Ms (α (1) , α (2) ) such that (α (1) , α (2) ) is dominant generate the group K 0 (Ars ). Proof. The proof is analogous to (6.5.10). We proceed by induction on s := s(α (1) , α (2) ) = α1(2) − αr(1) . Proof of Theorem (6.6.4). Part (a) is a consequence of the fact that the characters of M s (α)(−i) are linearly independent in Rep(GL(E))[[q]][q −1 ]. To prove (b) we deﬁne the homomorphism of abelian groups : Rep(GL(E))[q][q −1 ] → K 0 (Ars ) by sending the class [(L α E)q i ] to [M s (α)(−i)]. By Theorem (6.6.3) the homomorphism is an epimorphism. It is also a monomorphism, because the characters of modules M s (α)(−i) are linearly independent in Rep(GL(E)) [[q]][q −1 ] and therefore in K 0 (Ars ).

6.7. Modules Supported in Skew Symmetric Determinantal Varieties 213

We ﬁnish this section with a criterion for a module M s (α) to be maximal Cohen–Macaulay. We use Corollary (5.1.5). Before we start, let us work out the duality statement for the variety Z 1 . To this end we need to calculate the bundle m ∗ ξ ⊗ K , where m = rank ξ . Using (3.3.5), we get K V = ( r Q)−n+r n−r Vr m ∗ r (r +1)/2 n(n+1)/2 ⊗( R) . Similarly, ξ = S2 Q ⊗ S2 F ∗ . Putting these facts together, we conclude that the dualizing bundle is −n+r +1 r ∗ n . ω Z 1 = K (−n−1+r ) F ⊗ Q Therefore, for the weight (2) ), α = (α1(1) , . . . , αr(1) , α1(2) , αn−r

the dual weight is given by the formula (2) , . . . , −α1(2) ). α ∨ = (−n + r + 1 − αr(1) , . . . − n + r + 1 − α1(1) , −αn−r

(6.6.10) Proposition. The module M s (α) is maximal Cohen–Macaulay provided that l(α) = l(α ∨ ) = 0. Proof. Let’s denote V(α) = K α(1) Q ⊗ K α(2) R. One applies Corrolary (5.1.5) to the complexes F(V(α))• and F(V(α ∨ ))• . The vanishing of the higher cohomology is assured by Proposition (6.6.6). (6.6.11) Example. Let us assume that α (2) = 0. Then M s (α) has nonvanishing higher cohomology by Proposition (6.6.6). The condition l(α ∨ ) = 0 is true when the inequality −n + r + 1 − α1(1) ≥ −n + r is satisﬁed, i.e. when α1(1) ≤ 1. This proves that the modules u s u 0 M (1 ) = H Grass(n − r, F), Q ⊗ Sym(S2 Q) are maximal Cohen–Macaulay for 0 ≤ u ≤ r .

6.7. Modules Supported in Skew Symmetric Determinantal Varieties We preserve the notation of section 6.4. To construct a family of modules supported in Yra we use the ﬁrst incidence variety from section 6.4.

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Let us ﬁx n and r , with r even. We take Y = Yra and V = Grass(n − r, E) with the tautological sequence 0 → R → E × V → Q → 0. Here dim R = n − r and dim Q = q = r . For a subspace R in E we denote by i the embedding of R into E. We consider the variety Z 1 = { (φ, R) ∈ X × V | φ i = 0 }. Let Cra (E) be the category of graded Ara : = K[Yra ]-modules with rational GL(E)-action compatible with the module structure, and equivariant degree 0 maps. We denote by K 0 (Ara ) the Grothendieck group of the category Cra (E). For an equivariant graded module M ∈ Ob(Cra (E)) and for q ∈ Z we denote by M(q) the module M with gradation shifted by q, i.e. M(q)n = Mq+n . For M ∈ Ob(Cra (E)) we deﬁne the graded character of M, char(M) = char(Mn ) q n ∈ Rep(GL(E))[[q]][q −1 ], n∈Z

where Rep(GL(E)) denotes the representation ring of GL(E) and char denotes the character map deﬁned in (2.2.10). Let α = (α1 , . . . , αn ) be an integral weight for GL(E). We set α (1) = (α1 , . . . , αr ), α (2) = (αr +1 , . . . , αn ). Let α = (α (1) , α (2) ). Assume that both α (1) and α (2) are dominant. We deﬁne a sheaf Ma (α) = p ∗ (L α(1) Q ⊗ L α(2) R) ⊗ O Z 1 of graded modules on Z 1 . We deﬁne the equivariant graded Ara -modules M a (α) = H 0 (Z 1 , Ma (α)). (6.7.1) Example. The simplest examples of modules M a (α) are provided by the pushdowns of line bundles on Grass(n − r, E). The line bundles on Grass(n − r, E) correspond to tensor powers OGrass(n−r,E) (m) = ( r Q)⊗m for m ∈ Z. For m ≥ 0 the corresponding modules occur as a subset of a family from the next example. (6.7.2) Example. Let α (2) = (0). The sheaf Ma (α) equals L α(1) Q ⊗ O Z 1 . Therefore the direct image p∗ Ma (α) equals L α(1) Q ⊗ Sym(S2 Q). The higher cohomology of Ma (α) vanishes by Bott’s theorem (4.1.9). The j-th

6.7. Modules Supported in Skew Symmetric Determinantal Varieties 215

homogeneous component M aj (α) decomposes as follows: (L α(1) E ⊗ L µ E)≤r , |µ|= j, µi even for all i, µ1 ≤r

where (L α(1) E ⊗ L µ E)≤r denotes a factor of (L α(1) E ⊗ L µ E) consisting of all L ν E in the tensor product with ν1 ≤ r . (6.7.3) Theorem. The Grothendieck group K 0 (Ara ) is generated by the classes of the modules M a (α)(q), where α is a dominant weight and q ∈ Z. (6.7.4) Theorem. (a) The Grothendieck group K 0 (Ara ) is isomorphic to the additive subgroup of the ring Rep(GL(E))[[q]][q −1 ] generated by shifted graded characters of modules M a (α) for α dominant. (b) The group K 0 (Ara ) is isomorphic to the additive group of the ring Rep(GL(E))[q][q −1 ]. Again the proofs of (6.7.3) and (6.7.4) follow the same steps as the proofs of (6.5.3) and (6.5.4). We just give the statements, leaving the proofs to the reader. (6.7.5) Proposition. Let α be a dominant weight for GL(E). Then H i (Z 1 , Ma (α)) = 0 for i > 0. Proof. This is the analogue of (6.5.5). Let α be a weight with α (1) , α (2) dominant. We deﬁne the number l(α) as follows. Consider the weight α + ρ, where ρ = (n − 1, n − 2, . . . , 1, 0) = (u 1 , . . . , u n ). Deﬁne, by reverse induction on s (from s = r to s = 1) the numbers δs = min{t | t ≥ δs+1 , t +n−s ∈ / {δs+1 + n − s − 1, . . . , δr + n − r, αr +1 + n − r − 1, . . . , αn }}. By construction the weight (δ1 , . . . , δr , αr +1 , . . . , αn ) + ρ is not orthogonal to any root. By (4.1.9) there exists a unique l such that H l (Grass(n − r, E), K δ Q ⊗ K α(2) R) = 0. We deﬁne l(α) := l. (6.7.6) Proposition. Let K be a ﬁeld of characteristic zero. Let α = (α (1) , α (2) ) be a weight for GL(F) with α (1) , α (2) dominant. Let l(α) be deﬁned as above.

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Then H i (Z 1 , Ma (α)) = 0

for i > l(α).

Proof. This is the analogue of the ﬁrst statement of (6.5.6). Notice that the analogue of the second statement need not be true. Each equivariant sheaf Ma on Z 1 has its Euler characteristic class (−1)i [H i (Z 1 , Ma )] ∈ K 0 (Ara ). χ (Ma ) = i≥0

(6.7.7) Proposition. The group K 0 (Ara ) is generated by the Euler characteristic classes χ(Ma (α)) for (α (1) , α (2) dominant). Proof. This is the analogue of (6.5.8). We use a similar induction, using the ﬂag variety Flag(n − r, n − r + 2; E). (6.7.8) Remarks. Notice that, as in section 6.5, we can also show that if ˆ a (α (1) , α (2) ) is (α (1) , α (2) ) is dominant, then the Euler characteristic of M a in the subgroup of K 0 (Ar ) spanned by the Euler characteristics of sheaves Ma (α, β) with α dominant. Theorem (6.7.3) follows from the following statement (6.7.9) Proposition. The classes χ (Ma (α (1) , α (2) ) such that (α (1) , α (2) ) is dominant generate the group K 0 (Ara ). Proof. This is the analogue of (6.5.10). Again we induct on s := s(α (1) , α (2) ) = α1(2) − αr(1) . Proof of Theorem (6.7.4). Part (a) is a consequence of the fact that the characters of M a (α)(−i) are linearly independent in Rep(GL(E))[[q]][q −1 ]. To prove (b) we deﬁne the homomorphism of abelian groups : Rep(GL(E))[q][q −1 ] → K 0 (Ara ) by sending the class [(L α E)q i ] to [M a (α)(−i)]. By Theorem (6.7.3) the homomorphism is an epimorphism. It is also a monomorphism, because

6.7. Modules Supported in Skew Symmetric Determinantal Varieties 217

the characters of modules M a (α)(−i) are linearly independent in Rep(GL(E)) [[q]][q −1 ] and therefore in K 0 (Ara ). Next we give a criterion for a module M a (α) to be maximal Cohen– Macaulay. We use Corollary (5.1.5). Before we start, let us work out the duality statement for the variety Z 1 . To this end we need to calculate the bundle m ∗ ξ ⊗ K where m = rank ξ . Using (3.3.5), we get K V = ( r Q)−n+r n−r Vr m ∗ r (r −1)/2 2 n(n−1)/2 2 ∗ ⊗( R) . Similarly, ξ = S Q⊗ F . Putting these facts together, we conclude that the dualizing bundle is ωZ1

−n+r −1 r = K (−n+1+r )n F ⊗ . Q ∗

Therefore for the weight (2) α = (α1(1) , . . . , αr(1) , α1(2) , αn−r ),

the dual weight is given by the formula (2) , . . . , −α1(2) ). α ∨ = (−n + r − 1 − αr(1) , . . . − n + r − 1 − α1(1) , −αn−r

(6.7.10) Proposition. The module M a (α) is maximal Cohen–Macaulay provided that l(α) = l(α ∨ ) = 0. Proof. Let’s denote V(α) = K α(1) Q ⊗ K α(2) R. One applies Corrolary (5.1.5) to the complexes F• (V(α)) and F• (V(α ∨ )). The vanishing of the higher cohomology is assured by Proposition (6.7.6). (6.7.11) Example. Let us assume that α (2) = 0. Then M a (α) has nonvanishing higher cohomology by Proposition (6.6.6). Let us also assume α1(1) ≤ 1. This means we consider the modules v 2 Q ⊗ Sym Q . M a (1v ) = H 0 Grass(n − r, F), Then condition l(α ∨ ) = 0 is equivalent to v ≥ 3 (which implies r = 2u ≥ 4). This proves that the modules M a (1v ) are maximal Cohen–Macaulay for 3 ≤ v ≤ r.

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Exercises for Chapter 6 Analogues of Determinantal Varieties for Other Classical Groups The Symplectic Group Let F be a symplectic space of dimension 2n, let G be a vector space, dim G = m. Consider X = HomK (F, G). This space can be identiﬁed with the set of m-tuples of vectors from F. 1.

For 1 ≤ r ≤ n deﬁne Yr = {φ ∈ X | ∃R ∈ IGrass(r, F), φ(R) = 0 }. Prove that Yr has a resolution of singularities which is a total space of a vector bundle over IGrass(r, F)

2.

Prove, by using Theorem (5.1.2)(b), that in that case Yn is normal and has rational singularities.

3.

Let m ≤ n. Calculate the complex F• , and use it to show m that the variety Yn is a complete intersection given by the vanishing of 2 Sp(F)-invariants of degree 2, given by the representation 2 G ∗ .

4.

Use exercises 2 and 3 to show that there exists an acyclic complex K (n, λ; F)• resolving the irreducible representation Vλ F of the group Sp(F) in terms of Schur functors, with the i-th term K λ/µ F. K (n, λ; F)i := µ∈Q −1 (2i)

5.

Let r = n, dim G = m > n. Calculate the terms of the complex F• . Prove in a determinantal variety that the variety Yn is a complete intersection m of matrices of rank ≤ n, given by 2 Sp(F)-invariants.

6.

Let r < n. Prove that Yr is normal and has rational singularities. Assume that m ≤ 2n − r + 1. Then the terms of F• contain only trivial Sp(F)representations. More precisely, F• is a specialization of the resolution of 2(n − r + 1) Pfafﬁans of the generic skew symmetric m × m matrix (described in section 6.3), where our m × m skew symmetric matrix is a matrix of Sp(F)-invariants in A = Sym(F ⊗ G ∗ ), given by the repre sentation 2 G ∗ ⊂ A2 .

7. For the example 2n = 6, r = 2, m = 6 calculate the terms of the complex F• and prove that Yr is not a complete intersection in a determinantal variety.

Exercises for Chapter 6

219

8. Prove that the deﬁning ideal of Yr is generated by 2(n − r + 1) Pfafﬁans of the skew symmetric m × m matrix of Sp(F)-invariants in A = Sym(F ⊗ G ∗ ), given by the representation 2 G ∗ ⊂ A2 . 9. Let 1 ≤ r < n. Deﬁne the variety Y2n−r = {φ ∈ X | ∃R ∈ IGrass(r, F), φ(R ∨ ) = 0}. Prove that Y2n−r has a resolution of singularities which is a total space of a vector bundle over IGrass(r, F). Prove that Y2n−r is normal , with rational singularities. 10. Prove that the deﬁning ideal of Y2n−r is generated by Sp(F)-invariants in A2 (given by the representation 2 G ∗ ) and the (r + 1) × (r + 1) minors of the matrix φ. The Orthogonal Group We will formulate the exercises for the even orthogonal group. The formulations for the odd orthogonal group are left to the reader. Let F be an orthogonal space of dimension 2n; let G be a vector space, dim G = m. Consider X = HomK (F, G). This space can be identiﬁed with the set of m-tuples of vectors from F. 11. For 1 ≤ r ≤ n let Yr = {φ ∈ X | ∃R ∈ IGrass(r, F), φ(R) = 0 }. Prove that Yr has a resolution of singularities which is a total space of a vector bundle over IGrass(r, F). (Hint: ξ = R ⊗ G ∗ .) 12. Prove, by using Theorem (5.1.2)(b), that in that case Yn is normal and has rational singularities. 13. Let m ≤ n. Calculate the complex F• to show that the variety Yn is a complete intersection given by the vanishing of m+1 O(F)-invariants 2 ∗ of degree 2, given by the representation S2 G . 14. Let λ be a dominant weight for SO(F) with integer coordinates. Use exercises 12 and 13 to show that there exists an acyclic complex K (n, λ; F)• resolving the irreducible representation Vλ F of the group SO(F) in terms of Schur functors, with the i-th term K λ/µ F. K (n, λ; F)i := µ∈Q 1 (2i)

15. Let r = n, dim G = m > n. Calculate the terms of the complex F• . Prove that the variety Yn is a complete intersection in a determinantal variety SO(F)-invariants. of matrices of rank ≤ n, given by n+1 2

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16. Let r < n. Prove that Yr is not normal, but that its normalization has rational singularities. Assume that m ≤ 2n − r . Then the terms of F• contain only trivial SO(F)-representations. 17. For the example 2n = 8, r = 2, m = 6 calculate the terms of the complex F• . 18. Prove that the deﬁning ideal of Yr is generated by (n − r + 1) × (n − r + 1) minors of the symmetric m × m matrix of SO(F)-invariants in A = Sym(F ⊗ G ∗ ), given by the representation S2 G ∗ ⊂ A2 . 19. Let 1 ≤ r ≤ n. Deﬁne the variety Y2n−r = { φ ∈ X | ∃R ∈ IGrass(r, F), φ(R ∨ ) = 0 }. Prove that Y2n−r has a resolution of singularities which is a total space of a vector bundle over IGrass(r, F). (Hint: ξ = R∨ ⊗ G ∗ .) Prove that Y2n−r is normal, with rational singularities. 20. Prove that the deﬁning ideal of Y2n−r is generated by O(F)-invariants in A2 (given by the representation S2 G ∗ ) and the (r + 1) × (r + 1) minors of the matrix φ. The First Fundamental Theorem for the General Linear Group Let E be a vector space of dimension n. Let X = E ⊗m ⊕ E ∗⊗ p . We identify X with HomK (G, E) ⊕ HomK (E, H ) where G, H are two vector spaces, dim G = m, dim H = p. We have A = Sym(G ⊗ E ∗ ⊕ E ⊗ H ∗ ). We can identify X with the set of m-tuples of vectors from E and p-tuples of covectors from E ∗ . 21. For each pair (r, s), r + s = n, r ≤ m, s ≤ p, consider the variety Yr,s = {(φ, ψ) ∈ X | ψφ = 0, rank φ ≤ r, rank ψ ≤ s }. Prove that Yr,s has a desingularization Z r,s with V = Grass(r, E), ξ = G ⊗ Q∗ ⊕ R ⊗ H ∗ . Prove that Yr,s is normal and has rational singularities. 22. Choose m + p = n, r = m, s = p. Prove that in this case the complex F• is a Koszul complex on the GL(E)-invariants in A of bidegree (1, 1) which correspond to the representation G ⊗ H ∗ . 23. Let λ = (λ1 , . . . , λr ), µ = (µ1 , . . . , µs ) be two partitions. Take the isotypic component of type K λ G ⊗ K µ H ∗ to obtain the complex K (r, s, λ, µ; E)• with the following properties: (a) K (r, s, λ, µ; E)• is acyclic.

Exercises for Chapter 6

221

(b) The i-th term of K (r, s, λ, µ; E)• is K (r, s, λ, µ; E)i = K λ/ν E ∗ ⊗ K µ/ν E. |ν|=i

(c) The complex K (r, s, λ, µ; E)• resolves the representation K (µ,−λ) E, where (µ, −λ) = (µ1 , . . . , µs , −λr , . . . , −λ1 ). Isotropic Grassmannians Revisited 24. Prove that the equations of isotropic Grassmannians deﬁned in exercises 1, 2, 3 in chapter 4 generate (together with the Pl¨ucker relations) the deﬁning ideals of the cones over isotropic Grassmannians. Differentials in the Resolutions of Ideals of Minors of a Generic Matrix 25. We work with the notation of section 6.1. Recall that by (6.1.3) the i-th term in the Lascoux complex equals Fi = L P1 (α,β) F ⊗ L P2 (α,β) G ∗ ⊗K A. s≥0

(α,β)∈Q(s), i=s 2 +|α|+|β|

Denote Fi(s) =

(α,β)∈Q(s),

L P1 (α,β) F ⊗ L P2 (α,β) G ∗ ⊗K A,

i=s 2 +|α|+|β|

so Fi = s≥0 Fi(s) . Prove that the differential di : Fi → Fi−1 has only (s) and components of degree components of degree 1 taking Fi(s) to Fi−1 (s) (s−1) r + 1 taking Fi to Fi−1 . 26. Prove that the only possible nonzero components of the degree 1 part of (s) restricted to L P1 (α,β) F ⊗ L P2 (α,β) G ∗ the differential di1,s : Fi(s) → Fi−1 ⊗K A go to the terms L P1 (γ ,δ) F ⊗ L P2 (γ ,δ) G ∗ ⊗K A with γ ⊂ α, δ ⊂ β, |α/γ | = 1, |β/δ| = 1. 27. Prove that the only possible nonzero components of the degree r + 1 part (s−1) of the differential dir +1,s : Fi(s) → Fi−1 are as follows. The map dir +1,s restricted to the term L P1 (α,β) F ⊗ L P2 (α,β) G ∗ ⊗K A is zero unless αs = 0, α1 ≤ s − 1, βs = 0, β1 ≤ s − 1. If these conditions are satisﬁed, the only nonzero component of d r +1,s restricted to L P1 (α,β) F ⊗ L P2 (α,β) G ∗ ⊗K A (s−1) , where γ = (α1 + 1, . . . , goes to L P1 (γ ,δ) F ⊗ L P2 (γ ,δ) G ∗ ⊗K A from Fi−1 αs−1 + 1), δ = (β1 + 1, . . . , βs−1 + 1). The coefﬁcients of that component are the linear combinations of (r + 1) × (r + 1) minors of .

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Differentials in the Resolutions of Ideals of Minors of a Generic Symmetric Matrix 28. We work with the notation of section 6.3. By Theorem (6.3.1) the i-th term G i of the resolution of the ideal of (r + 1) × (r + 1) minors of a generic symmetric matrix is given by Gi = L λ(α,u) E ⊗K As . u≥0

λ(α,u); i=|α|+2u 2 −u

Denote

G i(u) = λ(α,u);

L λ(α,u) E ⊗K As ,

i=|α|+2u 2 −u

so G i = u≥0 G i(u) . Prove that the differential di : G i → G i−1 has only (u) components of degree 1 taking G i(u) to G i−1 and components of degree (u) (u−1) r + 1 taking G i to G i−1 . 29. Prove that the only possible nonzero components of the degree 1 part of (u) the differential di1,u : G i(u) → G i−1 restricted to L λ(α,u) ⊗K As go to the s terms L λ(β,u) ⊗K A with β ⊂ α, |α/β| = 1. 30. Prove that the only possible nonzero components of the degree r + 1 part (u−1) of the differential dir +1,u : G i(u) → G i−1 are as follows. The map dir +1,u s restricted to the term L λ(α,u) ⊗K A is zero unless α2u−1 = α2u = 0, α1 ≤ 2u − 1. If these conditions are satisﬁed, the only nonzero component of d r +1,u restricted to L λ(α,u) ⊗K As goes to L λ(β,u−1) ⊗K As , where β = (α1 + 2, . . . , α2u−2 + 2). The coefﬁcients of that component are the linear combinations of (r + 1) × (r + 1) minors of .

Differentials in the Resolutions of Ideals of Pfafﬁans of a Generic Skew Symmetric Matrix 31. We work with the notation of section 6.4. By Theoerm (6.4.1) the i-th term G i of the resolution of the ideal of (2r + 2) × (2r + 2) Pfafﬁans of a generic skew symmetric matrix is given by Gi = L λ(α,v) E ⊗K Aa . v≥0

Denote G i(v) =

λ(α,v); i=|α|+ 12 v(v+1)

λ(α,v); i=|α|+ 12 v(v+1)

L λ(α,v) E ⊗K Aa ,

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223

so G i = v≥0 G i(v) . Prove that the differential di : G i → G i−1 has only (v) components of degree 1 taking G i(v) to G i−1 and components of degree (v) (v−1) r + 1 taking G i to G i−1 . 32. Prove that the only possible nonzero components of the degree 1 part of (v) the differential di1,v : G i(v) → G i−1 restricted to L λ(α,v) ⊗K Aa go to the a terms L λ(β,v) ⊗K A with β ⊂ α, |α/β| = 1. 33. Prove that the only possible nonzero components of the degree r + 1 (v−1) part of the differential dir +1,v : G i(v) → G i−1 are as follows. The map r +1,v a restricted to the term L λ(α,v) ⊗K A is zero unless αv = 0, α1 ≤ di v − 1. If these conditions are satisﬁed, the only nonzero component of d r +1,v restricted to L λ(α,v) ⊗K Aa goes to L λ(β,v−1) ⊗K Aa , where β = (α1 + 1, . . . , αv−1 + 1). The coefﬁcients of that component are the linear combinations of (2r + 2) × (2r + 2) Pfafﬁans of . Maximal Cohen–Macaulay Modules with Linear Resolutions 34. Consider the twisted sheaf M((n − r )r , (0)) = K (n−r )r Q ⊗ O Z r(1) deﬁned in section 6.5. (a) Prove that the sheaf M((n − r )r , (0)) has no higher cohomology, so the twisted complex F(K (n−r )r Q)• provides a minimal resolution of M((n − r )r , (0)). (b) Show that the complex F(K (n−r )r Q)• has length (m − r )(n − r ) and that it has a linear differential. More precisely, the only nonvanishing terms of the complex F(K (n−r )r Q)• are i 0 F(K (n−r )r Q)i = H Grass(m − r, F), K (n−r )r Q ⊗ ξ for 0 ≤ i ≤ (m − r )(n − r ). (c) Use the duality from exercise 18, chapter 2, changing the Schur functors on F to the Schur functors on F ∗ , to identify the complex F(K (n−r )r Q)• with the Schur complex L (m−r )(n−r ) (∗ ). (d) Use Buchsbaum–Eisenbud acyclicity criterion (1.2.12) to prove that over a ﬁeld K of arbitrary characteristic the Schur complex L (m−r )(n−r ) (∗ ) is acyclic. Note that the Euler characteristic of this complex in the relative situation (i.e. when F, G are replaced by vector bundles over some scheme) gives the class occurring in the Porteous formula for the cohomology class of the degeneracy locus X r . The module M((n − r )r , (0)) is a maximal Cohen–Macaulay module supported in X r with a linear resolution.

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35. Consider the twisted sheaf M((n − r )r , n − r − 1, n − r − 2, . . . , 1, 0) deﬁned in section 6.6. It is supported in the symmetric determinantal variety X rs of symmetric n × n matrices of rank ≤ r . (a) Prove that the sheaf M((n − r )r , n − r − 1, n − r − 2, . . . , 1, 0) has no higher cohomology, so the twisted complex F(K (n−r )r Q ⊗ K (n−r −1,n−r −2,...,1,0) R)• provides a minimal resolution of M((n − r )r , n − r − 1, n − r − 2, . . . , 1, 0). (b) Show that the complex F(K (n−r )r Q ⊗ K (n−r −1,n−r −2,...,1,0) R)• has length n−r2+1 and that it has a linear differential. More precisely, the only nonvanishing terms of the complex F(K (n−r )r Q ⊗ K (n−r −1,n−r −2,...,1,0) R)• are i 0 H Grass(n − r, F), K (n−r )r Q ⊗ K (n−r −1,n−r −2,...,1,0) R ⊗ ξ . for 0 ≤ i ≤ n−r2+1 . Conclude that M((n − r )r , n − r − 1, n − r − 2, . . . , 1, 0) is a maximal Cohen–Macaulay module supported in X rs with a linear resolution. 36. Consider the twisted sheaf M((n − r − 1)r , n − r − 1, n − r − 2, . . . , 1, 0) deﬁned in section 6.7. It is supported in the skew symmetric determinantal variety X ra of skew symmetric n × n matrices of rank ≤ r . Here we assume that r = 2u is even. (a) Prove that the sheaf M((n − r − 1)r , n − r − 1, n − r − 2, . . . , 1, 0) has no higher cohomology, so the twisted complex F(K (n−r −1)r Q ⊗ K (n−r −1,n−r −2,...,1,0) R)• provides a minimal resolution of M((n − r − 1)r , n − r − 1, n − r − 2, . . . , 1, 0). (b) Show that the complex F(K (n−r −1)r Q ⊗ K (n−r −1,n−r −2,...,1,0) R)• has length n−r and that it has a linear differential. More precisely, 2 the only nonvanishing terms of the complex F(K (n−r −1)r Q ⊗ K (n−r −1,n−r −2,...,1,0) R)• are i H 0 Grass(n − r, F), K (n−r −1)r Q ⊗ K (n−r −1,n−r −2,...,1,0) R) ⊗ ξ for 0 ≤ i ≤ n−r . Conclude that M((n − r − 1)r , n − r − 1, n − r − 2 2, . . . , 1, 0) is a maximal Cohen–Macaulay module supported in X ra with a linear resolution. Resolutions of K λ (Φ) 37. Consider the Grassmannian Grass(m − n, F). Consider the incidence variety Z m = {(φ, R) ∈ X × Grass(m − r, F) | φ | R = 0}.

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225

Denote by ˆ → F × Grass(m − n, F) → Qˆ → 0 0→R the tautological sequence on Grass(m − n, F). Deﬁne X (λ) := K λ Qˆ ⊗ O Z m The corresponding modules of sections are X (λ) := H 0 (Z m , K λ Qˆ ⊗ O Z m ). (a) Prove that H i (Z m , X (λ)) = 0 for i > 0, (b) Prove that p∗ X (λ) := K λ Qˆ ⊗ Sym(Qˆ ⊗ G ∗ ) with Ri p∗ X (λ) = 0 for i > 0. ˆ • gives a minimal free resolution of the (c) Conclude that F(K λ (Q)) module X (λ). (d) Assume that the partition λ has i boxes on the diagonal, i.e. λi+1 ≤ i ≤ λi . Let us also assume that λ has exactly s nonzero parts. Show that the module X (λ) has a minimal presentation K (λ,1n+1−s ) F ⊗

n+1−s

G ∗ ⊗K A(−n − 1 + s)

→ K λ F ⊗K A → X (λ) → 0. (e) Show that the projective dimension of X (λ) is equal to (m − n)i. ˆ • can be aug(f) Assume that i = 1. Show that the complex F(K λ Q) ∗ mented by one more map K λ F ⊗K A → K λ G ⊗K A(|λ|) to get a longer minimal free resolution. In other words, the module X (λ) turns out to be the ﬁrst syzygy of the module C(λ) := Coker(K λ ()). The modules C(λ) are therefore perfect modules supported in In (φ). 38. Let i = 1. Write λ = (q, 1 p−1 ). (a) Use the formula (2.32)(b) and exercise 5 of chapter 4 to prove that the argument from exercise 37 can be made characteristic free. Writing λ = (q, 1 p−1 ), the terms at the right end of the characteristic free version of the resolution of C(λ) (deﬁned as the cokernel of L ( p,1q−1 ) (φ)) are as follows: F0 = L ( p,1q−1 ) G ⊗K A( p + q − 1), F1 = L ( p,1q−1 ) F ⊗K A, Fi = L (i+n−1,1q−1 ) F ⊗ (L (n− p+1,1i−2 ) G)∗ ⊗K A(−n + p + i − 1) for 2 ≤ i ≤ m − n + 1,

226

The Determinantal Varieties

(b) Use exercise 17 of chapter 2 to describe the differential in the resolution of C( p, 1q−1 ) explicitly. Prove that C( p, 1q−1 ) is a perfect module.This resolution was ﬁrst described in [BE2]. The free resolutions of the modules C(λ) for partitions with more boxes on the diagonal were analyzed in [Ar1] and [Ar2]. None of these modules are perfect. 39. Let λ = (k n ). (a) Prove that the module X (λ) ⊗K ( n G ∗ )⊗k is isomorphic (as an equivariant GL(F) × GL(G ∗ )-module) to the k-th power of the ideal In (φ) of maximal minors of φ. The projective dimension of In (φ)k equals (m − n) min(k, n). ˆ • are characteristic free and linear (b) Prove that the complexes F(L λ Q) when there are no terms coming from higher cohomology, i.e. when λi+1 = . . . = λn = i. Resolutions of Powers of the Ideal of 2t × 2t Pfafﬁans of a (2t + 1) × (2t + 1) Skew Symmetric Matrix 40. Take dim E = 2t + 1, V = Grass(1, E) with the tautological sequence ˆ → E × V → Qˆ → 0. 0→R Deﬁne Z 2n = {(φ, R) ∈ X × V | φ |R = 0}, ˆ into E × V . Denote by p, q, q the where i denotes the embedding of R usual projections. Then Xλa = K λ Qˆ ⊗ O Z 2n on Z 2n , where λ = (λ1 , . . . , λ2t ) is a partition into at most 2t parts. Denote X a (λ) = H 0 (Z 2t , X a (λ)). (a) H i (Z 2t , X a (λ)) = 0 for i > 0. ˆ with Ri p∗ X a (λ) = 0 for i > 0. (b) p∗ X a (λ) := K λ Qˆ ⊗ Sym( 2 Q) ˆ • gives a minimal free resolution of the module (c) The complex F(K λ (Q) a X (λ). (d) Let λ = (k 2t ). Show that the module X a (λ) = I2ta (φ)k has a linear free resolution for k even and has one dimensional representation outside the linear strand for k odd, k < 2t + 1. (e) Show k for k even, 2t pd A (X (k )) = k + 1 for k odd.

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227

More precisely, the terms in the resolution are Fi = L (k 2t ,i) F ⊗K A(−i) for 0 ≤ i ≤ min(k, 2t), with the additional term k + 1 + 2t Fk+1 = L ((k+1)2t+1 ) F ⊗K A − 2 occurring for k odd, k < 2t. (f) Show, using Kempf’s vanishing theorem and exercise 5 of chapter 4, that the Betti numbers of powers of I2ta (φ)k do not depend on the characteristic of K. The differentials in these resolutions were described explicitly in [BS] and [KU].

7 Higher Rank Varieties

In this chapter we investigate the higher rank varieties. They are the analogues of determinantal varieties for more complicated representations L λ E. They were ﬁrst considered in the paper [Po] of Porras. In section 7.1 we look at the general case. We prove that higher determinantal varieties have rational singularities, and we ﬁnd equations deﬁning them set-theoretically. We also classify the rank varieties whose deﬁning ideals are Gorenstein. In section 7.2 we investigate the rank varieties for symmetric tensors of degree bigger than two. We prove that in this case the deﬁning equations described in section 7.1 generate the radical ideal. We also analyze the cases of tensors of rank one, which correspond to the cones over multiple embeddings of projective spaces. In section 7.3 we look at rank varieties for skew symmetric tensors of degree bigger than two. An interesting feature is that the normality of these rank varieties depends on the characteristic of the base ﬁeld. We pay particular attention to the special case of syzygies of Pl¨ucker ideals deﬁning the cones over Grassmannians embedded into projective space by Pl¨ucker embeddings.

7.1. Basic Properties Let λ be a partition. Let E be a vector space of dimension n over K. Consider the representation X = K λ E ∗ as an afﬁne space over K. Its coordinate ring can be identiﬁed with Aλ = K[X ] = SymK (L λ E). For λ1 ≤ r < n we deﬁne the rank variety Yrλ ⊂ X of tensors of rank ≤ r , Yrλ = {φ ∈ K λ E | ∃S ⊂ E, dim S = r, φ ∈ K λ S ⊂ K λ E }. This means the tensor φ has a rank ≤ r if there exists a basis {e1 , . . . , en } of E such that φ can be written using the tensors involving e1 , . . . , er . The condition r ≥ λ1 assures that X r = ∅. 228

7.1. Basic Properties

229

(7.1.1) Examples. (a) Let λ = (2). Then Aλ = Sym(S2 E), and Yrλ is the rank variety Yrs for symmetric matrices analyzed in section 6.3. (b) Let λ = (12 ). Then Aλ = Sym( 2 E). Assume that r is even. Then Yrλ is the rank variety Yra for skew symmetric matrices considered in section 6.4. If r is odd, we get Yrλ to be Yra−1 . In order to analyze the variety Yrλ we use the obvious incidence variety Z rλ = {(φ, S) ∈ K λ E × Grass(r, E ∗ ) | φ ∈ K λ S ⊂ K λ E ∗ }. We can identify the Grassmannian Grass(r, E ∗ ) with Grass(n − r, E). We write the tautological sequence 0 → R → E × Grass(n − r, E) → Q → 0 with dim R = n − r , dim Q = r . The subspace S becomes a ﬁber of Q∗ . The variety Z rλ is an analogue of varieties Z 1 from sections 6.3, 6.4. We can use our incidence variety in the same way as in chapter 6. (7.1.2) Proposition. Let K be a ﬁeld of characteristic 0. (a) The coordinate ring K[Yrλ ] is normal and has rational singularities. In particular, K[Yrλ ] is Cohen–Macaulay. (b) The ideal Irλ of functions vanishing on Yrλ is a span of all representations L µ E with µ1 > r inside of Sym(L λ E). Proof. Let us use the notation from section 5.1, denoting by p : Z rλ → Grass (n − r, E), q : X × Grass(n − r, E) → X , and q : Z rλ → Yrλ the projections. We will write ξ λ for ξ and ηλ for η. The bundle ηλ can be identiﬁed with L λ Q, and ξ λ ﬁts into an exact sequence 0 → ξ λ → L λ E → L λ Q → 0. Using Theorem (5.1.2) (b) we see that it is enough to show that H i (Grass (n − r, E), Sym(ηλ )) = 0 for i > 0. But since the ﬁeld K has characteristic zero, it is clear that for each j, Sym j (ηλ ) = Sym j (L λ Q) decomposes to a direct sum (with multiplicities) of Schur functors L µ Q. By Corollary (4.1.9) we get the vanishing. It is also clear that the ring H 0 (Grass(n − r, E), Sym(ηλ )) is a factor of Sym(L λ E) obtained by factoring out all representations L µ E with µ1 > r . This proves the normality of H 0 (Grass(n − r, E), Sym(ηλ )) and part (b).

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Higher Rank Varieties

It is an interesting but difﬁcult problem to determine the deﬁning equations of varieties Yrλ . It turns out that in general case we have an easy set of equations deﬁning Yrλ set theoretically. Let us look at the map 1⊗Tr

m⊗1

L λ/(1) E → L λ/(1) E ⊗ E ⊗ E ∗ → L λ E ⊗ E ∗ ,

n ei ⊗ ei∗ , and m is where 1 ⊗ Tr is the multiplication by the trace element i=1 an epimorphism m : L λ/(1) E ⊗ E → L λ E sending the element T ⊗ u where T is a tableau of shape λ/(1) to a tableau T with u inserted in the upper left corner. The presentation of L λ/(1) E by generators and relations from section 2.1 implies the existence of such an epimorphism. The map above induces the map of free Aλ -modules λ : L λ/(1) E ⊗K Aλ (−1) → E ∗ ⊗K Aλ . (7.1.3) Proposition. The variety Yrλ is deﬁned set-theoretically by (r + 1) × (r + 1) minors of λ . Proof. Let φ ∈ K λ E ∗ . Denote by λ (φ) the linear map L λ/(1) E → E ∗ obtained from λ by substituting for each linear function on K λ E ∗ (identiﬁed with an element of degree one in Aλ ) its value on a tensor φ. If a tensor φ ∈ K λ S ⊂ K λ E for some subspace S of dimension r in E ∗ , then the image Im λ (φ) is clearly contained in S. Indeed, choose a basis {e1 , . . . , en−r } of Ker(E → S ∗ ), and complement it to the basis {e1 , . . . , en } ∗ ∗ of E, so en−r +1 . . . . , en are a basis of S. Consider the basis of L λ E consisting of standard tableaux with respect to this basis. If a tableau contains a number ≤ n − r , it is zero when evaluated on φ. Therefore the image of λ (φ) is ∗ ∗ contained in the span of en−r +1 , . . . , en which is S. The other implication follows similarly. If rank λ (φ) ≤ r , then there exists a subspace S of dimension r in E ∗ containing the image of λ (φ). Choosing a basis {e1 , . . . , en } as above, we can conclude that if a standard tableau T contains a number ≤ n − r , then the number in the upper left corner is ≤ n − r , so T was gotten from inserting that number in an upper left corner of a tableau T of shape λ/(i). If T evaluated on φ were not zero, then the image λ (φ)(T ) would not be contained in S. Next we give a criterion for the projection q : Z rλ → Yrλ to be a birational isomorphism. (7.1.4) Proposition. Let λ be a partition, r a number such that λ1 ≤ r < n. Assume that λ is not one of the partitions (2), (r − 2), (1), (r − 1). Then the projection q : Z rλ → Yrλ is a birational isomorphism.

7.1. Basic Properties

231

Proof. Let λ be a partition with λ1 ≤ r < n. First we notice that if there is a tensor φ ∈ K λ E such that rank λ (φ) = r , then the projection q : Z rλ → Yrλ is a birational isomorphism. Indeed, the map sending φ to (φ, Im λ (φ)) is the inverse map to q on an open subset of X r consisting of tensors φ for which rank λ (φ) = r . This subset is nonempty by our assumption. The proposition follows now from the next statement. (7.1.5) Lemma. Let λ be a partition with λ1 ≤ n. Assume λ is not one of the partitions (1), (2), (n − 2), (n − 1). Then there exists in K λ E a tensor φ of rank n. Proof. Let X n−1 be the subset of K λ E of tensors of rank ≤ n − 1. It is enough to show that if λ is not one of the partitions listed in the proposition, then X n−1 = K λ E. If λ = (d), then the tensor e1d + . . . + end is easily seen to be of rank n. If λ1 = n, it is clear that every tableau of shape λ in order to be nonzero has to have all n numbers from [1, n] in the ﬁrst row, so every nonzero tensor φ ∈ K λ E has rank n. This means that when writing λ = (λ1 , . . . , λs ) we can assume 2 ≤ λ1 ≤ n − 1. λ λ Consider the modiﬁcation Z n−1 . It is enough to show that dim Z n−1 < λ λ dim K λ E. The dimension of Z n−1 equals dim Z n−1 = n − 1 + dim K λ E , where E is a vector space of dimension n − 1. It is therefore enogh to show that if λ is not one of the partitions listed in the proposition then dim K λ E − dim K λ E > n − 1. First consider the case λ = (t). Since t t n n−1 n−1 E − dim E = − = , dim t t t −1 we see that we are done in the case 2 < t < n − 1. Notice that for t = 2 and t = n − 2 we have the quality above. Let λ = (λ1 , . . . , λs ) be an arbitrary partition with s ≥ 2, 2 ≤ t = λ1 ≤ n − 1. It is enough to show that dim K λ E − dim K λ E > dim

t

E − dim

t

E .

(∗)

In order to show this fact, we recall that the dimension of K λ E is the number of standard tableaux of shape λ with entries from [1, n]. For every tableau S of shape (t) with number n occurring in S we construct the standard tableau T (S) by setting T (S)(i, j) = S(1, j) for all (i, j) ∈ D(λ). This proves

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that the weak inequality (∗) holds. In order to prove that the inequality is sharp, we produce one more standard tableau of shape λ with entries [1, n], containing n. This will be the tableau U given by setting U (i, λi + 1 − u) = n + 1 − u in the case when the partition λ is not rectangular. If λ is rectangular, then it is easy to produce a standard tableau U containing n which is not constant in columns by taking U (i, λi + 1 − u) = n + 1 − u for i ≥ 2 and U (1, j) = j. Lemma 7.1.5 is proved. (7.1.6) Remark. Propositions (7.1.2) and (7.1.3) generalize to several tensors. If X = K λ(1) E ∗ ⊕ . . . ⊕ K λ(t) E ∗ , we could deﬁne the subvariety Yr to be the set of t-tuples of tensors (φ1 , . . . , φt ) ∈ X which can be simultaneously expressed using tableaux involving r basis vectors. The role of the map λ is played by the map φλ(1) ,...,λ(t) : L λ(1) E ⊗K A ⊕ . . . ⊕ L λ(t) E ⊗K A → E ∗ ⊗K A deﬁned on the j-th component using the tensor φ j . Finally we address the question when the deﬁning ideal of Yrλ is Gorenstein. (7.1.7) Theorem. The variety Yrλ is deﬁned by Gorenstein ideals in the following cases: (a) n =

|λ| dim L λ Q . r

In the remaining cases n > (|λ| dim L λ Q)/r : (b1) λ = (r k , 12 ), r > 1, and n − (|λ| dim L λ Q)/r is positive, divisible by |λ|/2, (b2) λ = (r k , 2), r > 2 is even, and n − (|λ| dim L λ Q)/r is positive, divisible by |λ|, (b3) λ = (r k , (r − 1)2 ), r ≥ 1, and n − (|λ| dim L λ Q)/r is divisible by |λ|/2, (b4) λ = (r k , r − 2), r > 2 is even, and n − (|λ| dim L λ Q)/r is divisible by |λ|, (b5) λ = (r k ), n > k, and n is divisible by k. Proof. We use the duality statement (5.1.4). Denote t = dim ξ . We cal culate the bundle t ξ ∗ ⊗ ωV . By (3.3.5) ωV = OV (−n). Also we have t ∗ ∼ ξ = OV ((|λ| dim L λ Q)/r ). Therefore the dualizing bundle is given by OV ((|λ| dim L λ Q)/r − n). This means that in the case n = (|λ| dim L λ Q)/r the deﬁning ideal is Gorenstein.

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233

Also in the case n < (|λ| dim L λ Q)/r it cannot happen that the deﬁning ideal is Gorenstein, because the module of sections of the sheaf OV ((|λ| dim L λ Q)/r − n) ⊗ Sym(L λ Q) has a representation of dimension > 1 in degree 0, so it cannot be isomorphic to K [Yrλ ]. However, for n > (|λ| dim L λ Q)/r it can still happen that the deﬁning ideal is Gorenstein. It happens when the module of sections of OV ((|λ| dim L λ Q)/r − n) ⊗ Sym(L λ Q) is isomorphic to K [Yrλ ]. The sections of OV ((|λ| dim L λ Q)/r − n) ⊗ Sym(L λ Q) are given by all representations L µ Q from Sym(L λ Q) such that for the conjugate partition µ = (µ1 , . . . , µr ) we have |λ| dim L λ Q . r Such situation can occur only when the ideal consisting of representations L µ Q satisfying the above condition is generated by the one dimensional bundle L µ Q with µ = (r x ), where x = n − (|λ| dim L λ Q)/r . This can happen only when for L λ F, with dim F = r , the variety of tensors of rank < r has codimension 1. Let us classify such cases. µr ≥ n −

(7.1.8) Lemma. Let dim F = r . Let λ be a partition with λ1 < r . Then the subvariety Yrλ−1 has codimension 1 in the following cases: (b1) (b2) (b3) (b4)

λ = (12 ), λ = (2), r even, λ = ((r − 1)2 ), λ = (r − 2), r even.

Proof. Let us assume that codim Yrλ−1 = 1. Then we have the inequality dim L λ F ≤ 1 + r − 1 + dim L λ F or, equivalently, dim L λ F − dim L λ F ≤ r, where F is a vector space of dimension r − 1. But setting F = F ⊕ K we get that dim L µ F = r. µ,λ/µ∈VS, λ/µ=∅

The case r = 1 is trivial, as λ = ∅. In the case r = 2 we have λ = (1d ) and the set of tensors of rank ≤ 1 is a cone over a rational normal curve of

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Higher Rank Varieties

degree d by exercise 5 of chapter 5. Its codimension is equal to d − 1. The case d = 2 is covered under (b1). Let us assume r ≥ 3. If λ has at least two rows of different lengths, then there are at least three partitions µ on the left hand side of the last formula. Also, observe that if a partition µ has a row of length < r − 1, then dim L µ F ≥ r − 1. Thus our equality cannot happen. Thus all rows of λ have the same length. However if there are at least three of them, we still have at least three possible partitions on the left hand side, so the equality cannot happen. Thus λ has two rows or one row. Now we can analyze the situation directly to see that if there are two rows, they have to have length r − 1 or 1, giving cases (b1) and (b3). If there is one row, we can see that the only possibilities are that the length is 1, 2, r − 2, r − 1. But the cases r − 1 and 1 are eliminated because there Yrλ−1 has codimension zero. This completes the proof of the lemma. Now we can conclude the proof of Theorem (7.1.7). Indeed, we have four possible cases, but we have to take into account that the partition λ can have some rows of length r . Moreover, we have an additional case λ = (r k ) where also the codimension of tensors of rank < r is one. This leads to the cases (b1)–(b5) of Theorem (7.1.7). The divisibility condition comes from the fact that we need x to be divisible by the degree of the generating invariant of the subring of SL(F)-invariants in Sym(L λ F). This completes the proof of Theorem (7.1.7).• 7.2. Rank Varieties for Symmetric Tensors In this section we consider rank varieties for symmetric tensors. Some of the results of the previous section can be strengthened in this case. Again E denotes a vector space over K of dimension n. We take λ = (1d ) and so d X = Dd E ∗ , A(1 ) = Sym(Sd E). We assume that d ≥ 3, because in the case d = 2 we get the determinantal varieties for symmetric matrices which were discussed in section 6.3. First we consider the case r = n − 1. Following the paper [Po] of Porras, (1d ) . The incidence we describe the whole minimal free resolution of the ideal In−1 variety deﬁned in section 7.1 becomes d

(1 ) = {(φ, S) ∈ X × Grass(n − 1, E ∗ ) | φ ∈ Dd S ⊂ Dd E ∗ }. Z n−1

The tautological sequence we use is 0 → R → E × Grass(1, E) → Q → 0. We recall that η = Sd Q.

7.2. Rank Varieties for Symmetric Tensors

235 d

(7.2.1) Proposition. In the case of symmetric tensors we have ξ (1 ) = R ⊗ Sd−1 E. d

Proof. The bundle ξ (1 ) ﬁts into the exact sequence d

0 → ξ (1 ) → Sd E → Sd Q → 0. This means we need to show the exactness of the sequence 0 → R ⊗ Sd−1 E → Sd E → Sd Q → 0. The maps are easily deﬁned. The composition of both maps is zero. To check the exactness we need to do it locally. There the sequence is exact because of the direct sum decomposition (2.3.1) for the symmetric power. This means the calculation of the cohomology for the exterior powers of d ξ (1 ) reduces to the calculation of cohomology of line bundles on the projective space. This is provided by Serre’s theorem [H1, chapter III, Theorem 5.1]. Let us summarize. (7.2.2) Proposition. The complex F• has terms given by d

F0 = A(1 ) , Fi = K i,1n−1 E ⊗

n−1+i

d

(Sd−1 E) ⊗K A(1 ) (−i − n + 1) −n+ for i ≥ 1. The length of the complex is dim Sd−1 E − n + 1 = n+d−2 d−1 1. In fact, F• is just an Eagon–Northcott complex associated to the maximal minors of the map (1d ) : Sd−1 E ⊗K A(1 ) (−1) → E ∗ ⊗K A(1 ) . d

d

d

(7.2.3) Corollary. The ideal Ir(1 ) is generated by (r + 1) × (r + 1) minors of the matrix (1d ) . Proof. For r = n − 1 it follows from Proposition (7.2.2). To prove the general d case we use descending induction on r . Let Ir (1 ) be the ideal generated by (r + d d 1) × (r + 1) minors of the matrix (1d ) . Assume that Ir(1+1) = Ir+1 (1 ) . Let us d also assume that A/Ir (1 ) contains nilpotents. The set of nilpotents in this ring is a GL(E)-stable subspace. Therefore it has to contain a U+ -invariant. But the weight of this U+ -invariant contains ≤ r + 1 basis vectors, otherwise it d d would be in Ir(1+1) = Ir+1 (1 ) . This means such U+ -invariant (which is nilpotent d modulo Ir (1 ) ) exists already for dim E = r + 1. This is a contradiction with Proposition (7.2.2).

236

Higher Rank Varieties

(7.2.4) Remark. Proposition (7.2.2) and Corollary (7.2.3) are true for several symmetric tensors of degrees d1 , . . . , dt . The role of the map (1d ) is played by the map φ1d1 ,...,1dt : Sd1 −1 E ⊗K A(1 ) ⊕ . . . ⊕ Sdt −1 E ⊗K A(1 ) → E ∗ ⊗K A(1 ) , d

d

d

where the j-th component is deﬁned using the j-th tensor. We also state the criterion for rank varieties of symmetric tensors to be Gorenstein. d

(7.2.5) Theorem. The variety Yr(1 ) is Gorenstein in the following cases: (a) n = r +d−1 , d−1 r +d−1 (b) n > d−1 , r = 1, n divisible by d. Proof. This follows from Theorem (7.1.7). Analyzing all cases, we see that λ = (1d ) can occur only in cases (a) and (b3) (with r = 1). This last case gives case (b) of our statement. The most interesting varieties are those of tensors of rank ≤ 1. The variety Y1(d) is the cone over the d-tuple embedding of Pn−1 = P(E) into the projective space P(Sd E). There is a lot of interest in the resolutions of these varieties, especially trying to determine for which p they satisfy the property Np of Green and Lazarsfeld introduced in [GL]. This is equivalent to asking about the smallest i for which H 2 ( i ξ ) = 0. The reader may consult [B1], for some recent results. In general the description of the resolution seems to be rather difﬁcult, as it is connected to the problem of inner plethysm. The composition d series of the bundle ξ (1 ) induced by the tautological exact sequence will always involve the term Sd R, so the exterior powers are related to higher plethysm. One might hope, however, to determine all pairs (i, j) for which the d cohomology group H i (Grass(n − r, E ∗ ), i+ j ξ (1 ) ) = 0. One might hope that for every such pair (i, j) there will be a representation that will occur only few times in the spectral sequence and thus will allow one to determine that d H i (Grass(n − r, E ∗ ), i+ j ξ (1 ) ) = 0. Below we do it for the embeddings of P2 . We use the approach of Ottaviani and Paoletti [OP]. Let us look at the resolutions of d-tuple embeddings of projective spaces in more detail. In section 6.3 we exhibited the resolutions of such embeddings for d = 2. Proposition (7.2.2) provides the answer for the embeddings of P1 . d d It is not difﬁcult to locate the top part of the resolution of A(1 ) /Ir (1 ) .

7.2. Rank Varieties for Symmetric Tensors

237

(7.2.6) Proposition ([OP]). Let us ﬁx n, d. We choose the number j to be the minimal number such that ( j + 1)d ≥ n. Then the top part of the complex F• is ⊗(n+d−1 (d+n−1 )−1− j n n−1 n ) d n− j (1 ) ∗ ξ E . Grass(1, E), = S( j+1)d−n E ⊗ H

Proof. Using the duality statement (5.1.4) and taking into account that the canonical bundle on Pn−1 is O(−n), we see that it is enough to locate the rightmost part of the complex F(Sd−n−1 Q)• . But this is equivalent by (5.1.2) (b) to ﬁnding the cohomology of j≥0 Sd−n+ jd Q. It is now clear by (4.1.9) that the higher cohomology has to vanish and that the module H 0 (Grass(1, E), j≥0 Sd−n+ jd Q) is generated by the representation S( j+1)d−n V in degree j described in our statement. For the remainder of this section we assume that char K = 0. (7.2.7) Example. Let us take d = n = 3. Then j = 0, and Proposition (7.2.6) says that the top of the resolution is one dimensional. This means that the 3 resolution in question is self-dual and the ideal I1(1 ) is Gorenstein. But this means that H 0 and H 2 strands of the resolution consist of one copy of one dimensional representation each. This allows to describe the terms of the d resolution by calculating Euler characteristics of exterior powers of ξ (1 ) . The terms of the complex F• are F0 = (0, 0, 0), F1 = (4, 2, 0), F2 = (4, 3, 2) ⊕ (5, 3, 1) ⊕ (5, 4, 0) ⊕ (6, 2, 1), F3 = (5, 4, 3) ⊕ (5, 5, 2) ⊕ (6, 3, 3) ⊕ (6, 4, 2) ⊕ (6, 5, 1) ⊕ (7, 3, 2) ⊕(7, 4, 1), F4 = (6, 5, 4) ⊕ (6, 6, 3) ⊕ (7, 4, 4) ⊕ (7, 5, 3) ⊕ (7, 6, 2) ⊕ (8, 4, 3) ⊕(8, 5, 2), F5 = (7, 6, 5) ⊕ (8, 6, 4) ⊕ (8, 7, 3) ⊕ (9, 5, 4), F6 = (9, 7, 5), F7 = (9, 9, 9), d

where we write (a, b, c) instead of K a,b,c V ⊗ A(1 ) . The homogeneous degree is easily seen from the size of each partition.

238

Higher Rank Varieties

Let us ﬁx n = 3. We want to determine the minimal j for which the d cohomology modules H 2 (Grass(1, E) j+2 ξ (1 ) ) = 0. (7.2.8) Proposition (Ottaviani–Paoletti, [OP, Theorem 2.1]). Let n = 3. d We have H 2 (Grass(1, E), j+2 ξ (1 ) ) = 0 for j ≥ 3d − 2. d Proof. It is enough to show that H 2 (Grass(1, E), 3d ξ (1 ) ) = 0. Indeed, applying the duality (5.1.4) takes the H 2 strand to the dual of the H 0 strand, and since the H 0 strand has to be linearly exact we know that if the j-th term in this strand is nonzero, then all terms in degrees ≤ j also have to be nonzero. By Serre duality it is enough to show that H 0 (Grass(1, E), Sd−3 Q ⊗ d(d−3)/2 (1d ) ξ ) = 0. Now everything follows from the following j (1d ) ⊗ St Q has nonzero sections for 1 ≤ j ≤ (7.2.9) n+d−1Lemma. The sheaf n+t−1 ξ − 1, j + 1 ≤ and t ≥ 1. d n−1 But we have an exact sequence 0→

j

d

ξ (1 ) →

j j−1 d (Sd V ) × Grass(1, E) → Sd Q ⊗ ξ (1 ) → 0.

d This means that the sections of j ξ (1 ) ⊗ St Q can be identiﬁed with the kernel j j−1 at (Sd V ⊗ St V → (Sd V ) ⊗ St+d V ). Ker We use a Koszul complex ... →

j+1 j j−1 (Sd V ) ⊗ St−d Q → (Sd V ) ⊗ St Q→ (Sd V ) ⊗ St+d Q → . . .

with the differential being a composition j+1

→

j

(Sd V ) ⊗ St−d Q →

(Sd V ) ⊗ Sd Q ⊗ St−d Q →

j

(Sd V ) ⊗ Sd V ⊗ St−d Q

j

(Sd V ) ⊗ St Q.

Notice that if t = pd + q, this is just a twisted symmetric power Sq Q ⊗ S j+ p (Sd V → Sd Q). The existence of this complex means that for t ≥ d the j+1 j (1d ) sections of (Sd V ) ⊗ St−d Q give the sections of ξ ⊗ St Q. In particular, for d = t we get for each family of polynomials s0 , . . . , s j an

7.3. Rank Varieties for Skew Symmetric Tensors

239

element j

(−1)i s0 ⊗ . . . ⊗ sˆi ⊗ . . . ⊗ s j ⊗ si

i=0

in Ker at . Let 1 ≤ t < d. If we can factor si = uwi with deg u = d − t, we see that the element j

(−1)i s0 ⊗ . . . ⊗ sˆi ⊗ . . . ⊗ s j ⊗ wi

i=0

d gives a nonzero section of j ξ (1 ) ⊗ St Q. This construction is possible as soon as we can ﬁnd j + 1 linearly independent polynomials of degree ≤ t, .• i.e. if j + 1 ≤ n+t−1 n−1

7.3. Rank Varieties for Skew Symmetric Tensors In this section we consider rank varieties for skew symmetric tensors. The results on the equations are less precise than for symmetric tensors because the ideal of minors of the map λ is not radical in this case. We follow the paper [Po] of Porras in extracting information about the resolutions and equations of the deﬁning ideals of tensors of rank ≤ n − 1. One can describe the generators of the deﬁning ideals quite precisely in the case of skew symmetric tensors of degree 3. We also pay special attention to the skew symmetric tensors of degree d of minimal possible rank d. The variety Yd(d) in this case is the cone over a Grassmannian Grass(r, E) embedded via Pl¨ucker embedding. The problem of ﬁnding higher syzygies of these ideals was posed by Study over 100 years ago. As before E denotes a vector space over K of dimension n. We take λ = (d) and so X = d E ∗ , A(d) = Sym( d E). We assume that n − 3 ≥ d ≥ 3, because in the remaining cases the only possible varieties we can get are the determinantal varieties for skew symmetric matrices which were considered in section 6.4. For d ≤ r < n we deﬁned in 7.1 the rank varieties Yr(d) . First we consider the case r = n − 1. The incidence variety deﬁned in section 7.1 becomes (d) = {(φ, S) ∈ X × Grass(n − 1, E ∗ ) | φ ∈ Z n−1

d

S⊂

d

E ∗ }.

240

Higher Rank Varieties

The tautological sequence we use is 0 → R → E × Grass(1, E) → Q → 0. We recall that η(d) = d Q. (7.3.1) Proposition. In the case of skew symmetric tensors of rank ≤ n − 1 we have ξ (d) = R ⊗ d−1 Q. Proof. The bundle ξ (d) ﬁts into the exact sequence 0 → ξ (d) →

d

E→

d

Q → 0.

This means we need to show the exactness of the sequence 0→R⊗

d−1

Q→

d

E→

d

Q → 0.

The maps are easily deﬁned. The composition of both maps is zero. To check the exactness we need to do it locally. There the sequence is exact because of the direct sum decomposition (2.3.1) for the exterior power. We can also easily determine the top term of the resolution. (7.3.2) Proposition([Po]). = n − 1. De Let us assume 3 ≤ d ≤ n − 3. Let rn−1 n−1 note m = dim ξ = d−1 . The top term of the complex F• is H (Grass(n − n−2 1, F), m ξ ) = K (m−n+1,q+1,...,q+1) E ⊗ A(d) (−m), where q = d−2 . In particular the only case in which In−1 is a Gorenstein ideal is when d = 3, n = 6. Proof. The term in question comes from the cohomology of the top exterior power of ξ . It occurs in Fm−n+1 . It is clear by (5.1.6) (a) and by (7.1.3) that Fi = 0 for i > m − n + 1. It remains to show that the term listed above is the only term in Fm−n+1 . The only possible cohomology groups are m− j n−1− j ξ Grass(n − 1, F), H for j > 0. But by (7.3.1), using the isomorphism we see that we look at the weights

m− j

ξ=

m

ξ⊗

j

ξ ∗,

(q − µn−1 , . . . , q − µ1 , m − j), where µ is a partition such that K µ Q∗ occurs in j ( d−1 Q∗ ). We also assume that the weight in question does not have a nonzero (n − 1)st cohomology

7.3. Rank Varieties for Skew Symmetric Tensors

241

group. This means q − µn−1 ≥ m − j − n + 2. This implies

n−2 j ≥m−q −n+2= − n + 2. d −1

If we can show that the inequality above implies j ≥ n − 1, we are done, as we have eliminated all possibilities for j. However the inequality n−2 −n+2≥n−1 d −1 fails only for n = 6, d = 3 and for n = 7, d = 3, 4. These three cases can be handled explicitly. Proposition (7.3.1) means that the calculation of the cohomology for the exterior powers of ξ (d) is not as easy as in the symmetric case. In order to perform the explicit calculation we would need to know the decomposition of i ( d−1 Q) into Schur functors. This, as explained in section 2.3, is a very difﬁcult problem. Still, we have an explicit formula for plethysm when d − 1 = 2. It also allows us to describe in some cases, as for symmetric tensors, the pairs (i, j) for which H i (Grass(1, E), i+ j ξ (d) ) = 0. Let us assume d = 3. Then the plethysm formula (2.3.9) (b) makes the problem of calculating a complex F• a combinatorial exercise. Still, the analysis of the whole complex F• is quite complicated. Here we just analyze the (d) . Interested readers should consult [Po]. deﬁning equations of the ideal of Yn−1 (d) of the variety (7.3.3) Proposition ([Po]). Let us assume d = 3. The ideal In−1 (d) has generators in all degrees i satisfying [n/2] ≤ i ≤ (1 + 2n − (1+ Yn−1 8n)1/2 )/2.

Proof. We need to show that the terms occurring in the term F1 of the complex F• appear in homogeneous degrees i satisfying the above inequalities and that for each such i we get a nonzero contribution. We know that i i−1 Grass(n − 1, E), F1 = H ξ ⊗ A(d) (−i). i>0

We also have by (2.3.9) (b) i

ξ=

µ∈Q −1 (2i)

K µ Q ⊗ Si R.

242

Higher Rank Varieties

This means we need to apply (4.1.9) to the weights (µ1 , . . . , µn−1 , i) with µ ∈ Q −1 (2i). Notice that such term gives a contribution to H i−1 if and only if µn−i = 0 and µn−i+1 = 0. Therefore we need to estimate for which i we have a partition in Q −1 (2i) with exactly n − i parts. Let us look for a lower bound for i. For n = 2t even the smallest partition of this kind is clearly µ = (t − 1, 22 , 1t−3 ) ∈ Q −1 (2t). For n = 2t + 1 the smallest such partition is clearly µ = (t, 1t ) ∈ Q −1 (2t). This proves the lower bound of the proposition. Let us seek the biggest possible partition in Q −1 (2i) with n − i parts. Any such partition has to be contained in the rectangle ((n − i − 1)n−i ), so we must have the inequality 2i ≤ (n − i)(n − i + 1), which gives the upper bound in the proposition. Of course, for any i satisfying the inequalities of the proposition we can ﬁnd the appropriate µ ∈ Q −1 (2i) by choosing any partition from Q −1 (2i) containing the partition giving the lower bound and contained in the rectangle ((n − i + 1)n−i ). (7.3.4) Corollary ([Po]). Let d, n be as above, with 3 ≤ d ≤ n − 3. For any r satisfying 3 ≤ r ≤ n − 1 the ideal of (r + 1) × (r + 1) minors of (d) is not radical. Proof. It is enough to show that Ir(d) has some nonzero elements in degrees ≤ r . This is clear, since the representations generating Ir(d) for r + 1 dimensional space will give the representations from Ir(d) for n dimensional space. By (7.3.3) they occur in degrees 1 + 2(r + 1) − (1 + 8(r + 1))1/2 1 + 2(r + 1) − 3 ≤ = r. 2 2 The corollary follows. Let us also state when the ideal Ir(d) is Gorenstein. (7.3.5) Theorem. The ideal Ir(d) is Gorenstein in the following cases: r −1 (a) n = d−1 , (b) d = 2, and r is even, r −1 r −1 , , d = r − 2, and n − d−1 is divisible by r −2 (c) n > d−1 2 (d) d = r . Proof. This is a special case of Theorem (7.1.7). Cases (b2), (b4), and (b5) of (7.1.7) lead to cases (b), (c), and (d) of our statement.

7.3. Rank Varieties for Skew Symmetric Tensors

243

For the remainder of this section we investigate the case r = d. In this case η = d Q is one dimensional and therefore K (n d ) E. A(d) /Id(d) = n≥0

Let us describe the top term of the resolution. (7.3.6) Proposition. The top term in the resolution F• is H

d(n−d)−n+1

(Grass(n − d, E),

n ( d )−n

ξ ) = K (((n−1)−1)n ) E. d−1

Proof. Let us use the duality statement (5.1.4). The canonical n sheaf K Grass(n−d,E) is equal to OGrass(n−d,E) (−n). The top exterior power (d )−1 ξ is isomorphic (up to the twist by a power of determinant of E) to OGrass(n−d,E) (1). Therefore the dualizing bundle will be OGrass(n−d,E) (−n + 1) or, in terms of tautological bundles d Q⊗(−n+1) . Calculating the cohomology of the sheaf d ⊗(−n+1) Q ⊗ Sym(η), we see that the generator occurs in degree n − 1 and is a trivial representation. This means that by (5.1.4) the top term of the resolution occurs in homogeneous degree dim ξ − n + 1, and it is a one dimensional representation. Since the variety Yd(d) is normal with rational singularities by (5.1.2) (b) and (5.1.3), we see n that this term has to occur in (d) the term Fi with i = dim X − dim Yd = d − d(n − d) − 1. Calculating the powers of the determinants involved, one gets the proposition. (7.3.7) Remark. Note that for d = 2 we constructed the resolution of the Pl¨ucker ideal in section 6.4. It is the resolution of 4 × 4 Pfafﬁans of a generic skew symmetric n × n matrix. Indeed, the cone over the Grassmannian Grass (2, E) can be identiﬁed with the set of skew-symmetric matrices of rank ≤ 1. We ﬁnish this section with the analysis of the case d = 3, n = 6. We will analyze the rank variety Y5(3) . The main goal is to show that this variety is not normal over a ﬁeld K of characteristic 2, so the conclusion of Proposition (7.1.2) fails in positive characteristic. Thus E is a vector space of dimension 6, and we work over the Grassmannian Grass(1, E) with tautological sequence 0 → R → E × Grass(1, E) → Q → 0. We have η(3) =

3

Q, ξ (3) = R ⊗

2

Q.

244

Higher Rank Varieties

(7.3.8) Proposition. Let char K = 0. The complex F• has the following nonzero terms: F0 = A(d) ,

F1 = L (6,3) E ⊗ A(d) (−3),

F3 = L (62 ,5,1) E ⊗ A(d) (−6),

F2 = L (6,5,1) E ⊗ A(d) (−4),

F4 = L (63 ,3) E ⊗ A(d) (−7),

F5 = L (65 ) E ⊗ A(d) (−10). (7.3.9) Proposition. Let K be a ﬁeld of characteristic 2. Then the only nonzero cohomology groups of S2 R ⊗ 2 Q are 2 2 1 H Grass(1, E), S2 R ⊗ Q 2 2 6 = H Grass(1, E), S2 R ⊗ Q = E. 2

Proof. We notice that in characteristic 2, L (3,1) Q is not irreducible. In fact we have the natural exact sequence 0 → M(2,1,1) Q → L (3,1) Q →

4

Q → 0.

Also, we have a natural map ψ : L 3,1 Q →

2 2 Q

induced by the composition map 3

⊗1

Q⊗Q →

2

1⊗m

Q⊗Q⊗Q →

2

Q⊗

2

Q.

This map, however, is not an isomorphism in characteristic zero. Its image is isomorphic to 4 Q; its kernel, to M(2,1,1) Q. Next we notice that H s (Grass(1, E), S2 R ⊗ L 3,1 Q) = 0 for all s ≥ 0, by exercise 5 of chapter 4. Also, the only nonzero cohomology group of S2 R ⊗ 4 Q is H 1 (Grass(1, E), S2 R ⊗ 4 Q) = 6 E. This we can deduce from identifying S2 R with D2 R and using the *-acyclic resolution 2 (E → Q) of D2 R, tensored with 4 Q. The sections of this resolution

Exercises for Chapter 7

give a complex 0 → D2 E ⊗

4

E→E⊗E⊗

4

E→

2

245

E⊗

4

# E

6

E,

which has only one homology – 6 E in degree 1. Now, using the short exact sequences 0 → M(2,1,1) Q → L (3,1) Q → and 0→

4

4

Q→0

2 2 Q→ Q → M(2,1,1) Q → 0

tensored with S2 R and the long exact sequences of cohomology they induce, we deduce the proposition. (7.3.10) Proposition. Let char K = 2. Then the ring A(3) /J5 is not normal. Proof. Recall that H 0 (Grass(1, E), Sym( 3 Q)) can be identiﬁed with the normalization of A/J5 . By Proposition (7.3.9) the complex F• has a nontrivial term in homological degree 0 and in homogeneous degree 2. It follows that the natural map 3 3 0 S2 ( E) → H Grass(1, E), Sym2 Q is not onto, and therefore the normalization of A(3) /J5 is not generated as an A(3) -module by a unit in degree 0.

Exercises for Chapter 7 Minimal Resolutions of the Ideal I3(3) for n = 6, 7 1. Let X = d E ∗ be the set of skew symmetric tensors. Denote by Y ⊂ X the set of 1-decomposable tensors, i.e. the set of tensors φ in X such that φ = ψ ∧ l where l ∈ E ∗ is a linear form and ψ ∈ d−1 E ∗ . Prove that Y has a desingularization which is a total space of a vector bundle over the Grassmannian Grass(n − 1, F). Identify ξ = d R and η = d−1 Q⊗ R. Prove that Y is normal and has rational singularities.

246

Higher Rank Varieties

2. In the situation of exercise 1, consider the twisted complex F(Sd Q∗ )• . Prove that its homology modules are H−i (F(Sd Q∗ )• ) = ⊕ j≥0 H i Grass(n − 1, E), d−1 Sd Q∗ ⊗Sym j Q ⊗ R . Prove that the nonzero homology occurs for i = −d + 1, . . . , 0. Prove that H−d+1 (F(Sd Q∗ )• ) = A/Id(d) (−1). 3. In the situation of exercise 2, specialize to d = 3. Prove that the only nonzero homology modules of F(S3 Q∗ )• are H−2 and H0 . Writing N = H0 (F(S3 Q∗ )• ), prove that the j-th graded component of N is Nj = K ( j−3,a,a,b,b,...) E. a+b+...= j, j−3≥a

4. In the situation of exercise 3, specialize to n = 6. We have K ( j−3,a,a,b,b,0) E. Nj = a+b+...= j, j−3≥a≥b

Prove that the complex F(S3 Q∗ )• has the following nonzero terms (06 ) 4

(2, 14 , 0) ↑ (3, 2, 14 ) ⊕ (24 , 1, 0) ↑ 4 (32 , 22 , 12 ) 4

(5, 25 ) ↑ (5, 33 , 22 ) ⊕ (43 , 23 ) ↑ (5, 42 , 32 , 2) ↑ (52 , 43 , 2) ↑ (55 , 2)

where we just write the partitions of the occurring Weyl functors. The vertical arrows denote the maps of degree 1 and skew arrows denote the maps of degree 2.

Exercises for Chapter 7

247

5. We still work with X = 3 E ∗ , dim E = 6. Consider the variety Y5(3) of tensors of rank ≤ 5 in X . Let Z 5(3) be its desingularization considered in section 7.3: $ 3 3 (3) ∗ ∗ S⊂ E Z 5 = (φ, S) ∈ X × Grass(5, E ) | φ ∈ . We write 0 → R →

3

E × Grass(5, E) → Q → 0

for the tautological sequence on Grass(5, E). We have η = 3 Q , ξ = R ⊗ 2 Q . Consider the twisted complex F ( 5 Q⊗3 )• . Prove that it is acyclic, and show that the A-module H0 (F ( 5 Q⊗3 )• ) is isomorphic to N twisted by 6 E ⊗−3 . Prove that the complex F ( 5 Q⊗3 )• has the 6 ⊗3 E ): following nonzero terms (after twisting back by (35 , 0) ↑ 2 3 (4 , 3 , 1) ↑ (5, 42 , 32 , 2) ↑ 3 2 (6, 4 , 3 ) ⊕ (53 , 33 ) ↑ 4 (7, 45 ) 4

(62 , 52 , 42 ) ↑ 4 (6 , 5, 4) ⊕ (7, 6, 54 ) ↑ (7, 64 , 5)

4 (76 )

The vertical arrows denote the maps of degree 1, and the skew arrows denote the maps of degree 2. 6. Prove that there exists a map of the complex constructed in exercise 5 to the complex constructed in exercise 4 induced by the isomorphism H0 (F(S3 Q∗ )• ) = N . Prove that the cone of gives a minimal resolution of the A(3) -module A(3) /I3(3) .

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Higher Rank Varieties

7. Let us specialize the situation from exercise 3 to n = 7. We have Nj = K ( j−3,a,a,b,b,c,c) E. a+b+c...= j, j−3≥a≥b≥c

The components N j are nonzero for j ≥ 5. Let N be the span of all summands with c ≥ 1. Prove that N is an A(3) -submodule of N . Denote N = N /N . Prove that the minimal resolution of N equals F(Q∗ )• ⊗ 7 F, in the notation of exercise 1. 8. Prove that the minimal resolution of the A(3) -module N from exercise 7 can be obtained as follows. Take V = Grass(, E) with tautological sequence 0 → R → E × Grass(2, E) → Q → 0. Here dim R = 2, dim Q = 5. Take the bundle ξ to be the kernel 0 → ξ →

3

3

E × Grass(2, E) →

Q → 0.

The bundle ξ also ﬁts into the exact seuence 0→

2

R ⊗ Q → ξ → R ⊗

2

Prove that the minimal resolution of N is F ((

Q → 0.

5

Q )⊗3 )• .

9. Use the information obtained in exercise 8 to calculate the terms in the linear strand of the minimal resolution of A(3) /I3(3) for n = 7. 10. Consider the twisted sheaf M = K ((r −1)(n−r −1))r Q ⊗ K ((r −1)(n−r −1),(r −1)(n−r −2),...,(r −1),0) R ⊗ O Z r(r ) supported in the rank variety Yr(r ) . Prove that M has no higher cohomology. Show that M := H 0 (Z r(r ) , M) is a maximal Cohen–Macaulay module with a linear resolution supported in Yr(r ) . 11. Finish the proof of Proposition (7.3.2). Calculate explicitly the resolution in the case n = 6, d = 3 and in the cases n = 7, d = 3, 4. 12. Let X = Sd E ∗ be the set of symmetric tensors. Denote by Y ⊂ X the set of 1-decomposable tensors, i.e. the set of tensors φ in X such that φ = ψ ◦ l where l ∈ E ∗ is a linear form. Prove that Y has a desingularition which is a total space of a vector bundle over the Grassmannian Grass(n − 1, F). Identify ξ = Sd R and η = Q ⊗ Sd−1 E. Prove that Y is not normal but its normalization has rational singularities. Identify the normalization of Y .

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249

Higher Rank Varieties for Orthogonal and Symplectic Groups 13. Let F be an orthogonal space of dimension m = 2n + 1 or m = 2n. We denote the nondegenerate symmetric form on F by ( , ). We choose a standard hyperbolic basis of F, denoted {e1 , . . . , en , e, e¯ n , . . . , e¯ 1 } in the odd case and {e1 , . . . , en , e¯ n , . . . , e¯ 1 } in the even case. The only nonzero values of the form ( , ) are (ei , e¯ i ) = 1, (e, e) = 1. Let λ be a partition, and let Vλ F be a representation of SO(F) of highest weight λ. This is a space of tensors from L λ F modulo the tensors containing a trace element (compare exercise 14 of chapter 6). For λ1 ≤ n we deﬁne the rank variety Yrλ as the set of tensors that (after the change of basis) can be written only in terms of e1 , . . . , er . Construct the desingularization Z rλ of the variety Yrλ using the isotropic Grassmannian IGrass(r, F). Prove that the variety Yrλ is normal and has rational singularities. 14. Let F be an symplectic space of dimension 2n. We denote the skew symetric nondegenerate form on F by ( , ). We choose a standard symplectic basis of F, denoted {e1 , . . . , en , e¯ n , . . . , e¯ 1 }. The only nonzero values of the form ( , ) are (ei , e¯ i ) = 1, (e, e) = 1. Let λ be a partition, and let Vλ F be a representation of Sp(F) of highest weight λ. These are tensors from L λ F modulo the tensors containing a trace element (compare exercise 4 of chapter 6). For λ1 ≤ n we deﬁne the rank variety Yrλ as the set of tensors that (after the change of basis) can be written only in terms of e1 , . . . , er . Construct the desingularization Z rλ of the variety Yrλ using the isotropic Grassmannian IGrass(r, F). Prove that the variety Yrλ is normal and has rational singularities.

The Isotropic Grassmannian IGrass(3, 6) 15. Let F be a symplectic space of dimension 6. Consider the representation V13 F. It ﬁts into the exact sequence t

0→F →

3

F → V13 F → 0,

where the map t is the multiplication by the element t = 1≤i≤n ei ∧ e¯ 1 2 from F given by the form. Prove that the representation V13 F has four Sp(F)-orbits (except the zero orbit). There is a general orbit, a hypersurface given by the vanishing of (the unique up to scalar) invariant of degree 4, the orbit X given by the tensors where the partial derivatives of (forming a representation V13 F in degree 3) vanish, and the orbit Y which is the cone over IGrass(3, F). Prove that codim X = 4, codim Y = 7.

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Higher Rank Varieties

16. Find the desingularization of X which is a homogeneous bundle over some isotropic ﬂag variety. Conclude that the minimal rsolution of K[X ] has the following terms: F0 = A,

F1 = V13 F ⊗ A(−3),

F3 = V12 F ⊗ A(−6),

F2 = V2 F ⊗ A(−4),

F4 = V1 F ⊗ A(−7).

17. Calculate the minimal resolution of K[Y ]. Show that its terms are F0 = A,

F1 = V2 F ⊗ A(−2),

F2 = V2,1 F ⊗ A(−3),

F3 = V2,1,1 F ⊗ A(−4), F4 = V2,1,1 F ⊗ A(−6), F6 = V2 F ⊗ A(−8),

F5 = V2,1 F ⊗ A(−7), F7 = A(−10).

This representation is one of the so-called subexceptional series, corresponding to the entries in the row of the Freudenthal magic square. These representations are (using Bourbaki’s notation for fundamental weights) (a) G = SL(2), V = V (3ω1 ), (b) G = SL(2) × SL(2) × SL(2) × 3 , V = V (ω1 ) ⊗ V (ω1 ) ⊗ V (ω1 ), where 3 denotes a permutation group, (c) G = SP(6), V = V (ω3 ), (d) G = SL(6), V = V (ω3 ), (e) G = (spinor group.)(12), V = V (ω5 ) (highest weight ( 12 , 12 , 12 , 12 , 1 1 , )), 2 2 (f) G = E 7 , V = V (ω6 ). The uniformity, described by Landsberg and Manivel in [LM], is with respect to the parameter m, which in the above cases takes values − 23 , 0, 1, 2, 4, 8. The dimension of the representation V is 6m + 8. Again there are four orbits: the general one, the hypersurface deﬁned by the vanishing of a (unique up to scalar) invariant of degree 4, X , and Y . The codimension of X is m + 3; the codimension of Y is 3m + 4. Notice that item (d) on the list is the representation 3 F for dim F = 3. The resolution of X in that case is described in (7.3.8). The resolution of Y is calculated in exercises 4, 5, 6.

8 The Nilpotent Orbit Closures

In this chapter we deal with another important class of varieties – the nilpotent orbit closures of the adjoint action of a simple algebraic group on its Lie algebra. These varieties play an important role in representation theory. All such orbit closures have desingularizations which are total spaces of vector bundles over homogeneous spaces. We describe the applications of the geometric method. The vector bundles involved in the construction of these desingularizations are more complicated than in the case of determinantal varieties. The explicit formula for the terms of complexes F(L)• is not known in general. Still, one can prove some interesting results. The ﬁrst two sections of the chapter are devoted to the nilpotent orbit closures for the general linear group. In section 8.1 we describe the desingularizations of these orbit closures explicitly. We apply theorems from chapter 5 to prove that all orbit closures are normal, are Gorenstein, and have rational singularities. We also describe the combinatorial way of estimating the terms of the complexes F• in this case. This method is then used in section 8.2 to describe the generators of the deﬁning ideals of nilpotent orbit closures. In section 8.3 we treat the case of general simple groups. We prove a theorem of Hinich and Panyushev saying that the normalization of every nilpotent orbit closure is Gorenstein and has rational singularities. Finally, in sections 8.4 and 8.5 we look at the case of classical groups. We mainly work with examples showing how geometric method can be applied in special cases. We give examples of nonnormal orbit closures and discuss some special cases. In the case of the symplectic groups, for orbits corresponding to partitions with even parts, we prove the estimate on the weights of representations generating the deﬁning ideals. We give conjectures for such estimates for all nilpotent orbits for classical groups.

251

252

The Nilpotent Orbit Closures

8.1. The Closures of Conjugacy Classes of Nilpotent Matrices Let E be a vector space of dimension n over a ﬁeld K. We consider the afﬁne space X = HomK (E, E) of n × n matrices. We identify the space X with E ∗ ⊗ E. The general linear group GL(E) acts on X by conjugation; the element g ∈ GL(E) sends the matrix φ ∈ Hom(E, E) to g −1 φg. The coordinate ring of X can be identiﬁed with the symmetric algebra A = Sym(E ⊗ E ∗ ). We start with the brief analysis of the ring of invariants AGL(E) . Let us denote by vi (i = 1, . . . , n ) the unique (up to scalar) GL(E) invariant in i E ⊗ i E ∗ . To ﬁx the scalar we can choose φ I,I vi = I ⊂[1,n], |I |=i

to be the sum of principal minors of the generic matrix φ. Notice that vi is the coefﬁcient of s n−i in the characteristic polynomial χ (φ, s) := det(φ − s Id) of φ. The following proposition is a special case of Chevalley’s theorem. (8.1.1) Proposition. The ring AGL(E) is a polynomial ring in v1 , . . . , vn . Proof. We notice that by Cauchy decomposition A= L α E ⊗ L α E ∗. α

Moreover, by the Littlewood–Richardson rule (2.3.4) we see that for each α the tensor product L α E ⊗ L α E ∗ contains precisely one trivial representation. This means that the dimension of the d-th graded component AdGL(E) is equal to the number of partitions of d with at most n parts. This in turn means that the Poincar´e series of AGL(E) is given by the formula 1 (dim AGL(E) )t d = PAGL(E) (t) := . d (1 − t) . . . (1 − t n ) d≥0 This suggests that AGL(E) is a polynomial ring in n variables, with generators of degree 1, . . . , n. To conclude the proof of (8.1.1) it is enough to show that the polynomials v1 , . . . , vn are algebraically independent. This is however clear: after substituting φi, j = 0 for i = j and φi,i = xi , we see that the polynomial vi specializes to the elementary symmetric function ei (x1 , . . . , xn ). The embedding AGL(E) → A induces the orbit map χ : X → X/GL(E) = Kn sending the matrix φ to its characteristic polynomial χ(φ, s).

8.1. The Closures of Conjugacy Classes of Nilpotent Matrices

253

This map has been analyzed in many contexts. We will take the point of view of geometric invariant theory and analyze the nullcone of X – the ﬁber χ −1 (0). This is the set of matrices with all eigenvalues equal to 0, i.e. the set of nilpotent matrices. By the Jordan canonical form we know that the set χ −1 (0) has ﬁnitely many GL(E)-orbits. They correspond to the partitions µ of n. For a partition µ = (µ1 , . . . , µr ) we denote by O(µ) the set of nilpotent matrices with Jordan blocks of dimensions µ1 , . . . , µr . If µ = (µ1 , . . . , µs ) is the partition conjugate to µ, we can describe the orbit O(µ) as a set of endomorphisms φ of E for which dim Ker φ i = µ1 + . . . + µi for i = 1, . . . , s. We denote by Yµ the closure of the orbit O(µ) in X . (8.1.2) Examples. (a) µ = (n). In this case the set Yµ is the set of all nilpotent matrices. (b) µ = (1n ). In this case Yµ is just a point 0. (c) More generally, let µ = ( p, 1n− p ). The variety Yµ is a set of nilpotent matrices of rank < p. (d) Let µ = (2i , 1 j ) where 2i + j = n. In this case Yµ is a set of matrices φ such that φ 2 = 0 and rank φ ≤ i. (8.1.3) Proposition. The closure Yµ is deﬁned set-theoretically by the conditions dim Ker φ i ≥ µ1 + . . . + µi for i = 1, . . . , s. Proof. The conditions of the Proposition are algebraic and satisﬁed on O(µ), so they have to be satisﬁed on Yµ . Let us denote by Yµ the closed subset of X of endomorphisms φ for which all the above inequalities are equalities. In order to show that Yµ = Yµ it will be enough to show that Yµ is irreducible, of the same dimension as O(µ). In order to do that we will construct a desingularization of Yµ . Let us consider the ﬂag variety Vµ = Flag(µ1 , µ1 + µ2 , . . . , µ1 + . . . + µs−1 ; E). We denote the typical point in Vµ by (Rµ1 , . . . , Rµ1 +...+µs−1 ), and we also write in that setup that Rµ1 +...+µs = E.

254

The Nilpotent Orbit Closures

Consider the incidence variety Z µ = {(φ, (Rµ1 , . . . , Rµ1 +...+µs−1 )) ∈ X × Vµ | φ(Rµ1 ) }. = 0, ∀2≤i≤s φ(Rµ1 +...+µi ) ⊂ Rµ1 +...+µi−1

Notice that the last conditions imply Rµ1 +...+µi ⊂ Ker (φ i ). Now we can consider the diagram Zµ ⊂ ↓ qµ Yµ ⊂

X × Vµ ↓ qµ X

We denote by pµ the projection of X × Vµ onto Vµ and its restriction to Z µ . It is clear that this makes Z µ a vector bundle on Vµ . Thus we have the exact sequence of vector bundles on Vµ 0 −→ Sµ −→ E ⊗ E ∗ −→ Tµ −→ 0, where E ⊗ E ∗ denotes a trivial bundle on Vµ with the ﬁber E ⊗ E ∗ , and Z µ is a total space of Sµ . The ﬁber of qµ over a point from O(µ) consists of one point. Indeed, if the pair (φ, (Rµ1 , . . . , Rµ1 +...+µs−1 )) ∈ Z µ and φ ∈ O(µ), we are forced to have Rµ1 +...+µi = Ker φ i . Since the variety Z µ is irreducible, we see that Yµ is irreducible and of the same dimension that O(µ). This completes the proof of Proposition (8.1.3). The construction of the desingularization Z µ places us in the situation of section 5.1. Thus we get a Koszul complex K(ξµ )• of O X ×Vµ -modules which is a Koszul complex resolving the structure sheaf O Z µ . The terms of K(ξµ )• are given by the formula K(ξµ )• : 0 →

t

( p ∗ ξµ ) → . . . →

2

( p ∗ ξµ ) → p ∗ (ξµ ) → O X ×Vµ

where ξµ = Tµ∗ . We also denote ηµ = Sµ∗ . For a vector bundle V on Vµ we denote by M µ (V) the sheaf O Z µ ⊗ p ∗ (V) of O X ×Vµ -modules. We also recall that by A we denote the polynomial ring Sym(E ⊗ E ∗ ) of regular functions on X . Applying Theorem (5.1.2) to our situation, we get (8.1.4) Basic Theorem for Nilpotent Orbits. For a vector bundle V on Vµ we deﬁne free graded A-modules i+ j µ j ξµ ⊗ V ⊗k A(−i − j). F (V)i = H Vµ , j≥0

8.1. The Closures of Conjugacy Classes of Nilpotent Matrices

255

(a) There exist minimal differentials µ

di (V) : F µ (V)i → F µ (V)i−1 of degree 0 such that F•µ (V) is a complex of graded free A-modules with H−i (F µ (V)• ) = Ri q∗ M(V). In particular the complex F µ (V)• is exact in positive degrees. (b) The sheaf Ri q∗ M µ (V) is equal to H i (Z , M µ (V)), and it can be also identiﬁed with the graded A-module H i (V, Sym(ηµ ) ⊗ V). (c) If φ : M µ (V) → M µ (V )(n) is a morphism of graded sheaves, then there exists a morphism of complexes f • (φ) : F µ (V)• → F µ (V )• (n). Its induced map H−i ( f • (φ)) can be identiﬁed with the induced map H i (Z , M µ (V)) → H i (Z , M µ (V ))(n). The calculation of cohomology groups H j (Vµ , i+ j ξµ ⊗ V) and H i (V, Sym(ηµ ) ⊗ V) is much more difﬁcult than in the case of determinantal varieties, or even in the case of hyperdeterminants, considered in chapter 9. The reason is that the bundle ξµ cannot be expressed conveniently as a tensor product of tautological bundles. In fact this calculation has not been done explicitly even for the case where V is trivial, i.e. for the case of syzygies of the coordinate ring of Yµ . This is a very interesting problem that should be a subject of further research. I believe the full solution is possible and should lead to very interesting combinatorics. In the remainder of this section we will describe the inductive procedure which allows to give the estimates on the terms of complexes F•µ in the case of nilpotent orbits. This will be the main tool in the next section when we describe the generators of their deﬁning ideals. Let us ﬁrst mention that the constructions of Z µ , ξµ , and ηµ can be done in a relative setting, i.e. when we replace the vector space E by the vector bundle E over some scheme. Therefore we can talk about the bundles ξµ and ηµ associated to the bundle E of dimension n. They are respectively a subbundle and a factorbundle of E ⊗ E ∗ . Since the bundles ξµ and ηµ do not change when we replace E by E ∗ , we will denote them by ξµ (E, E ∗ ) and ηµ (E, E ∗ ) respectively. ˆ the Now we ﬁx n and a partition µ = (µ1 , . . . , µr ). We denote by µ partition (µ1 − 1, . . . , µr − 1). The conjugate partition µˆ is the partition (µ2 , . . . , µs ).

256

The Nilpotent Orbit Closures

On the flag variety Vµ we have the tautological subbundles Rµ1 +...+µi and the corresponding factorbundles Qµi+1 +...+µs . The indices here denote the dimensions. We will denote Rµ1 by R and Qµ2 +...+µs by Q. In this setting we have an exact sequence of vector bundles on Vµ : 0 −→ R ⊗ E ∗ −→ ξµ −→ ξµˆ (Q, Q∗ ) −→ 0, where ξµˆ (Q, Q∗ ) denotes the bundle ξµˆ in the relative setting described above. For each t ≥ 0 this sequence induces a ﬁltration Fit , 0 ≤ i ≤ t, on t t t (ξµ ) such that for each i ≤ t − 1 Fit ⊂ Fi+1 and Fi+1 /Fit = t−i (R ⊗ E ∗ ) ⊗ i ξµˆ (Q, Q∗ ). Indeed, we can deﬁne F0 = 0 and, for each i ≤ t − 1, F i+1 to be the image of the map t−i

(R ⊗ E ∗ ) ⊗

i

ξµ →

s

ξµ

induced by exterior multiplication. This allows us to consider for 0 ≤ i ≤ t − 1 the exact sequences t 0 → Fit → Fi+1 →

t−i

(R ⊗ E ∗ ) ⊗

i

ξµˆ (Q, Q∗ ) → 0

(∗)

0≤i≤t

and the associated long sequences of cohomology groups. We will use these sequences to estimate the terms in the complex F•µ . We start with the introduction of the bundle ξµ = Rµ1 ⊗ E ∗ ⊕ (Rµ1 +µ2 /Rµ1 ) ⊗ Q∗n−µ ⊕ . . . ⊕ 1

(E/Rµ1 +...+µs−1 ) ⊗ Q∗µs The consecutive sequences of type (∗) deﬁne a ﬁltration on the bundle ξµ whose associated graded is the bundle ξµ . We will need a recursive form of (∗), which is ξµ = R ⊗ E ∗ ⊕ ξµˆ (Q, Q∗ ),

(∗∗)

where ξµˆ (Q, Q∗ ) is a bundle of type ξ constructed in a relative situation. We introduce GL(E)-modules µ

G i :=

i+ j H j Vµ , ξµ .

j≥0

It is clear from the long exact sequences of cohomology associated to se quences (∗) that the groups H j (Vµ , i+ j ξµ ) are smaller than the groups

8.1. The Closures of Conjugacy Classes of Nilpotent Matrices

257

H j (Vµ , i+ j ξµ ). Therefore our ﬁrst step will be to give a procedure to µ calculate the terms G i . It is based on the formula (∗∗), which implies •

ξµ =

• • (R ⊗ E ∗ ) ⊗ ξµˆ (Q, Q∗ ).

µ

The terms of G i can be calculated in the following way. Let us choose the term of G µ•ˆ (Q, Q∗ ) which is a representation of the type K α Q where α is an integral dominant weight for the group GL(n − µ1 ). Let us assume that this term occurs in homogeneous degree t and in homological degree u. For this term we calculate G(K α Q) =

H j (Grass(µ1 , E), K α Q) ⊗

i+ j

(R ⊗ E ∗ ),

(?)

i, j≥0

where the terms corresponding to a given i, j will appear in homogeneous degree t + i + j and in homological degree u + i. Notice that the terms of the collection G(K α Q) will appear as tensor products K β E ⊗ K γ E ∗ , but we can decompose them into irreducible representations to make the next step. Now all the collections G(K α Q) give us the terms of G µ• . Notice that the collection (?) comprises the terms of the complex F(V)• associated to the bundle ξ = R ⊗ E ∗ with a twist V = K α Q. These are the twisted complexes with the support in the determinantal variety of n × n matrices of rank ≤ n − µ1 . Such complexes were considered in section 6.5. These complexes are GL(E) × GL(E ∗ )-equivariant. Now we will use our procedure to estimate the weights of the terms of µ G • . First we notice that all terms in G µ• are the representations K β E where the sum of all entries in the weight β is equal to 0. To separate positive, zero, and negative entries of β, let us write β = (σ, 0c , \τ ) where σ and τ are two partitions. Here for τ = (τ1 , . . . , τt ) we let \τ = (−τt , . . . , −τ1 ). (8.1.5) Lemma. Let n and µ be as above. Let us consider the term K α Q of G µ•ˆ (Q, Q∗ ) where α is such that the sum of its entries equals 0 and its last entry is ≥ −µ1 . (a) We consider the twisted complex F(V)• associated to the bundle ξ = R ⊗ E ∗ , with the twist V = K α Q. Then F(V)i = 0 for i < 0. Moreover, F(V)0 = K σ E ⊗ K τ E ∗ ⊗ A(−|σ |) and F(V)1 = K σ,1c+1 E ⊗ K τ,1c+1 E ∗ ⊗ A(−|σ | − c − 1).

258

The Nilpotent Orbit Closures

All terms of F(V)• are of the form K β E ⊗ K γ E ∗ where β and γ are partitions of the same number, and γ1 ≤ µ1 . µ ˆ (b) Let us assume that the term K α Q occurs in G i (Q, Q∗ ). Then all terms µ from G(K α Q) occur in G j with j ≥ i. Proof. First of all, let us notice that part (b) follows instantly from part (a). We will prove part (a). Let us write in this proof t = µ1 for short. By the Cauchy decomposition (3.2.5) we have •

(R ⊗ E ∗ ) =

Kβ R ⊗ Kβ E ∗.

β

This means that for each partition β we have to calculate the cohomology groups of K α Q ⊗ K β R ⊗β E ∗ , which are the cohomology groups of K α Q ⊗ K β R tensored with K β E ∗ . To calculate this cohomology for each β = (β1 , . . . , βt ) we have to apply Bott’s theorem (4.1.4) to the sequence z(β) = (α, β) = (σ, 0c , \τ, β). Let us consider the weight z(β) + ρ. This is a sequence (σ1 + n − 1, . . . , σu + n − u, t + v + c − 1, . . . , t + v, t + v − 1 − τv , . . . , t − τ1 , β1 + t − 1, . . . βt ), where we write σ = (σ1 , . . . , σu ), τ = (τ1 . . . , τv ) and keep in mind that n = u + c + v + t. By (4.1.9) the partitions β giving nonzero contributions to cohomology are these for which the sequence z(β) + ρ has no repetitions. For such β we have to reorder our sequence to make it decreasing, and subtract ρ from it. We get a weight γ (β). The resulting cohomology group will then be K γ (β) E ⊗ K β E ∗ occurring in the complex F(V)• in the place p(β) = |β| − l(w) where w is a permutation needed to reorder z(β) + ρ. First we notice that since t ≥ τ1 then all the entries in z(β) + ρ are positive which means that all the entries of γ (β) are nonnegative. Let us consider two partitions: β and γ = (β1 , . . . , βs + j, βs+1 , . . . , βt ), both giving nonzero contributions to the terms of F• (V). We will show that p(γ ) > p(β). Indeed, the sequences z(β) + ρ and z(γ ) + ρ differ only in one place, and the term βs + j + t − s in z(γ ) + ρ can be exchanged with at most j − 1 additional numbers compared to the corresponding term βs + t − s in z(β) + ρ. This will account for an increase of at most j − 1 in l(w).

8.1. The Closures of Conjugacy Classes of Nilpotent Matrices

259

Starting with an arbitrary β we can now use the steps described above to produce terms with smaller β that lie in smaller homological degree. Continuing like this we can make β satisfy the condition c + v ≥ β1 . On the other hand if c + v ≥ β1 , then {t + v + c + 1, . . . , t + v, t + v − 1 − τv , . . . , t − τ1 , β1 + t − 1, . . . , βt } are t + v + c numbers belonging to {0, 1, . . . , t + v + c − 1}. They can be distinct for a unique partition β and it is easy to see that β = τ . It is also clear that β = τ is the smallest partition giving a nonzero contribution the terms of F(V)• . By the previous argument p(β) is the smallest for β = τ . It is very easy to check that in fact p(τ ) = 0. Similarly we can identify γ = (τ, 1c ) as the only partition with p(γ ) = 1. It remains to prove the last statement in (a). But this follows if we take into account that the term of F• (V) corresponding to β is K γ (β) E ⊗ K β E ∗ and that β1 ≤ t. (8.1.6) Theorem. Let µ be a partition of n. Let F•µ be a complex of A-modules described in (8.1.4). (a) (b) (c) (d) (e)

µ

The terms Fi are zero for i < 0. µ The term F0 equals A(0). For any representation K α E occurring in F•µ we have αn ≥ −µ1 . The varieties Yµ are normal and they have rational singularities. The coordinate rings of varieties Yµ are Gorenstein.

Proof. First we notice Theorem (5.1.3) (c) implies that (d) follows from (a) and (b). Also we notice that (e) follows from Theorem (5.1.4). The point is that the maximal exterior power of ξµ is isomorphic to the canonical bundle on Vµ , by Exercise 13, chapter 3. Therefore it is enough to prove the ﬁrst three statements of the theorem. We will actually deal with the spaces G µ• . We will prove the following statements: µ

(a ) The terms G i are zero for i < 0. µ (b ) The term G 0 equals k. (c ) For any representation K α E occurring in G µ• we have αn ≥ −µ1 . The statements for F•µ follow because the cohomology groups of exterior powers of ξµ are smaller than those of exterior powers of ξµ . We argue by induction on the number of parts in µ .

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If µ has only one part, i.e. µ = (n), then Yµ is the origin and ξµ = ξµ , µ so G i = i (E ⊗ E ∗ ). Then obviously (a ) and (b ) are satisﬁed, and (c ) follows from the Cauchy formula (2.3.3) (b) and the Littlewood–Richardson rule (2.3.4). Let us therefore assume that µ = (µ1 , . . . , µs ) and that all three statements are true for partitions with at most s − 1 parts. Let µˆ = (µ2 , . . . , µs ). We consider the terms of the complex G µ•ˆ (Q, Q∗ ). By the inductive assumption, all such terms K α Q occur in nonnegative homological degrees and they satisfy the assumption αn−µ1 ≥ −µ2 . This means each term satisﬁes the assumptions of Lemma (8.1.5). Now statement (a ) for µ follows from (8.1.5) (b). We also µ see that the contribution to G 0 can come only from the 0th term of G µ•ˆ (Q, Q∗ ). But this consists of one copy of the trivial representation. Therefore by (8.1.5) (a) statement (b ) is satisﬁed for µ. Finally (c ) also follows from the last part of (8.1.4) (a) and from the Littlewood–Richardson rule (2.3.4). In the sequel we will denote by Aµ the coordinate ring of Yµ . Theorem (8.1.6) implies that the complex F•µ is a minimal free resolution of Aµ treated as an A-module. It also says that the rings Aµ are normal, are Gorenstein, and have rational singularities. We will denote by Jµ the deﬁning ideal of Yµ . This means that Aµ = A/Jµ . Thus the ﬁrst term of F•µ gives us the information about the minimal generators of the ideal Jµ . Let us note the following corollary from our calculation of G µ• . µ

(8.1.7) Corollary. Let µ be a partition of n. The term G 1 contains only the representations K (1 j ,0n−2 j ,(−1) j ) E for 0 ≤ j ≤ n2 . Proof. Again we use the induction on the number of parts of µ . For µ having one part, again the result is true, because G i(n) = i (E ⊗ E ∗ ). To make an inductive step from µ ˆ to µ, we again use Lemma (8.1.5). We ﬁnish this section with some examples where the complete calculation of the complex F•µ is possible. (8.1.8) Examples. (a) Let µ = (n). Then Yµ = {0}. The bundle ξµ = E ⊗ E ∗ , and therefore Fi(n) =

i

(E ⊗ E ∗ ) ⊗ A(−i)

8.1. The Closures of Conjugacy Classes of Nilpotent Matrices

261

and F•(n) is the Koszul complex associated to the ideal generated by all variables φi, j . (b) Let µ = (1n ). The variety Y(1n ) is the set of all nilpotent matrices. We n will show that the complex F•(1 ) is the Koszul complex associated to the ideal generated by the basic invariants v1 , . . . , vn . This means that n (1n ) A − j j Fi = {(1 ,...,n ) | j ∈{0,1},

n

j=1

j =i}

j=1 n

) We show ﬁrst that the cohomology groups G (1 • consist of trivial representations only. We proceed by induction on n. For n = 1, the statement is obviously true. Let us assume the statement is true for µ = (1n−1 ). Using the formula for G(K α Q), we see that each trivial representation n−1 n which is a term of G •(1 ) (Q1 , Q∗1 ) contributes to G •(1 ) the terms

H

j

Grass(1, E),

i+ j

∗

(R ⊗ E ) ,

i, j≥0

where R is the tautological subbundle on Grass(1, E), i.e. the bundle O(−1). Now it follows from Serre’s theorem ([H1, chapter III, Theorem 5.1]) that only two cohomology groups in the above formula are nonzero. These are 0 0 ∗ H Grass(1, E), (R ⊗ E ) = K (0) E and

n H n−1 Grass(1, E), (R ⊗ E ∗ ) = K (0) E.

Therefore we get two copies of the trivial representation. Our staten ment is proved. It follows that the terms of the complex F•(1 ) consist entirely of trivial representations. This implies that the ideal of functions vanishing on Y(1n ) is generated by GL(E)-invariants, i.e. by v1 , . . . , vn (in view of the fact that B = AGL(E) is a polynomial ring in v1 , . . . , vn ). It remains to show that v1 , . . . , vn form a regular sequence in A. This follows from dimensional considerations because dim Y(1n ) = dim Z (1n ) = n 2 − n. (c) Let µ = (n − p, 1 p ) be a hook. The variety Yµ is the set of nilpotent matrices of rank ≤ p. We again use the formulas for G(K α Q). p ) ∗ Each term of G (1 • (Q, Q ) (which has to be a trivial representation)

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contributes

i+ j H j Grass(n − p, E), (R ⊗ E ∗ ) ,

i, j≥0

where R and Q denote respectively the tautological subbundle and factorbundle on Grass(n − p, E). This gives us the terms of the Lascoux complex resolving p + 1 order minors of φ, described in (6.1.3). Lemma (8.1.5) now shows that the generators of the deﬁning ideal of Yµ are a subrepresentation of a direct sum of p copies of trivial representations (in homogeneous degrees 1, . . . , p) and of the copy of p+1 E ⊗ p+1 E ∗ in homogeneous degree p + 1. On the other hand, we have some obvious polynomials vanishing on Yµ : the minors of degree p + 1 of the matrix φ, and the invariants v1 , . . . , v p . Moreover, these equations obviously deﬁne Yµ set-theoretically. If we consider the determinantal variety Y p of matrices of rank ≤ p, the dimension count shows that dim Y(n− p,1 p ) = dim Y p − p. This means that the invariants v1 , . . . , v p form a regular sequence in the coordinate ring of the determinantal variety Y p . Now we identify the generators of the deﬁning ideal of Yµ by induction on the homogeneous degree. In degrees 1, . . . , p the only generators are v1 , . . . , v p , because they have to come from trivial representations. In degree p + 1 the minors of degree p + 1 of φ have to be among the minimal generators (because no linear combinations of these minors with coefﬁcients in K can be in the ideal generated by v1 , . . . , v p ). Our estimate from above of the p ﬁrst term of F•(n− p,1 ) shows now that the deﬁning ideal of Y(n− p,1 p ) is generated by v1 , . . . , v p and by p + 1 order minors of φ. The complex p F•(n− p,1 ) being a minimal resolution of this ideal, has to be the tensor product of the Lascoux’s resolution (6.1.3) of the ideal of p + 1 order minors, and of the Koszul complex on v1 , . . . , v p . (d) Let µ = (n − p, p) be the partition with two columns. The variety Yµ consists of matrices φ such that rank φ ≤ p and φ 2 = 0. These varieties were ﬁrst considered by Strickland in [S], where they were called the projector varieties, because of the property φ 2 = 0. Here we just identify the homogeneous components of the coordinate ring of Yµ . We notice that the ﬂag variety Vµ we are using is in this case just Grass(n − p, E). It is easy to identify the bundle ηµ with Q ⊗ R∗ , where as usual R and Q denote respectively the tautological subbundle and factorbundle on Grass(n − p, E). Applying Theorem (8.1.4)(b) and Theorem (8.1.6), we see that F•µ is a minimal resolution of the coordinate ring of Yµ which can be identiﬁed with

8.2. The Equations of the Conjugacy Classes of Nilpotent Matrices 263

H 0 (Grass(n − p, E), Sym• (Q ⊗ R∗ )). Thus Cauchy’s formula (3.2.5) gives Symd (Q ⊗ R∗ ) = K α Q ⊗ K α R∗ . |α|=d

Using Corollary (4.1.9), we get that for α = (α1 , . . . , α p ) H 0 (Grass(n − p, E), K α Q ⊗ K α R∗ ) = K (α1 ,...,α p ,0n−2 p ,−α p ,...,−α1 ) E. The result is that (A(n− p, p) )d =

K (α1 ,...,α p ,0n−2 p ,−α p ,...,−α1 ) E.

|α|=d

This allows us to see that A(n− p, p) is a factor of A. Cauchy’s formula for A gives Ad = Kα E ⊗ Kα E ∗. |α|=d

We see now that if α = (α1 , . . . , αs ) with s ≥ p + 1, then the summand K α E ⊗ K α E ∗ is contained in J(n− p, p) . If s ≤ p, the only surviving part of K α E ⊗ K α E ∗ in A(n− p, p) is the Cartan representation K (α1 ,...,α p ,0n−2 p ,−α p ,...,−α1 ) E from K α E ⊗ K α E ∗ . The kernel of the epimorphism K α E ⊗ K α E ∗ → K (α1 ,...,α p ,0n−2 p ,−α p ,...,−α1 ) E consists of all polynomials having a trace component, i.e., combi n nations of polynomials divisible by ws,t = i=1 φi,s φt,i or by v1 =

n i=1 φi,i . Therefore the ideal J(n− p, p) is spanned by the quadratic polynomials of that kind, by the invariant v1 , and by the p + 1 order minors of φ.

8.2. The Equations of the Conjugacy Classes of Nilpotent Matrices We use the inductive procedure from the previous section to get the information about the generators of the deﬁning ideals Jµ of the varieties Yµ . We preserve the notation from the previous section. In particular, µ = (µ1 , . . . , µr ) is a partition of n, and µ ˆ = (µ1 − 1, . . . , µr − 1). For such a partition µ we denote the coordinate ring of Yµ by Aµ . Thus Aµ = A/Jµ . We start by exhibiting some explicit polynomials vanishing on Yµ . By Corollary (8.1.7) we know that the generators of the ideals Jµ consist of the representations of type K (1 j 0n−2 j ,(−1) j ) E for 0 ≤ j ≤ n2 . The representations of this type are exactly the ones occurring as composition factors of repre sentations p E ⊗ p E ∗ which are linear spans of p × p minors of the

264

The Nilpotent Orbit Closures

matrix φ. The straightening law (3.2.5) tells us that there is a basis of the polynomial ring A = Sym(E ⊗ E ∗ ) consisting of the products of minors of the generic n × n matrix φ = (φi, j )1≤i, j≤n . This gives the idea of looking for the polynomials vanishing on Yµ which are linear combinations of the minors of various sizes of φ. Let p be a number such that 1 ≤ p ≤ n, and let I = (i 1 , . . . , i p ) , J = ( j1 , . . . , j p ) be two multiindices with entries in the set [1, n]. We denote by φ I,J the p × p minor of φ corresponding to rows i 1 , . . . , i p and columns j1 , . . . , j p . The polynomial φ(I |J ) is obviously antisymmetric in the entries i 1 , . . . , i p and in the entries j1 , . . . , j p . We consider the linear span of p × p minors of φ. This is a linear supspace of A p , which can be identiﬁed with p E ⊗ p E ∗ . If {e1 , . . . , en } is a basis of E, and {e1∗ , . . . , en∗ } is a dual basis of E ∗ , the tensor ei1 ∧ . . . ∧ ei p ⊗ e∗j1 ∧ . . . ∧ e∗j p is identiﬁed with φ I,J . Using the Littlewood–Richardson rule (2.3.4), we see the following decomposition: p

E⊗

p

E∗ =

K (1i ,0n−2i ,(−1)i ) E.

0≤i≤ min( p,n− p)

We will denote the copy of the representation K (1i ,0n−2i ,(−1)i ) E inside the span p E ⊗ p E ∗ of p × p minors by Ui, p . If i > min( p, n − p), we set Ui, p = 0. This means we can rewrite the previous decomposition as p

p

E⊗

E∗ =

Ui, p .

0≤i≤ min( p,n− p)

Next, we denote by Vi, p the subspace of the span minors deﬁned as

p

E⊗

p

E ∗ of p × p

Vi, p = U0, p ⊕ U1, p ⊕ . . . ⊕ Ui, p . The point of introducing the spaces Vi, p is that they have a simple description in terms of minors of φ. The space Vi, p is isomorphic as a GL(E)-module to i E ⊗ i E ∗ , and it can be identiﬁed with the image of the map h i, p :

i

E⊗

m⊗m

E ∗ −→

p

i

1⊗1⊗t p−i

E ∗ −→

E⊗

p

E ∗, p−i

i

E⊗

i

E∗ ⊗

p−i

E⊗

p−i

where t p−i : K → p−i E ⊗ E ∗ is the map sending 1 to the GL(E)invariant t p−i , and m ⊗ m denotes exterior multiplication of the ﬁrst component with 3rd, and of the second with 4-th.

8.2. The Equations of the Conjugacy Classes of Nilpotent Matrices 265

Let us ﬁx two multiindices P = ( p1 , . . . , pi ) and Q = (q1 , . . . , qi ). We have φ(P,J |Q,J ) , h i, p (e p1 ∧ . . . ∧ e pi ⊗ eq∗1 ∧ . . . ∧ eq∗i ) = |J |= p−i

and so Vi, p is the span of such elements for different choices of P and Q. (8.2.1) Lemma. Let µ be a partition of n. The elements of the space Vi, p vanish identically on the variety Yµ if and only if p > µ1 + . . . + µi − i. Proof. Let us ﬁrst assume that the condition of Lemma (8.2.1) is satisﬁed. We will show that Vi, p vanishes on Yµ . Let us consider the typical generator φ(P,J |Q,J ) | j|= p−i

for ﬁxed subsets P and Q of cardinality i. Since these elements span a GL(E)stable subspace in A p , it is enough to show that they vanish on a single matrix from O(µ). Choosing a matrix from O(µ) in a canonical Jordan form, it is easy to see that in fact all summands φ(P,J |Q,J ) are zero when evaluated on that matrix. To prove the other implication, let us assume that the condition of (8.2.1) is not satisﬁed. This means that p ≤ µ1 + . . . + µi − i. Let us choose j for which µ j > 1 and µ j+1 = 1 (if µi > 1, we choose j = i). Now we set P = (1, µ1 + 1, . . . , µ1 + . . . + µ j + 1). We can choose the numbers w1 , . . . , w j in such way that 1 ≤ wm ≤ µm for m = 1, . . . , j, and w1 + . . . + w j = p + i. We set Q = (w1 , µ1 + w2 , . . . , µ1 + . . . + µ j−1 + w j ).

We consider the polynomial |J |= p− j φ(P,J |Q,J ) . Clearly its value on a matrix from O(µ) in Jordan canonical form is not zero. Moreover, our element is in V j, p , which is contained in Vi, p by deﬁnition. This completes the proof of the lemma. Our goal in this section is to prove that the representations Vi, p satisfying the condition (8.2.1) are the generators of the ideal Jµ . It is worthwhile to point out right away that they do not deﬁne a minimal set of generators of

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The Nilpotent Orbit Closures

the ideals Jµ . For example, the following result is a simple consequence of Laplace expansion. (8.2.2) Lemma. For i ≥ 1 the representation Vi, p+1 is contained in the ideal generated by the representation Vi, p . Let us denote by Jµ the ideal generated by the representations Vi, p satisfying the condition in (8.2.1). Lemma (8.2.2) immediately implies (8.2.3) Proposition. The ideal Jµ is generated by the spaces Ui,µ(i) (1 ≤ i ≤ n) where µ(i) = µ1 + . . . + µi − i + 1 (which are zero if i > min(µ(i), n − µ(i))), and by the spaces U0, p (for 1 ≤ p ≤ n) which correspond to the invariants t p . We will use the following graphical representation of the representations Ui, p . We will represent them by an (n + 1) × ([ n2 ] + 1) matrix whose rows are indexed by 0, 1, . . . , n and columns are indexed by 0, 1, . . . , [ n2 ]. The ( p, i)th entry corresponds to Ui, p . We will treat the entries corresponding to the spaces Ui, p which are zero as empty. For a given partition µ we will represent the generators of Jµ by a matrix M(µ) whose entry equals 1 if the corresponding Ui, p vanishes on Yµ , equals 0 if the corresponding Ui, p does not vanish on Yµ , and is empty if the corresponding Ui, p = 0. (8.2.4) Example. Let us take n = 12, µ = (3, 3, 2, 2, 1, 1). We have µ(1) = 3, µ(2) = 5, µ(3) = 6, µ(4) = µ(5) = µ(6) = 7. This means we have 0 1 1 1 1 1 M(µ) = 1 1 1 1 1 1 1

0 0 1 1 1 1 1 1 1 1 1

0 0 0 1 1 1 1 1 1

0 0 0 1 1 1 1

0 0 0 0 0 0. 1 1 1

8.2. The Equations of the Conjugacy Classes of Nilpotent Matrices 267

The fourth row of the matrix tells us that U0,3 and U1,3 are in Jµ and that U2,3 and U3,3 are not. Lemma (8.2.2) tells us that Jµ is generated by the Ui, p ’s corresponding to the 1’s in the ﬁrst column of M(µ) and by the ones corresponding to the highest 1’s in each other column. In our exam is generated by U0, p (1 ≤ p ≤ 12) and by ple we would conclude that J(6,4,2) U1,3 , U2,5 , U3,6 , U4,7 , U5,7 . Now we state the main result of this section. (8.2.5) Theorem. For each partition µ the ideals Jµ and Jµ are equal, i.e., Jµ is generated by the spaces Ui,µ(i) (for 1 ≤ i ≤ n ) and by the spaces U0, p (for 1 ≤ p ≤ n). (8.2.6) Remark. The set of generators given in (8.2.5) is not claimed to be minimal. We will discuss the minimal sets of generators at the end of this section. Proof of Theorem (8.2.5). We proceed by induction on the number s of parts in µ . If s = 1, we have µ = (n) and Y(1n ) = 0. Therefore the ideal Jµ is generated by the entries φi, j , i.e., by the representations U0,1 and U1,1 . The , so we combinatorial condition in (8.2.1) tells us that U0,1 and U1,1 are in J(n) are done. ˆ = Let us consider the partition µ = (µ1 , . . . , µr ). As before we denote µ (µ1 − 1, . . . , µr − 1). By induction we know that the generators of Jµˆ are (1 ≤ i ≤ n − µ1 ) and by the spaces U0, p (1 ≤ p ≤ given by the spaces Ui,µ(i) ˆ n − µ1 ). Let us consider the complex F•µˆ . We can construct this complex in a relative situation, taking the bundle Q on the Grassmannian Grass(µ1 , E) instead of a vector space E. We get a complex F•µˆ (Q, Q∗ ) of locally free sheaves over the sheaf of algebras A := Sym(Q ⊗ Q∗ ) deﬁned over the Grassmannian Grass(µ1 , E). The terms of this complex are the sheaves of type K α Q ⊗ A where α is a dominant integral weight for the group GL(n − µ1 ). Moreover, by (8.1.6)(c) we see that only the weights satisfying α1 ≤ n − µ1 occur. Let us ﬁx two terms of the complex F•µˆ (Q, Q∗ ): the term K α Q ⊗ A occurring in homological dimension t and the term K β Q ⊗ A occurring in homological degree t − 1. The component dα,β of the differential from the term K α Q ⊗ A to the term K β Q ⊗ A comes from the natural map K α Q → K β Q ⊗ S j (Q ⊗ Q∗ ). Let B denote the sheaf of algebras Sym(Q ⊗ E ∗ ) on Grass(µ1 , E). We can deﬁne the complex F˜ µˆ (Q, Q∗ )• of B-modules to be the complex with the same

268

The Nilpotent Orbit Closures

terms as F•µˆ (Q, Q∗ ) but replacing the differential dα,β with its composition with the natural embedding of K β Q ⊗ S j (Q ⊗ Q∗ ) into K β Q ⊗ S j (Q ⊗ E ∗ ). The complex F˜ µˆ (Q, Q∗ )• is acyclic. Indeed, the complex F•µˆ (Q, Q∗ ) is, and locally (on Grass(µ1 , E)) the complex F˜ µˆ (Q, Q∗ )• has the same differentials as F•µˆ (Q, Q∗ ), though it is a complex over a polynomial ring with some additional irrelevant variables. Let us identify the only homology group of F˜ µˆ (Q, Q∗ )• . We deﬁne the subvariety Wµ of X × Grass(µ1 , E) as follows. First of all Wµ ⊂ {(φ, R) ∈ X × Grass(µ1 , E) | φ| R = 0}. An endomorphism φ such that φ| R = 0 induces the morphism of bundles φ : Q → E over X × Grass(µ1 , E). We deﬁne φˆ : Q → Q to be the composition of φ with the natural epimorphism E → Q. Now we set Wµ = {(φ, R) ∈ X × Grass(µ1 , E) | φ| R = 0, φˆ ∈ Yµˆ (Q, Q∗ )}, where Yµˆ (Q, Q∗ ) is the variety Yµˆ constructed in relative situation. For the remainder of this section we will denote by p (by q) the projection from X × Grass(µ1 , E) onto Grass(µ1 , E) (onto X ). (8.2.7) Proposition. (a) The homology sheaf H0 ( F˜ µˆ (Q, Q∗ )• ) is equal to the direct image p∗ OWµ of the structure sheaf OWµ . (b) The higher direct images Ri q∗ OWµ are 0 for i > 0 and q∗ OWµ = OYµ . Proof. It is clear that Wµ is a reduced variety in X × Grass(µ1 , E). The calculation can be done locally on Grass(µ1 , E). The term F˜ µˆ (Q, Q∗ )0 equals B, which accounts for the condition φ R = 0. The reduced equations giving the condition φ ∈ Y µˆ (Q, Q∗ ) are given by images of the elements in the term F˜ µˆ (Q, Q∗ )1 . This proves part (a). The map p is afﬁne, so to prove part (b) it is enough to show that the cohomology groups H i (Grass(µ1 , E), p∗ OWµ ) are 0 for i > 0 and H 0 (Grass(µ1 , E), p∗ OWµ ) = OYµ . The complex F˜ µˆ (Q, Q∗ )• is an acyclic complex of locally free B-modules with the terms being direct sums of sheaves of the form K α ⊗ BQ. Let us consider the subvariety Z = Z µ1 ⊂ X × Grass(µ1 , E), which is a total space of the vector bundle Q∗ ⊗ E. In other words, Z = {(φ, R) ∈ X × Grass(µ1 , E) | φ| R = 0}.

8.2. The Equations of the Conjugacy Classes of Nilpotent Matrices 269

This is a desingularization of the determinantal variety of matrices φ of rank ≤ n − µ1 . We considered such varieties in section 6.1. Notice that by Proposition (5.1.1)(b) the sheaf of algebras B is just the direct image p∗ O Z . Therefore, each term K α Q ⊗ B is the direct image p∗ M(K α Q) of the corresponding twisted module M(K α Q) associated to the variety Z . Such modules were considered in section 6.5. We also notice that Lemma (8.1.5) says that for all modules M(K α Q) occurring in the complex F˜ µˆ (Q, Q∗ )• we have Ri q∗ (M(K α Q)) = 0 for i > 0. This proves the ﬁrst part of (b). To prove the second part of (b), let us calculate the graded Euler characteristic of F˜ µˆ (Q, Q∗ )• . By part (a) it is equal to the graded Hilbert function of q∗ OWµ . On the other hand, the graded Euler characteristic of F˜ µˆ (Q, Q∗ )• can be calculated in another way. By (8.1.6) we know that the graded Euler characteristic of OYµˆ can be calculated as the graded Euler characteristic of the complex F µˆ (Q, Q∗ ), i.e. as the Euler characteristic of the Koszul complex on the bundle p ∗ (ξµˆ ) on X × Vµ . Thus the Euler characteristic of OWµ can be calculated as the Euler characteristic of the Koszul complex on the bundle p ∗ (ξµˆ ⊕ (R ⊗ E ∗ )) on X × Vµ . But by the exact sequence 0 −→ R ⊗ E ∗ −→ ξµ −→ ξµˆ (Q, Q∗ ) −→ 0, this is the same as the graded Euler characteristic of F•µ , i.e., by (8.1.6), the graded Hilbert function of OYµ . This concludes the proof of part (b) of the lemma. Let us concentrate on two ﬁrst terms of the complex F˜ µˆ (Q, Q∗ )• . The term F˜ µˆ (Q, Q∗ )0 is just B. To identify the term F˜ µˆ (Q, Q∗ )1 , we notice that by the inductive assumption we can assume that the ideal Jµˆ is generated by the representations U0, p (1 ≤ p ≤ n ) and by the representations Ui,µ(i) (1 ≤ i ≤ n ), where we denote ˆ n := n − µ1 . This means that Jµˆ is minimally generated by a subset of these representations. Let us denote this subset by C. This means that F˜ µˆ (Q, Q∗ )1 = K (1i ,0n −2i ,(−1)i ) Q ⊗ B(− p). (i, p)∈C

The map ∂(i, p) , from the summand corresponding to Ui, p to F˜ µˆ (Q, Q∗ )0 , is induced by Ui, p . We consider the map q∗ (∂(i, p) ) : q∗ (M(K (1i ,0n −2i ,(−1)i ) Q)) → q∗ (M(K (0n ) Q)). To make our notation more transparent we will denote N0 = q∗ (M(K (0n ) Q)) and N(i, p) = q∗ (M(K (1i ,0n −2i ,(−1)i ) Q)).

270

The Nilpotent Orbit Closures

By Lemma (8.1.5) we know that the higher direct images Ri q∗ applied to the modules N0 and N(i, p) are 0 for i > 0. Moreover, we know that the module N0 is a coordinate ring of determinantal variety of matrices φ of rank ≤ n − µ1 . The module N(i, p) has the presentation i+c+1

E⊗

i+c+1

E ∗ ⊗ A(−i − c − 1) →

i

E⊗

i

E ∗ ⊗ A(−i)

→ N(i, p) → 0, where c = n − 2i. Since N0 is the coordinate ring of the determinantal variety, and its deﬁning ideal (generated by the (n + 1) × (n + 1) minors of φ) is clearly contained in Jµ . We denote the image of Jµ in N0 by Jµ . Theorem (8.2.5) clearly follows from the following lemma. (8.2.8) Lemma. Let (i, p) ∈ C. The image of the map q∗ (∂(i, p) ) is contained in the ideal Jµ . Proof. Let us consider two cases. Either (i, p) = (0, p) or (i, p) = (i, µ(i)). ˆ In the ﬁrst case the map ∂(0, p) is a GL(E)-equivariant map from N0 to itself. This means it has to be a multiplication by a GL(E)-invariant. Therefore its image is contained in Jµ . Let us assume that (i, p) = (i, µ(i)). ˆ According to the Theorem (5.1.2)(b) the modules N0 and N(i, p) can be identiﬁed as follows: H 0 (Grass(µ1 , E), Sd (Q ⊗ E ∗ )), N0 = d≥0

N(i, p) =

H 0 (Grass(µ1 , E), K (1i ,0n −2i ,(−1)i ) Q ⊗ Sd (Q ⊗ E ∗ )).

d≥0

We know that the module N(i, p) is generated by its component in homogeneous degree i. We calculate the action of ∂(i, p) on the generators of N(i, p) . The key statement is (8.2.9) Lemma. The map ∂(i, p) factors as follows: i E ⊗ i E∗ ↓ t p ⊗1 p E ⊗ p E∗ ⊗ i E ⊗ i E∗ ↓w p p ∗ i E⊗ E ⊗ E ⊗ i E∗ ↓ ζ p ⊗ζi Si+ p (E ⊗ E ∗ )/(Iµ1 +1 )i+ p ,

8.2. The Equations of the Conjugacy Classes of Nilpotent Matrices 271

where w is the identity on the components involving E ∗ tensored with a GL(E)-equivariant map on the components involving E, and ζ p ⊗ ζi is the product of maps coming from straightening formula. The composition of the last two maps in the composition (8.2.9) is a GL(E) × GL(E ∗ )-equivariant map. The rest of the proof of Lemma (8.2.8) (and thus of Theorem (8.2.5)) is based on the following idea. We exhibit an explicit set of generators of the group p p i i HomGL(E)×GL(E ∗ ) E⊗ E∗ ⊗ E⊗ E ∗, Si+ p (E ⊗ E ∗ )/(Iµ1 +1 )i+ p , and we show that the image of each of them composed with the map t p ⊗ 1 from (8.2.9) is in Jµ . The precise statements we need are (8.2.10) Lemma. The vector space p p i i E⊗ E∗ ⊗ E⊗ E ∗, HomGL(E)×GL(E ∗ ) ∗ Si+ p (E ⊗ E )/(Iµ1 +1 )i+ p is generated by the elements w j deﬁned as compositions p E ⊗ p E∗ ⊗ i E ⊗ i E∗ ↓ 1⊗1⊗ˆ p ∗ j p E⊗ E ⊗ E ⊗ j E ∗ ⊗ i− j E ⊗ i− j E ∗ ˆ ↓ m⊗1⊗1 p+ j E ⊗ p+ j E ∗ ⊗ i− j E ⊗ i− j E ∗ ↓ ζ p+ j ⊗ζi− j Si+ p (E ⊗ E ∗ )/(Iµ1 +1 )i+ p ˆ denotes the product of two for j satisfying 0 ≤ j ≤ i and p + j ≤ n. Here diagonal maps composed with the appropriate permutation of the factors. Similarly mˆ is the permutation of the factors composed with the product of two exterior multiplications. (8.2.11) Lemma. Let t p ⊗ 1 be the ﬁrst map in the composition (8.2.9). Let w j be the maps deﬁned in the statement of (8.2.10). Then for each j satisfying 0 ≤ j ≤ i, p + j ≤ n, the image of the composition w j (t p ⊗ 1) is contained in the ideal Jµ .

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The Nilpotent Orbit Closures

To conclude the proof of Theorem (8.2.5) it remains to show the statements (8.2.9), (8.2.10), (8.2.11). Proof of (8.2.10). We will show that the morphisms w j are a basis of p p i i ∗ ∗ ∗ ∗ HomGL(E)×GL(E ) E⊗ E ⊗ E⊗ E , Si+ p (E ⊗ E ) . Indeed, decomposing two sides into irreducibles by Pieri’s formula and by Cauchy’s formula, we see that the common irreducibles are L ( p+ j,i− j) E ⊗ L ( p+ j,i− j) E ∗ for each j satisfying 0 ≤ j ≤ i, p + j ≤ n. Each of these irreducibles occurs with multiplicity 1 in both modules. Therefore the basis of p p i i ∗ ∗ ∗ ∗ E⊗ E ⊗ E⊗ E , Si+ p (E ⊗ E ) HomGL(E)×GL(E ) is given by the following compositions v j : p

E⊗

p

E∗ ⊗

i

E⊗

i

pr

E ∗ →L ( p+ j,i− j) E ⊗ L ( p+ j,i− j)

incl

E ∗ →Si+ p (E ⊗ E ∗ ), where pr and incl denote respectively the GL(E) × GL(E ∗ )-equivariant projection and inclusion. Now it is a direct calculation that the matrix expressing the elements w j as combinations of v j ’s is triangular with nonzero diagonal entries. The lemma follows. Proof of (8.2.11). Let us consider the composition w j (t p ⊗ 1). The image is a combination of products of ( p + j) × ( p + j) and (i − j) × (i − j) minors of φ, with ( p + j) × ( p + j) minors containing the traces on p components. It is enough to show that each such combination of ( p + j) × ( p + j) minors is in Jµ . In order to establish this we need to check the condition of (8.2.1), which in this case reads p + j > µ1 + . . . + µ j − j. However, we know that our pair (i, p) corresponds to the generators of Jµˆ , so p = µ(i) ˆ =µ ˆ1 + ... + µ ˆ i − i + 1 and by deﬁnition of U(i, p) we have p ≥ i. We know also that µ ˆ k = µk − 1 as long as k ≤ µ1 . This has to happen for all i in question, because otherwise p + j ≥ i + j > µ1 and we are in Jµ anyway. Therefore we have p+ j =µ ˆ1 + ... + µ ˆi −i +1+ j > µ ˆ1 + ... + µ ˆi −i + j = µ1 + . . . + µi + j ≥ µ1 + . . . + µ j + j, and the condition above is proved.

8.2. The Equations of the Conjugacy Classes of Nilpotent Matrices 273

Proof of (8.2.9). It follows from the deﬁnition that the map ∂(i, p) is the map induced on the sections by the following composition of maps: K (1i ,0n −2i ,(−1)i ) Q ↓ i Q ⊗ i Q∗ ↓ i Q ⊗ i Q∗ ⊗ p−i Q ⊗ p−i Q∗ ↓ p Q ⊗ p Q∗ ↓ p Q ⊗ p E∗ ↓ B p = S p (Q ⊗ E ∗ ). Therefore the action of ∂(i, p) on the generators of N(i, p) is given by a composition i E ⊗ i E ∗ = H 0 (K (1i ,0n −2i ,(−1)i ) Q ⊗ i Q ⊗ i E ∗ ) ↓ H 0 (U ⊗1) p p ∗ i,ip 0 H ( Q⊗ Q ⊗ Q ⊗ i E ∗) ↓ H 0 (1⊗i⊗1⊗1) p p ∗ i 0 H ( Q⊗ E ⊗ Q ⊗ i E ∗) (∗) ↓ H 0 (ζ p ⊗ζi ) H 0 (Si+ p (Q ⊗ E ∗ )) ↓ Si+ p (E ⊗ E ∗ )/(Iµ1 +1 )i+ p . Here we denote by i the canonical inclusion of Q∗ into E ∗ , and by ζ p the embedding p F ⊗ p G → S p (F ⊗ G) (with F := E, G := E ∗ ) deﬁned in section 3.2. Let us notice that the last two maps in (∗) are GL(E) × GL(E ∗ )-equivariant. Let us describe more precisely the composition of the ﬁrst two maps in (∗). It comes from applying the functor H 0 to the composition i Q ⊗ i E∗ ↓ j1 ⊗1 K (1i ,0n −2i ,(−1)i ) Q ⊗ i Q ⊗ i E ∗ ↓ j2 ⊗1⊗1 i Q ⊗ i Q∗ ⊗ i Q ⊗ i E ∗ (∗∗) ∧t ⊗1⊗1 ↓ p−i p Q ⊗ p Q∗ ⊗ i Q ⊗ i E ∗ ↓ 1⊗i⊗1⊗1 p Q ⊗ p E ∗ ⊗ i Q ⊗ i E ∗, where j1 denotes and j2 denote the canonical inclusions.

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Let us look at the composition of all maps in (∗∗). We notice that all of them are identities on the factor i E ∗ appearing on the right hand side. This means our composition is a tensor product of a composition i ↓

Q j1

K (1i ,0n −2i ,(−1)i ) Q ⊗ i Q ↓ j ⊗1 i 2 Q ⊗ i Q∗ ⊗ i Q ↓ ∧t p−i ⊗1 p ∗ i p Q⊗ Q ⊗ Q ↓ 1⊗i⊗1 p Q ⊗ p E∗ ⊗ i Q tensored with the identity on i E ∗ . The map induced by the last composition by applying a functor H 0 is a GL(E)-equivariant map from i E = H 0 ( i Q) to H 0 ( p Q ⊗ p E ∗ ⊗ i Q). If p + i ≤ n , the latter group is p E ⊗ p E ∗ ⊗ i E. If p + i > n , the latter group is a factor of p E ⊗ p E ∗ ⊗ i E by the image of the map n +1

E⊗

p

∗

E ⊗

p+i−n −1

E→

p

E⊗

p

E∗ ⊗

i

E,

which is the identity on the second factor, and the composition n +1

E⊗ 1⊗m

E→ i

p+i−n −1

p

E⊗

p ⊗1

E→

i

E⊗

n +1− p

E⊗

p+i−n −1

E.

Let us consider the vector space HomGL(E) ( i E, p E ⊗ p E ∗ ⊗ E). We will exhibit an explicit basis of this vector space.

(8.2.12) Lemma. For all j satisfying 0 ≤ j ≤ i, p + j ≤ n we deﬁne the morphisms h j : i E → p E ⊗ p E ∗ ⊗ i E as the compositions

hj :

i

p t p ⊗1

E→

E⊗

p

E∗ ⊗

i

gj

E→

i

E⊗

p

E∗ ⊗

p

E,

8.2. The Equations of the Conjugacy Classes of Nilpotent Matrices 275

where g j is the identity on the second factor tensored with the map g j deﬁned as a composition p E⊗ iE ↓ 1⊗ p E ⊗ j E ⊗ i− j E ↓ m⊗1 p+ j E ⊗ i− j E ↓ ⊗1 p E ⊗ j E ⊗ i− j E ↓ 1⊗m p E ⊗ i E. Then the maps h j form a basis of HomGL(E) ( i E, p E ⊗ p E ∗ ⊗ i E). Proof. The vector space HomGL(E) ( i E, p E ⊗ p E ∗ ⊗ i E) is canonp i p i E⊗ E, E⊗ E). The identiically isomorphic with HomGL(E) ( p ﬁcation is done by associating to a morphism f ∈ HomGL(E) ( E ⊗ i E, i p E⊗ E) the composition h:

i

p t p ⊗1

E→

E⊗

p

E∗ ⊗

i

g

E→

p

E⊗

p

E∗ ⊗

i

E,

where g is the identity on the second factor tensored with the map f . It is therefore enough to show that the maps g j given in (8.2.12) form a basis of HomGL(E) ( p E ⊗ i E, p E ⊗ i E). This is however clear. p i Indeed, decomposing E⊗ E into irreducibles L ( p+ j,i− j) E, we see that the natural basis of p p i i HomGL(E) ( E ⊗ E, E⊗ E)

consists of the morphisms uj :

p

E⊗

i

pr

p incl

E →L ( p+ j,i− j) E →

E⊗

i

E

where pr and incl denote respectively the GL(E)-equivariant projection and inclusion, and j satisﬁes conditions 0 ≤ j ≤ i, p + j ≤ n. Now it is clear that the transition matrix expressing g j ’s as combinations of u j ’s is triangular with the nonzero entries on the diagonal. This concludes the proof of (8.2.12). Now we come back to the map h we get when applying the functor H 0 to the composition of the maps in (∗∗). Lemma (8.2.12) shows that the map h

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can be written as a linear combination of maps h j . This means that it can be written as a composition h:

i

p t p ⊗1

E→

E⊗

p

E∗ ⊗

i

h

E→

p

E⊗

p

E∗ ⊗

i

E,

where h is the identity on the second factor tensored with the GL(E)equivariant map on the remaining two factors. Since the composition of the ﬁrst two maps in (∗) is the map h tensored with the identity on the remaining factor i E ∗ , we can write it in the form (8.2.9) as required. This concludes the proof of (8.2.9) and therefore the proof of Theorem (8.2.5).• Let us illustrate the inductive step in the proof of Theorem (8.2.5) with the following example. (8.2.13) Example. Let us consider µ = (6, 4, 2). Then µˆ = (4, 2). We obviously have dim E = 12. The inductive step consists of pushing down from the Grassmannian Grass(6, E) the complex of sheaves F˜ µˆ (Q, Q∗ )• where Q is a tautological factorbundle on Grass(6, E). The generators of J(4,2) can be described as follows: 0 1 1 M((4, 2)) = 1 1 1 1

0 1 0 1 1 1. 1 1 1

This means that J(4,2) is generated by the invariants U0, p (1 ≤ p ≤ 3) and by the representations U1,2 , U2,3 , and U3,3 . The ﬁrst two terms of F˜ µˆ (Q, Q∗ )• are easy to describe. The term F˜ µˆ (Q, Q∗ )0 equals B = Sym(Q ⊗ E ∗ ), and F˜ µˆ (Q, Q∗ )1 is a direct sum of the six terms B ⊗ K (1i ,06−2i ,(−1)i ) Q corresponding to the six representations Ui, p listed. We see ﬁrst of all that the direct image q∗ (B) = A/I7 . This tells us that 7 × 7 minors of the generic matrix φ are in the ideal J(6,4,2) . The invariants U0, p lead to the generators that are the invariants of degrees 1, 2, 3 in A/I7 . The remaining three Ui, p ’s give the following representations. The term U1,2 gives a representation E ⊗ E ∗ in degree 3. The term U2,3 gives a rep resentation 2 E ⊗ 2 E ∗ in degree 5. The term U3,3 gives a representation 3 ∗ 3 E⊗ E in degree 6. The proof above shows that these three representations have to be contained in the ideal generated by I7 , the invariants U0, p

8.2. The Equations of the Conjugacy Classes of Nilpotent Matrices 277

(1 ≤ p ≤ 6), and the representations U1,3 , U2,5 , and U3,6 . In fact one would expect them to be respectively V1,3 , V2,5 , and V3,6 . The conclusion in any case . This follows from Example (8.2.4). is that all generators have to be in J(6,4,2) (8.2.14) Remark. In the above proof of Theorem (8.2.5) we did not identify precisely the generators coming from the terms Ui, p occurring in F˜ µˆ (Q, Q∗ )1 . The reader might worry that we did not show that some of them are dependent on others. However, this is not a concern. We are assured by the fact that the only homology of F˜ µˆ (Q, Q∗ )• is OWµ and by Proposition (8.2.7) that the factor of A/Iµ1 +1 by the images of the terms coming from F˜ µˆ (Q, Q∗ )1 is A/Jµ . Therefore all elements of Jµ have to be contained in that image. The only concern is whether we get some generators not contained in Jµ . We ﬁnish this section with an example related to rectangular partitions. (8.2.15) Proposition. Let n = r e and let us consider the rectangular partition µ = (er ). Then the ideal J(er ) is generated by the invariants v1 , . . . , ve−1 and by the entries of the matrix φ e . These polynomials form a minimal set of generators of J(er ) . Proof. Let us use the inductive procedure used in the proof of (8.2.5) for the ˆ is just ((e − 1)r ). Thus we can family of partitions µ = (er ). The partition µ assume by induction that (8.2.15) is true for µ. ˆ The inductive assumption means that the ﬁrst term of F•µˆ consists of the trivial representations in homogeneous degrees 1, . . . , e − 1 and the representation K 1,0,...,0,−1 Q in homogeneous degree e − 1. This means that the µ possible terms in F1 are the trivial representations in homogeneous degrees 1, . . . , e − 1, the representation E ∗ ⊗ E in the degree e and the terms coming from n − µ1 + 1 = n − r + 1 size minors of the matrix φ. The generators in degrees ≤ e have to consist of invariants v1 , . . . , ve−1 and the vanishing of µ entries of φ e , because these terms have to occur in F1 and they match the terms we described. Thus the induction implies that the minimal generators of J (µ) are those listed in (8.2.15) plus possibly some linear combinations of (n − r + 1) × (n − r + 1) minors of φ. However recall that by (8.2.5) the ideal J (µ) is generated (nonminimally) by the invariants U0, p (1 ≤ p ≤ n) and by the representations Ui,ie−i+1 (for 1 ≤ i ≤ r ). The only possible generator in degree n − r + 1 on the above list is the representation Ur,n−r +1 which does not occur in the span of n − r + 1 minors of . This concludes the proof.

278

The Nilpotent Orbit Closures

A slightly more general result is proved in [W7]. Let e < n and let us consider the division of n by e with the remainder, n = r e + f with 0 ≤ f ≤ e − 1. We deﬁne the partition µ(n, e) = (er , f ). Then the ideal Jµ(n,e) is generated by the invariants v1 , . . . , ve−1 and by the entries of the matrix φ e . These polynomials form a minimal set of generators of Jµ(n,e) .

8.3. The Nilpotent Orbits for Other Simple Groups In this section we investigate the closures of conjugacy classes for other simple groups. We give their explicit desingularization by the collapsing of a homogeneous bundle. We sketch the proof that the normalizations of the closures of conjugacy classes have rational singularities. We use the approach of Broer [Br5]. Let G be a simple algebraic group with Lie algebra g. The group G acts on g by conjugation. We will still denote this action as a left action. Let e be a nilpotent element in g. We denote by Ge its orbit under the conjugation action, and by Ge its closure in g. By the Jacobson–Morozov lemma ([Bou, chapter VIII, section 11]) there exist elements h, f ∈ g such that {e, h, f } forms an sl2 -triple in g, i.e. [h, e] = 2e,

[h, f ] = −2 f,

[e, f ] = h.

If {e, h , f } forms another sl2 -triple, then there exists g ∈ G (and even in the connected component of the identity, CG (e)0 , of the centralizer of e), centralizing e, such that g f = f , gh = h . Let us ﬁx e and the sl2 -triple {e, h, f }. Let V be a ﬁnite dimensional g-module. The action of h induces a grading V = Vi , Vi = {v ∈ V | h . v = iv}. i∈Z

The associated ﬁltration . . . ⊂ V≥i+1 ⊂ V≥i ⊂ . . . does not depend on the choice of h, f . Let us consider V = g. Then e ∈ g 2 , h ∈ g 0 , f ∈ g −2 . We also clearly have [g i , g j ] ⊂ g i+ j . This means that g ≥0 is a Lie algebra of a parabolic group P ⊂ G with a nilpotent radical n = g >0 and the Levi factor g 0 . The subgroup P depends only on the ﬁltration and therefore does not depend on the choice of f, h. We consider the subspace g ≥2 of g. This is clearly a P-submodule of g. We can consider an induced homogeneous vector bundle Z = G ×P g ≥2 over

8.3. The Nilpotent Orbits for Other Simple Groups

279

G/P. Let us take X = g, V = G/P. Then Z ⊂ X × V . We can consider the diagram Z ⊂ ↓ q Y ⊂

X×V ↓q X

where Y = Ge and q is the restriction of the ﬁrst projection q. The map q sends the coset of the pair (g, x) to gx. In this situation we have (8.3.1) Proposition ([KP2], section 7.4). The map q makes Z a resolution of singularities of Y . Proof. Let us recall that the irreducible sl2 -modules are just the symmetric powers Sd F where F = K2 . The Lie algebra sl2 = sl(F) acts by the derivative action of the conjugation action, i.e. the commutator action. The tangent space at e to Pe is therefore the commutator space [g ≥0 , e] = g ≥2 . It follows that Pe is an open set in g ≥2 . Since the group P is uniqely determined by e, we have P = gPg −1 for each g in the stabilizer Ge . Since each parabolic subgroup equals its normalizer ([Hu2], Corollary B, section 23), we have Ge = Pe . This means that q restricted to the open orbit G(1, e) is an isomorphism, and therefore q is a birational map. The variety Z is nonsingular, because it is a vector bundle over G/P. (8.3.2) Example. Consider the special linear group G = SL(n) = SL(E), for E = Kn . The Lie algebra g of G is the set of n × n matrices of trace 0. We identify the set of n × n matrices with HomK (E, E) = E ∗ ⊗ E. Let us start with the nilpotent element e, which in canonical Jordan form has one Jordan block of size n. Then we can choose a basis e1 , . . . , en of E such that e(e1 ) = 0 and e(ei ) = ei−1 for i = 2, . . . , n. Then the element f can be chosen as follows: f (ei ) = ei+1 for i = 1, . . . , n − 1, f (en ) = 0. The element h can be chosen to be the diagonal matrix h(ei ) = (n + 1 − 2i)ei . We introduce the grading on E by letting deg(ei ) = n + 1 − 2i. This induces a grading on HomK (E, E) = E ∗ ⊗ E deﬁned by letting deg(ei∗ ⊗ e j ) = deg(e j ) − deg(ei ). The module g ≥2 consists then of all upper triangular matrices, and therefore the desingularization constructed above is the same as one constructed in section 8.1. Let µ = (µ1 , . . . , µr ) be a partition of n. Let us consider the nilpotent element e from the nilpotent orbit O(µ). We choose a basis e1 , . . . , en of E in such way that e(ei ) = 0 if i ∈ {1, µ1 + 1, µ1 + µ2 + 1, . . .}, e(ei ) = ei−1

otherwise.

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The Nilpotent Orbit Closures

This means that for each j = 1, . . . , r the span of eµ1 +...+µ j−1 +1 , . . . , eµ1 +...+µ j is the Jordan block of size µ j . We can choose f to act as follows: f (ei ) = 0 if i = µ1 + . . . + µ j ,

f (ei ) = ei+1

otherwise.

The element h is a diagonal matrix acting in the j-th block by h(eµ1 +...+µ j−1 +i ) = (µ j + 1 − 2i)ei

for i = 1, . . . , µ j .

Let us introduce the grading on E by letting deg(eµ1 +...+µ j−1 +i ) = µ j + 1 − 2i. The grading induced on HomK (E, E) is then given by identifying HomK (E, E) with E ∗ ⊗ E and letting deg(ei∗ ⊗ e j ) = deg(e j ) − deg(ei ). This induces a grading on g. This allows us to deﬁne the module g ≥2 . Notice that the desigularization given by g ≥2 is almost never the same as the one considered in the section 8.1. To see this we consider n = 3 and µ = (2, 1). Then the degrees of ei are as follows: deg(e1 ) = 1, deg(e2 ) = −1, deg(e3 ) = 0. Reordering our basis, we can assume that deg(e1 ) = 1, deg(e2 ) = 0, deg(e3 ) = −1. We see that the module g ≥2 is spanned by e3∗ ⊗ e1 . This is a B-submodule for the group B of upper triangiular matrices. The parabolic subgroup P deﬁned above is obviously equal to B. The desingularization G ×P g ≥2 we deﬁned above is a line bundle over G/B. Let us denote this desingularization by Z 1 . Identifying G/B with the set of flags R1 ⊂ R2 ⊂ E, we can identify our desingularization Z 1 with the set of pairs Z 1 = {(φ, R1 , R2 ) ∈ HomK (E, E) × G/B | φ(R2 ) = 0, φ(E) ⊂ R1 }. The desingularization Z 2 constructed in section 8.1 was a two dimensional bundle over the Grassmannian Grass(2, E). It was deﬁned as the set of pairs Z 2 = {(φ, R2 ) ∈ HomK (E, E) × Grass(2, E) | φ(R2 ) = 0, φ(E) ⊂ R2 }. To see that these two desingularizations are different, let us observe that the map that forgets R1 deﬁnes a regular map Z 1 → Z 2 . This map is not an isomorphism, because its ﬁber over a point (φ, R2 ) such that φ = 0 is clearly isomorphic to P1 . Therefore the desingularizations are different.• The main result of this section is the following theorem: (8.3.3) Theorem (Hinich [Hi], Panyushev [Pa2]). Let G be a simple group with the Lie algebra g. Let e be a nilpotent element in g. We consider the desingularization Z of the closure Y of the orbit Ge constructed above. (a) For all i > 0 we have H i (Z , O Z ) = 0. (b) The normalization of Y is a Gorenstein variety with rational singularities.

8.3. The Nilpotent Orbits for Other Simple Groups

281

Before we prove (8.3.3) we need some preparatory statements. (8.3.4) Proposition. Let g be a simple Lie algebra with the bracket [ , ]. Then all adjoint orbits in g have even dimensions. Proof. We denote by ( , ) the Killing form on g. Every element z ∈ g deﬁnes an antisymmetric form ( , )z on g given by (x, y)z := (z, [x, y]). If x is in the radical of ( , )z , then for all y ∈ g we have (z, [y, x]) = ([z, x], y) = 0. Since the Killing form is nondegenerate, we have [z, x] = 0. This means that the radical of ( , )z equals the centralizer g z . The induced antisymmetric form on g/g z is nondegenerate, and therefore dim g/g z is even. However, dim Gz = dim g/g z . (8.3.5) Lemma. Let g and e be as above. Let us consider the grading g = i∈Z g i induced by e, and the associated parabolic subgroup P. We denote b = dim g ≥2 , c := dim g 1 , d = dim G/P. Then: (a) The number c is even. (b) There is a nonzero P-equivariant map s : g ≥2 →

d

(g/ p)∗ ⊗

b

g ∗≥2 .

(c) s(ux) = u c/2 s(x) for all u ∈ K, x ∈ g ≥2 . Proof. Let z ∈ g 2 . The form ( , )z deﬁned in the proof of (8.3.4) restricts to a form on g −1 , which can be identiﬁed with an element wz ∈ 2 g ∗−1 . Consider the Levi decomposition P = LPu of the parabolic group P. The linear map w• : g 2 →

2

g ∗−1

sending z to wz is L-equivariant. For any z ∈ Le ⊂ g 2 we have g z ⊂ p = g ≥0 . From the proof of (8.3.4) it follows that wz is nondegenerate. This means that ∧c/2 c is even and the top exterior power wz ∈ c g ∗−1 does not vanish. The map s : g2 → ∧c/2

c

g ∗−1

sending z into wz is an L-equivariant polynomial map of homogeneous degree 2c . It does not vanish on the open orbit Le. We extend the L-modules

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The Nilpotent Orbit Closures

c ∗ g 2 and c g ∗−1 to P-modules g˜ 2 and % g −1 by the trivial action of Pu . We have the obvious isomorphisms of P-modules g ≥2 /g >2 / g˜2 ,

c% d b g ∗−1 / (g/ p)∗ ⊗ g ∗≥2 .

We deﬁne s as a composition s

s : g ≥2 → g ≥2 /g >2 / g˜ 2 −→

c% d b g ∗−1 / (g/ p)∗ ⊗ g ∗≥2 .

Then s clearly satisﬁes (b) and (c). We need one more statement. (8.3.6) Proposition. Let P = LPu be a Levi decomposition of a parabolic subgroup in G. We consider a P-submodule m of n := Lie(Pu ). We can construct as above a homogeneous bundle Z = G ×P m over G/P, which projects onto Y = Gm. Let us assume that dim Z = dim Y . We consider a one dimensional P-module u and the associated line bundle L := G ×P u on G/P. Let p denote as usual the projection of Z onto G/P. Assume that the twisted sheaf O Z ⊗ p ∗ (L) has a global G-invariant section s0 . The section s0 induces the morphism of s˜0 : O Z -modules O Z → O Z ⊗ p ∗ (L) given locally by sending a section f to f s0 . The morphism s˜0 then induces the isomorphism *(O Z ) / *(O Z ⊗ p ∗ (L)). Proof. Let s be a section of O Z ⊗ p ∗ (L). Then s = f s0 where f is a rational function on Z with poles only along the zeros of s0 . We know that Y := Spec K[Z ] is normal because it has an open orbit whose complement has codimension ≥ 2, and the morphism Z → Y is birational. Therefore f can be considered as a rational function on Y without poles on the open orbit. This means f has no poles on Y , because on the normal variety the set of poles of a function has codimension 1. Thus f is a regular function on Y . Since by deﬁnition of Y the regular functions on Y and Z are the same, f can be treated as a regular function on Z . We conclude that the map *(˜s0 ) : *(O Z ) → *(O Z ⊗ p ∗ (L)) is surjective. This map is obviously injective, which ﬁnishes the proof. Proof of Theorem (8.3.3). Let G, e, P be as in the statement of the theorem. We denote d = dim G/P, c = dim g −1 and b = dim g ≥2 . We take

8.4. Conjugacy Classes for the Orthogonal Group

283

u := d (g/ p)∗ ⊗ b g ∗≥2 . Consider the associated line bundle L = G ×P u on G/P. There exists m ∈ Z such that the canonical sheaf ω Z is isomorphic to the line bundle O Z ⊗ p ∗ (L)(m) with the grading shifted by m. By Lemma (8.3.5) there exists a P-equivariant map s : g ≥2 → u of degree 2c . This means that the sheaf O Z ⊗ p ∗ (L) has a nonzero global G-invariant section of degree 2c (the image of a constant section of g ≥2 ). This implies that the sheaf ω Z has a global G-invariant section s0 . Now Proposition (8.3.6) gives a morphism of sheaves s˜0 : O Z → ω Z (−m + 2c ) which induces an isomorphism on global sections. Applying Proposition (1.2.32), we get that Spec K[Z ] is a Gorenstein variety with rational singularities. We know however that Spec K[Z ] is the normalization of the closure Y of the orbit Ge. This concludes the proof of the theorem. (8.3.7) Remarks. Broer in a series of beautiful papers [Br1, Br2, Br3] applied the geometric method to deal with the twisted modules supported in nilpotent orbit closures. He also applied the method to decide normality of several nilpotent orbit closures for the exceptional groups. Kraft ([Kr2]) classiﬁed normal orbit closures for the groups of type G 2 . Recently the normal nilpotent orbit closures were classiﬁed for groups of type F4 (Broer, [Br6]) and of type E 6 (Sommers).

8.4. Conjugacy Classes for the Orthogonal Group In this section F denotes a vector space of dimension n with a nondegenerate symmetric form ( , ). The special orthogonal group SO(F) is the set of linear automorphisms of F preserving ( , ), i.e. φ ∈ SO(F) if and only if for each x, y ∈ F we have (φ(x), φ(y)) = (x, y). By deﬁnition SO(F) is a subgroup of GL(F). The corresponding Lie algebra so(F) is a subalgebra of the Lie algebra gl(F). The morphism φ ∈ HomK (F, F) is in so(F) if and only if for each x, y ∈ F we have (φ(x), y) + (x, φ(y)) = 0. Let us choose the hyperbolic basis e1 , . . . , em , e¯ m , . . . , e¯ 1 of F (in the case of n = 2m), and e1 , . . . , em , e0 , e¯ m , . . . , e¯ 1 . This means (ei , e j ) = (¯ei , e¯ j ) = 0

(ei , e¯ j ) = δi, j ,

(e0 , ei ) = (e0 , e¯ i ) = 0,

(e0 , e0 ) = 1,

where δi, j denotes the Kronecker delta.

284

The Nilpotent Orbit Closures

We can write φ as a matrix, writing in consecutive columns the images of vectors −¯e1 , . . . , −¯em , e0 , em , . . . , e1 expanded in the basis e1 , . . . , em , e0 , e¯ m , . . . , e¯ 1 . Then φ ∈ so(F) if and only if the matrix of φ is skew symmetric. This allows us to identify the adjoint representation of so(F) with 2 F. Since so(F) is a Lie subalgebra of gl(F), we might expect that the nilpotent conjugacy classes in so(F) will be related to intersections of the nilpotent conjugacy classes in gl(F) with so(F). The following result is proved in [SS; IV. 2.15]. (8.4.1) Proposition. Let µ be a partition of n. Let us consider the nilpotent conjugacy class O(µ) of gl(F) corresponding to µ. Then O(µ) intersects so(F) if and only if every even part of µ occurs even number of times. In such case the intersection O(µ) ∩ so(F) consists of a single conjugacy class of SO(F). For the remainder of this section we denote Po (n) the set of partitions µ of n in which every even part occurs even number of times. For µ ∈ Po (n) we denote by C(µ) the corresponding conjugacy class in so(F), and by Yµ its closure. In order to list the representatives of conjugacy classes for the orthogonal group, we need to exhibit the blocks corresponding to odd rows in our partition and the blocks corresponding to the pairs of even rows. The canonical forms of nilpotent orthogonal endomorphisms corresponding to both kinds of blocks are as follows. The nilpotent e corresponding to the partition (n) = (2m + 1) has the form e(¯e1 ) = 0,

e(¯ei ) = e¯ i−1

for i > 0,

e(en ) = e0 ,

e(e0 ) = −¯en , e(ei ) = −ei+1

for i < n

We can express the action of e by the sequence of arrows ±e1 → ∓e2 → . . . → en−1 → −en → e0 → e¯ n → e¯ n−1 → . . . → e¯ 2 → e¯ 1 → 0. If we extend e to an sl2 triple {e, h, f }, the grading induced by e described in section 8.3 is as follows. For the endomorphism e corresponding to the partition 2n we have h(ei ) = (2n + 1 − 2i)ei , h(¯ei ) = (2i − 1 − 2n)¯ei . Therefore the degrees of the basis vectors are deg ei = 2n + 1 − 2i, deg e¯ i = 2i − 1 − 2n.

8.4. Conjugacy Classes for the Orthogonal Group

285

If n = 2t is even, then the nilpotent e corresponding to the partition (n, n) has the form e(ei ) = ei+2

for i < 2t,

e(¯ei ) = −¯ei−2

e(e2t ) = e¯ 2t+1 ,

for i > 2,

e(e2t+1 ) = e¯ 2t ,

e(¯e2 ) = e(¯e1 ) = 0.

We can express the action of e by two sequences of arrows e1 → e3 → . . . → e2t+1 → e¯ 2t → −¯e2t−2 → . . . → ±¯e2 → 0, e2 → e4 → . . . → e2t → e¯ 2t+1 → −¯e2t−1 → . . . → ∓¯e1 → 0. Extending e to the sl2 -triple {e, h, f }, we see that the element h acts as follows: h(e2i+1 ) = 2(t − i)e2i+1 , h(e2i ) = 2(t + 1 − i)e2i , and h(¯e2i+1 ) = −2(t − i)¯e2i+1 , h(¯e2i ) = −2(t + 1 − i)¯e2i . This gives deg e2i+1 = 2(t − i), deg e2i = 2(t + 1 − i), and deg e¯ 2i+1 = − 2(t − i), deg (¯e2i ) = − 2(t + 1 − i). In this setup the degrees of vectors ei are always nonnegative and we always have deg ei = −deg e¯ i , because h ∈ so(F). If a partition µ corresponds to the conjugacy class in so(F), then we can assign the grading separately in each block, as in Example (8.3.2). We also use the convention that when dealing with several blocks, after assigning the grading, we order ei ’s in such a way that deg ei ≥ deg ei+1 . (8.4.2) Example. Let us take n = 8, µ = (3, 2, 2, 1). The grading of basis elements is as follows: 2 0 −2 1 −1 1 −1 0 We order the elements e1 , . . . e4 , e¯ 4 , . . . , e¯ 1 so their grading is nonincreasing. We get deg e1 = deg e2 = 2, deg e3 = 1, deg e4 = deg e5 = 0, with deg e¯ i = −deg ei for i = 1, . . . , 5. This allows us to determine the grading on so(F) in all cases. Identifying the adjoint representation with 2 F, we can arrange the weight vectors in it in a triangular grid. In order to describe it, let us introduce the involution ()¯ of our symplectic basis by requiring that e¯i = ei . The elements of the grid correspond to the entries above the diagonal of our matrix representation of φ. If u, v are the elements of our symplectic basis, then the entry corresponding to the row u and the column ±v will correspond to the weight vector u v¯ .

286

The Nilpotent Orbit Closures

For a given conjugacy class C(µ) we will mark the element of the grid with X if the corresponding weight vector is in g ≥2 , and with O otherwise. We will denote this grid by GC(µ). (8.4.3) Example. Let n = 5, µ = (3, 2, 2, 1). Then the degrees of the elements ei are given in Example (8.4.2) and we have X

X X

X O O

GC(µ) =

X O O O

O O O O O

O O O O O O

O O O O. O O O

We continue with several examples of conjugacy classes. (8.4.4) Example. Consider the class C((2m )) for even n = 2m. This is a class analogous to the previous example. The associated grading gives deg ei = 1, deg e¯ i = −1 for i = 1, . . . , m. This means that in the grid GC((2m )) the entries marked by X correspond to the vectors ei e j for i ≤ j. For example, if n = 10 we get X

GC((25 )) =

X X

X X X

X X X X

O O O O O

O O O O O O

O O O O O O O

O O O O O O O O

O O O O O. O O O O

The parabolic subgroup P is the set of elements ﬁxing a given isotropic subspace of dimension n. The homogeneous space G/P is the connected component of the isotropic Grassmannian IGrass+ (m, F). The desingularization Z constructed in section 8.3 can be identiﬁed as Z = {(φ, R) ∈ so(F) × IGrass+ (m, F) | φ(F) ⊂ R, φ(R) = 0}. Let R be a tautological subbundle (of dimension m ) on IGrass+ (m, F). We apply the results of section 5.1 to Z . The vector bundles ξ and η are easily identiﬁed: η = 2 (F/R) and ξ = Ker( 2 F → 2 (F/R)).

8.4. Conjugacy Classes for the Orthogonal Group

287

Theorem (5.1.2)(b) implies now that 2 (F/R) . R i q∗ O Z = H i IGrass+ (m, F), Sym Using the Cauchy formula (2.3.8)(a), we see that if we denote by EC(d) the set of partitions of 2d with every part occurring even number of times, Sd

2

(F/R) =

K β (F/R).

β∈EC (d), β1 ≤n

Theorem (4.3.1) implies that R i q∗ O Z = 0 for i > 0 and that K[Y((2m )) ] = Vβ (F), d≥0 β∈EC (d), β1 ≤m

where we identify the partition with at most m parts with the dominant weight for the group SO(F). We have an exact sequence describing the representation Vβ (F) as a cokernel of a map of Schur functors K β/(2) F → K β F → Vβ (F) → 0 (compare Exercise 14 of chapter 6) with the left map being induced by the

trace element tr = 1≤i≤m ei e¯ i ∈ S2 F. This means that the deﬁning ideal of Y(2m ) consists of all polynomials which can be expressed as (x1 ∧ ei )(x2 ∧ x3 ) . . . (x2t−1 ∧ e¯ i ) + (x1 ∧ e¯ i )(x2 ∧ x3 ) . . . (x2t−1 ∧ ei ), i

i

where x 1 , . . . , x2t−1 ∈ F. Since any such polynomial is clearly a product of a polynomial of degree 2 and t − 2 polynomials of degree 1, we see that the deﬁning ideal of Y(2m ) is generated by elements of degree 2 of type (x1 ∧ ei )(x2 ∧ e¯ i ) + (x1 ∧ e¯ i )(x2 ∧ ei ). i

i

(8.4.5) The Nonnormal Orbit. An important phenomenon, discovered by Kraft and Procesi, is that some of the closures of nilpotent conjugacy classes for symplectic and orthogonal groups are not normal. The smallest example is the orbit Y(3,2,2) . Let us consider this case. The grid GC((3, 2, 2)) looks as

288

The Nilpotent Orbit Closures

follows: X GC((3, 2, 2)) =

X X

X O O

O O O O

O O O O O

O O O . O O O

Let us consider the second symmetric power S1 (η). By the Hinich– Panyushev theorem (8.3.3) we know that the only cohomology group that does not vanish is H 0 (G/P, S1 (η)). To calculate this group, it is enough to calculate the Euler characteristic of the bundle S1 (η). To do this one can replace the bundle S1 (η) with the direct sum of its composition factors of dimension 1. The whole matter becomes an exercise of using Bott’s theorem. The result is H 0 (G/P, S2 (η)) = V(1,1,0,0) (F) ⊕ V(1,0,0,0) (F). Applying (5.1.3)(b), we see that the group H 0 (G/P, S2 (η)) is the ﬁrst graded component of the normalization of the coordinate ring of Y(3,2,2) . If this variety were normal, H 0 (G/P, S1 (η)) would be a factor of 2 F). Since 2 F = V(1,1,0,0) (F), we see that our orbit closure is not normal. The normal orbit closures for the orthogonal groups were determined by Kraft and Procesi in [KP2]. They used the method of minimal degenerations. Their result in this case is (8.4.6) Theorem (Kraft–Procesi). Let µ ∈ Po (2n). The orbit closure Yµ is normal if and only if for every i < j such that µi and µ j are even, with µi > µ j , at least one of the parts µi+1 , . . . , µ j−1 is odd. All of the above describes the results for the orbits of the orthogonal group. However one can also consider the nilpotent conjugacy classes of the special orthogonal group. Kraft and Procesi prove in [KP2] (8.4.7) Proposition (Kraft–Procesi). The conjugacy class C(µ) for the orthogonal group is also a conjugacy class for the special orthogonal group unless µ is very even, i.e., all parts of µ and µ are even. For a very even partition µ the conjugacy class C(µ) for the orthogonal group is a disjoint union of two conjugacy classes C(µ)(1) and C(µ)(2) for the special orthogonal group. We ﬁnish by presenting a conjecture on the generators of the deﬁning ideals of orbit closures Yµ .

8.4. Conjugacy Classes for the Orthogonal Group

289

(8.4.8) Conjecture. Let µ ∈ Po (n). The deﬁning ideal of Yµ is generated by the representations V(β1 ,...,βn ) (F) with β1 ≤ 2. The generators can be chosen to be subrepresentations of the Schur functors K γ F with γ1 ≤ 2. For the remainder of this section we look at the very even conjugacy classes, i.e., we assume that the partitions µ and µ have only even parts. These orbit closures they have another desingularization which is more convenient for explicit calculations. Let µ = (µ1 , . . . , µr ) be a very even partition of n = 2m, m even, µi = 2νi for i = 1, . . . , r . We denote ν = (ν1 , . . . , νr ). Note that the parts of ν are even. Take V = IFlag(ν1 , ν1 + ν2 , . . . , n; F). The “desingularization” Zˆ ν of Yµ is deﬁned as follows Zˆ ν = {(φ, (Rν1 , Rν1 +ν2 , . . . , Rn )) ∈ sp(F) × IFlag(ν1 , ν1 + ν2 , . . . , n; F) | for i = 2, . . . , r, φ(Rν1 ) = 0, φ(Rν1 +...+νi ) ⊂ Rν1 +...+νi−1

φ(Rν∨ +...+ν ) ⊂ Rν∨ +...+ν for i = 2, . . . , r, φ(F) ⊂ Rν∨ } 1

i−1

i

1

1

The word desingularization is given above in quotation marks because the variety IFlag(ν1 , ν1 + ν2 , . . . , n; F) has two connected components IFlag+ (ν1 , ν1 + ν2 , . . . , n; F) and IFlag− (ν1 , ν1 + ν2 , . . . , n; F), depending on whether Rn is in IGrass+ (m, F) or in IGrass− (m, F). The corresponding components Zˆ ν+ and Zˆ µ− give desingularizations of the closures of conjugacy classes Cµ(1) and Cµ(2) for the special orthogonal groups. In the sequel we work with Zˆ ν+ and with Cµ(1) . (8.4.9) Example. n = 5, µ = (4, 4, 2, 2). The partition ν = (4, 2). The grid corresponding to Zˆ ν is X

& GC((4, 4, 2, 2)) =

X X

X X X

X X X X

X X X X X

X X O O O O

X X O O O O O

X X O O O O O O

X X O O O O O O O

O O O O O O O O O O

O O O O O O. O O O O O

Notice that in this grid all entries corresponding to ei ∧ e j are marked with X , all entries corresponding to e¯ i ∧ e¯ j are marked with O, and in the region

290

The Nilpotent Orbit Closures

corresponding to ei ∧ e¯ j we have the same pattern as for the desingularization of the conjugacy class O(ν) for GL(n). This rule is true for a general very even partition. The variety Zˆ ν+ again ﬁts all the assumptions of the setup from chapter 5. Denote the corresponding bundles by ξ := Sν , η := Tν . Deﬁne the partition ˆ = 2ˆν is again very even. νˆ = (ν1 − 1, . . . , ν j − 1). Notice that the partition µ The idea of the inductive procedure for very even conjugacy classes is to look at the bundle corresponding to the weights in our grid in the columns entirely ﬁlled by circles. There are j = ν1 such columns. The corresponding bundle S j lives on IGrass( j, F), and it can be best described by two exact sequences it ﬁts: 0 → S2 R → R ⊗ F → S j → 0, 0→

(∗)

2

(∗∗) R → S j → R ⊗ (F/R) → 0. Notice that the factor T j := 2 F/S j can be identiﬁed with 2 (F/R). The point now is that we have an exact sequence 0 → Sν1 → Sν → Sνˆ (R∨ /R) → 0, where Sνˆ (R∨ /R) is the bundle Sνˆ for the partition νˆ in the relative situation where the orthogonal space of dimension 2(n − j) is replaced by the bundle R∨ /R. Our strategy is again to take a complex F•νˆ in a relative situation and for each term of that complex (which is a special orthogonal irreducible V(α1 ,...,αn− j ) (R∨ /R)) to estimate the terms resulting in the cohomology of V(α1 ,...,αn− j ) (R∨ /R) ⊗ • S j . Notice that these cohomology groups are the terms of a twisted complex F(V(α1 ,...,αn− j ) (R∨ /R))• supported in the variety Y j which is the image of the incidence variety Z j = {(φ, R) ∈ X × IGrass( j, F) | φ(R) = 0}. We start with the results we need about the twisted complexes of that kind. (8.4.10) Proposition. Assume that α1 ≤ j. (a) The complex F(V(α1 ,...,αn− j ) (R∨ /R))• has terms in nonnegative degree, i.e., m i ∨ Sj = 0 H IGrass( j, F), V(α1 ,...,αn− j ) (R /R) ⊗ for i > m. (b) The terms of the complex F(V(α1 ,...,αn− j ) (R∨ /R))• contain only the representations V(β1 ,...,βn ) F with β1 ≤ j.

8.4. Conjugacy Classes for the Orthogonal Group

291

Proof. We start with property (a). We use the exact sequence (∗). The m-th exterior power of the two term complex S2 R → R ⊗ F gives by (2.4.7) an acyclic complex 0 → Sm (S2 R) → Sm−1 (S2 R) ⊗ (R ⊗ F) m−1 m → . . . → S2 R ⊗ (R ⊗ F) → (R ⊗ F)

(?)

resolving m S j . The complex (?) ⊗ V(α1 ,...,αn− j ) (R∨ /R) provides a resolum S j ⊗ V(α1 ,...,αn− j ) (R∨ /R). We split that complex into short exact tion of sequences and use the induced long sequences of cohomology groups. It follows that in order to show (a) it is enough to prove that H i (IGrass( j, F),

m−u

(R ⊗ F) ⊗ Su (S2 R) ⊗ V(α1 ,...,αn− j ) (R∨ /R)) = 0

for i > m + u. Decomposing into representations K γ R using the formula (2.3.8)(b), the Cauchy fomula (2.3.3), and the Littlewood–Richardson rule (2.3.4), we see that in order to prove part (a) of Proposition (8.4.10) we need to show (8.4.11) Lemma. Let α = (α1 , . . . , αn− j ) be a partition such that α1 ≤ j. Let γ be a partition of m. Then H i (IGrass( j, F), K γ R ⊗ V(α1 ,...,αn− j ) (R∨ /R)) = 0 for all i > m. Proof. We ﬁx α and look at the set of all γ such that the cohomology H l(α,γ ) (IGrass( j, F), K γ R ⊗ V(α1 ,...,αn− j ) (R∨ /R)) = 0 for a unique number l(α, γ ). We deﬁne N (α, γ ) = |γ | − l(α, γ ). We want to show that N (α, γ ) ≥ 0. We recall that by Corollaries (4.3.7) and (4.3.9) the cohomology of the vector bundle K γ R ⊗ V(α1 ,...,αn− j ) (R∨ /R) is calculated by looking at the sequence (−γ j + n − 1, −γ j − 1 + n − 2, . . . , −γ1 + n − j, α1 + n − j − 1, . . . , αn− j ) and trying to make it decreasing, with the sum of last two entries positive, using the Weyl group action. Since we have −γ j + n − 1 > −γ j−1 + n − 2 > . . . > −γ1 + n − j, there exists the smallest s for which −γs + n − j + s − 1 < 0. The number l(α, γ ) is calculated as follows. We move −γ1 + n − j to the end by a sequence of exchanges with numbers following it, change its value

292

The Nilpotent Orbit Closures

to its negative, and do the same with −γ2 + n − j + 1, . . . , −γs + n − j + s − 1. We get a sequence of positive numbers P(α, γ ), which we reorder to get a decreasing sequence R(α, γ ). Each exchange of neighboring elements in this process and change of the sign of the last number contributes 1 to N (α, γ ). Notice that in the sequence P(α, γ ) the numbers from the positions of γ1 , . . . , γs occupy the last s spots. Also notice that α1 + n − j, . . . , αn− j + 1 as well as −γs+1 + (n − j) + s + 1, . . . , −γ j + n are all numbers ≤ n. Therefore there is a unique γ1 , . . . , γs for which the resulting sequence is (n, n − 1, . . . , 1). We claim that the minimum of N (α, γ ) is achieved for such γ . Indeed, assume that some of the numbers in R(α, γ ) are bigger than n. Let u be the smallest number not occurring in R(α, γ ). We have u ≤ n. Let m be the smallest number ≥ u that occurs in R(α, γ ) and comes from among the positions corresponding to γ1 , . . . , γs , say from γt . Let u = m − l. Then there exists a unique partition δ such that δ ⊂ γ , |δ| = |γ | − l, R(α, δ) = R(α, γ ) ∪ {u} \ {m}. Now δ is obtained from γ in at most l steps. Each step consists of decreasing a part of γ by 1 (which forces some of the following parts to also decrease so we again get a partition), starting with γt . At each stage the new repetition in the reordered sequence appears. This repetition has to involve a position coming from γ1 , . . . , γs , so we have to decrease this part of γ next, and so on. We know that |δ| = |γ | − l. Comparing the reordering processes for (α, γ ) and (α, δ), we see that l(α, δ) ≥ l(α, γ ) − l + 1 and the decrease in the number of exchanges is equal to the number of entries {m − 1, . . . , u + 1} in R(α, γ ) that did not come from positions corresponding to −γs + n − j + s, . . . , −γ1 + n − j + 1. Thus N (α, δ) ≤ N (α, γ ) − 1, as desired. Assume that (α, γ ) is such that the reordered sequence is (n, n − 1, . . . , 1). We shall prove that for ﬁxed α and for any γ for which R(α, γ ) = (n, n − 1, . . . , 1), the partition γ with minimal N (α, γ ) is the one with s = 0. Indeed, let us assume that s > 0. Consider the sequence (−γ j + n, −γ j − 1 + n − 1, . . . , −γ1 + n − j + 1, α1 + n − j, . . . , αn− j + 1). We construct a new partition δ as follows. We modify the above sequence by changing the negative entry −γs + n − j + s to its negative x = γs − n + j − s > 0. Then we order the positive part of this sequence and see that it can be written as (−δ j + n, −δ j−1 + n − 1, . . . , −δ1 + n − j + 1, α1 + n − j, . . . , αn− j + 1) for a partition δ with |δ| = |γ | − 2x. The partition δ has smaller invariant s. We count the number l(α, γ ) − l(α, δ). The difference is that the number −γs + n − j + s gets exchanged twice with numbers from the set α1 + n − j, . . . , αn− j + 1 that are in [1, x − 1], then it is reﬂected to its negative and

8.4. Conjugacy Classes for the Orthogonal Group

293

then gets exchanged additionally with the numbers from the interval [1, x − 1] that preceed it. Therefore we have l(α, γ ) − l(α, δ) ≤ 2x − 1 and thus N (α, γ ) > N (α, δ). It remains to show that for ﬁxed α and for partitions γ such that R(α, γ ) = (n, n − 1, . . . , 1) and s = 0 we have N (α, γ ) ≥ 0. But now we see that the last reﬂection does not occur in exchanging entries. Therefore we are in the situation of Lemma (8.1.5). We see that the choice of γ minimizing N (α, γ ) is γ = α and that N (α, α ) = 0. This proves Lemma (8.4.11) and therefore part (a) of Proposition (8.4.10). We prove part (b) of Proposition (8.4.10). We use the exact sequence (∗∗). We will prove that the cohomology of •

(S2 R) ⊗

•

(R ⊗ (F/R)) ⊗ Vα (R∨ /R)

does not contain the representations Vβ F with β1 > j. Using the exact sequence 0 → R∨ /R → F/R → F/R∨ → 0, we see that it is enough to show that the cohomology of • • • (S2 R) ⊗ (R ⊗ (R∨ /R)) ⊗ (R ⊗ (F/R∨ )) ⊗ Vα (R∨ /R)

does not contain the representations Vβ F with β1 > j. Let us look at a typical term K β R ⊗ K γ R ⊗ K γ (R∨ /R) ⊗ K δ R ⊗ K δ (F/R∨ ) ⊗ Vα (R∨ /R). After decomposing the Weyl functors to the symplectic irreducibles and calculating tensor products, we see that the term above will decompose to the summands of the form K λ (F/R∨ ) ⊗ Vθ (R∨ /R). Such a bundle has a representation Vβ F with β1 > j in cohomology if a sequence (λ1 + n, λ2 + n − 1, . . . , λ j + n − j + 1, θ1 + n − j, . . . , θn− j + 1) contains either the number > n + j or a number < −n − j. Since θ is an integral dominant weight for Sp(2n − 2 j), i.e. a partition, this can happen if

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The Nilpotent Orbit Closures

one of the following cases occurs: (a) λ1 + n > n + j, i.e. λ1 > j, (b) θ1 + n − j > n + j, i.e. θ1 > 2 j, (c) λ j + n − j + 1 < −n − j, i.e. −λ j > 2n + 1. Case (a) cannot occur because K λ (F/R∨ ) is a summand in a tensor product K β R ⊗ K γ R ⊗ K δ R ⊗ K δ (F/R∨ ), and the only possible positive weights come from δ . However, δ1 ≤ j, since dim R = j. Case (b) cannot occur because the weight θ comes from a summand in K γ (R∨ /R) ⊗ Vα (R∨ /R) and α1 ≤ j and γ1 ≤ j (because the corresponding functor K γ R is nonzero). Thus K γ (R∨ /R) decomposes to the representations V (R∨ /R), and in the tensor product V (R∨ /R) ⊗ Vα (R∨ /R) all weights have to be ≤ α + by Klimyk’s formula (exercise 12 of chapter 4), and therefore the ﬁrst entry of each occurring weight is ≤ 2 j. Case (c) cannot occur because in the tensor product K β R ⊗ K γ R ⊗ K δ R ⊗ K δ (F/R∨ ) the last entries are −β1 ≥ − j − 1, −γ1 ≥ 2 j − 2n, and −δ1 ≥ − j (because K δ (F/R∨ ) has to be nonzero). This concludes the proof of part (b) of Proposition (8.4.10).• (8.4.12) Corollary. (a) The term F0 consists of one copy of the trivial representation in homogeneous degree 0. In particular the variety Z j is normal, with rational singularities. (b) The term F1 contains only the representations Vβ (F) with β1 ≤ 1. In particular the deﬁning ideal of Y j := q(Z j ) is generated by such representations. More precisely, the minimal set of generators of the deﬁning ideal of Y j consists of the vectors in the representation 2(n− j+1)

F ⊂ K (22(n− j+1) ) F ⊂ S2(n− j+1) (S2 F).

Proof. The ﬁrst claim follows from the proof of Lemma (8.4.11). Indeed, the only term contributing to F0 corresponds to γ = 0 which, looking at the sequences (?), can appear only for m = 0.

8.4. Conjugacy Classes for the Orthogonal Group

295

The proof of the second part is based on the reﬁnement of the Lemma (8.4.11). Claim. The only partition γ with |γ | = m such that H m−1 (IGrass( j, F), K γ R) = 0 is γ = (2(n − j + 1)), and the resulting cohomology group is a trivial representation of Sp(F). Proof. We repeat the proof of Lemma (8.4.11) with α = (0). Our partition γ has to lead in one step to the partition γ = 0, which gives the only partition with N (γ , (0)) = 0. The sequence R(γ , (0)) has to be (n, n − 1, . . . , 1), because if not, then s > 0 and the ﬁrst step in the procedure does decrease s. We see now that s = 1; otherwise we need s steps to reduce to γ = 0. Now we can do a case by case analysis, because the number of possibilities for γ corresponds to possibilities for the number γ1 − n + j − 1 ∈ {n, n − 1, . . . , n − j + 1}. The only viable possibility turns out to be γ1 − n + j − 1 = n − j + 1 which implies γt = 0 for t ≥ 2. Now we see, using the exact sequence (?) resolving m S j , that the only 2(n− j+1) possible contribution to F1 is F in homogeneous degree 2(n − j + 1). We easily identify this set of equations with the ones given in (8.4.12) (b) because they vanish on Y j . We can ﬁnally state the results of our induction. (8.4.13) Theorem. Let µ be an even partition, µ = 2ν. (a) The orbit closure Yµ is normal, with rational singularities, (b) The irreducible representations occurring in the terms of the complex F•µ have highest weights (β1 , . . . , βn ) with β1 ≤ ν1 . µ (c) The term F1 contains only the representations Vβ (F) with β1 ≤ 1. Therefore the deﬁning ideal of Yµ is generated by the representations of this kind. Proof. We use the induction described above, connecting the relative complex F•µˆ to the complex F•µ . We use the exact sequence 0 → Sν1 → Sν → Sνˆ (R∨ /R) → 0 to estimate the terms in the cohomology of Sν . They come from the terms of the cohomology in Vα (R∨ /R) ⊗ • (Sν1 ). By induction the terms Vα (R∨ /R) satisfy the assumption of Proposition (8.4.10), which shows that the i-th term of the complex F µˆ (R∨ /R)• can produce only terms in the homological degree ≥ i. Thus, by induction, only the terms in nonnegative homological

296

The Nilpotent Orbit Closures µ

degree occur, and by Corollary (8.4.12) the term F0 consists of one copy of the trivial representation in homogeneous degree 0. Parts (a) and (b) of the theorem follow. µ To prove part (c) let us analyze the term F1 . The terms there can come from the term F• , which has the required form by Corollary (8.4.12)(b), and from µ ˆ the terms from F1 (R∨ /R), which by induction are the terms of homological degree zero in F(V(α1 ,...,αn− j ) (R∨ /R))• , with weight α for which α1 ≤ 1. By the proof of Lemma (8.4.11) the only such term comes from γ = α , so (by the use of the complexes Vα (R∨ /R) ⊗ (?)) the resulting contribution to the possible terms of F(V(α1 ,...,αn− j ) (R∨ /R))• is K α F ⊗ V(0) F, which decomposes to the representations of Sp(F) of the required form. 8.5. Conjugacy Classes for the Symplectic Group In this section, F denotes a vector space of dimension 2n with a nondegenerate antisymmetric form ( , ). The symplectic group Sp(F) is the set of linear automorphisms of F preserving ( , ), i.e., φ ∈ Sp(F) if and only if for each x, y ∈ F we have (φ(x), φ(y)) = (x, y). By deﬁnition Sp(F) is a subgroup of GL(F). The corresponding Lie algebra sp(F) is a subalgebra of the Lie algebra gl(F). The morphism φ ∈ HomK (F, F) is in sp(F) if and only if for each x, y ∈ F we have (φ(x), y) + (x, φ(y)) = 0. Let us choose the symplectic basis e1 , . . . , en , e¯ n , . . . , e¯ 1 of F. This means (ei , e j ) = (¯ei , e¯ j ) = 0,

(ei , e¯ j ) = δi, j ,

where δi, j denotes the Kronecker delta. We can write φ as a matrix, writing in consecutive columns the images of vectors −¯e1 , . . . , −¯en , en , . . . , e1 expanded in the basis e1 , . . . , en , e¯ n , . . . , e¯ 1 . Then φ ∈ sp(F) if and only if the matrix of φ is symmetric. This allows us to identify the adjoint representation of sp(F) with S2 F. Since sp(F) is a Lie subalgebra of gl(F), we might expect that the nilpotent conjugacy classes in sp(F) will be related to intersections of the nilpotent conjugacy classes in gl(F) with sp(F). The following result is proved in [SS; IV.2.15]. (8.5.1) Proposition. Let µ be a partition of 2n. Let us consider the nilpotent conjugacy class O(µ) of gl(F) corresponding to µ. Then O(µ) intersects sp(F) if and only if every odd part of µ occurs an even number of times. In this case the intersection O(µ) ∩ sp(F) consists of a single conjugacy class of Sp(F).

8.5. Conjugacy Classes for the Symplectic Group

297

For the remainder of this section we denote by Ps (n) the set of partitions µ of 2n in which every odd part occurs even number of times. For µ ∈ Ps (n) we denote C(µ) the corresponding conjugacy class in sp(F), and by Yµ its closure. One might say that the Jordan blocks for the symplectic group are of two kinds: the blocks corresponding to even rows in our partition and the blocks corresponding to the pairs of odd rows. The canonical forms of nilpotent symplectic endomorphisms corresponding to both kinds of blocks are as follows. The nilpotent e corresponding to the partition (2n) has the form e(¯e1 ) = 0,

e(¯ei ) = e¯ i−1

e(en ) = −¯en ,

e(ei ) = −ei+1

for i > 0, for i < n.

We can express the action of e by the sequence of arrows ±e1 → ∓e2 → . . . → en−1 → −en → e¯ n → e¯ n−1 → . . . → e¯ 2 → e¯ 1 → 0. If we extend e to an sl2 triple {e, h, f }, the grading induced by e described in section 8.3 is as follows. For the endomorphism e corresponding to the partition 2n we have h(ei ) = (2n + 1 − 2i)ei , h(¯ei ) = (2i − 1 − 2n)¯ei . Therefore the degrees of the basis vectors are deg ei = 2n + 1 − 2i, deg e¯ i = 2i − 1 − 2n. If n = 2t + 1 is odd, then the nilpotent e corresponding to the partition (n, n) has the form e(ei ) = ei+2

for i < 2t,

e(¯ei ) = −¯ei−2

e(e2t ) = e¯ 2t+1 ,

for i > 2,

e(e2t+1 ) = e¯ 2t ,

e(¯e2 ) = e(¯e1 ) = 0.

We can express the action of e by two sequences of arrows e1 → e3 → . . . → e2t+1 → e¯ 2t → −¯e2t−2 → . . . → ±¯e2 → 0, e2 → e4 → . . . → e2t → e¯ 2t+1 → −¯e2t−1 → . . . → ∓¯e1 → 0. Extending e to the sl2 -triple {e, h, f }, we see that the element h acts as follows: h(e2i+1 ) = 2(t − i)e2i+1 , h(e2i ) = 2(t + 1 − i)e2i , and h(¯e2i+1 ) = −2(t − i)¯e2i+1 , h(¯e2i ) = −2(t + 1 − i)¯e2i . This gives deg e2i+1 = 2(t − i), deg e2i = 2(t + 1 − i), and deg e¯ 2i+1 = −2(t − i), deg e¯ 2i = −2(t + 1 − i). In this setup the degrees of vectors ei are always nonnegative and we always have deg ei = −deg e¯ i , because h ∈ sp(F). If a partition µ corresponds to the conjugacy class in sp(F), then we can assign the grading separately in each block, as in Example (8.3.2). We also use the convention that when dealing with several blocks, after assigning the grading, we order ei ’s in such a way that deg ei ≥ deg ei+1 .

298

The Nilpotent Orbit Closures

(8.5.2) Example. Let us take n = 5, µ = (3, 3, 2, 1, 1). The grading of basis elements is as follows: 2 0 −2 2 0 −2 1 −1 . 0 0 We order the elements e1 , . . . e5 , e¯ 5 , . . . , e¯ 1 so their grading is nonincreasing. We get deg e1 = deg e2 = 2, deg e3 = 1, deg e4 = deg e5 = 0, with deg e¯ i = −deg ei for i = 1, . . . , 5. This allows us to determine the grading on sp(F) in all cases. Identifying the adjoint representation with S2 F, we can arrange the weight vectors in it in a triangular grid. In order to describe it, let us introduce the involution ()¯ of our symplectic basis by requiring that e¯i = ei . The elements of the grid correspond to the entries on or above the diagonal of our matrix representation of φ. If u, v are the elements of our symplectic basis, then the entry corresponding to the row u and the column ±v will correspond to the weight vector u v¯ . For a given conjugacy class C(µ) we will mark the element of the grid with X if the corresponding weight vector is in g ≥2 , and with O otherwise. We will denote this grid by GC(µ). (8.5.3) Example. Let n = 5, µ = (3, 3, 2, 1, 1). Then the degrees of elements ei are given in Example (8.5.2), and we have X

GC(µ) =

X X

X X X

X X O O

X X O O O

X X O O O O

X X O O O O O

O O O O O O O O

O O O O O O O O O

O O O O O . O O O O O

We continue with several examples of conjugacy classes. (8.5.4) Example. Consider the class C((2n )). This is a very interesting conjugacy class. The associated grading gives deg(ei ) = 1, deg(¯ei ) = −1 for

8.5. Conjugacy Classes for the Symplectic Group

299

i = 1, . . . , n. This means that in the grid GC((2n )) the entries marked by X correspond to the vectors ei e j for i ≤ j. For example, if n = 5 we get X

GC((25 )) =

X X

X X X

X X X X

X X X X X

O O O O O O

O O O O O O O

O O O O O O O O

O O O O O O O O O

O O O O O . O O O O O

The parabolic subgroup P is the set of elements ﬁxing a given isotropic subspace of dimension n. The homogeneous space G/P is the isotropic Grassmannian IGrass(n, F). The desingularization Z constructed in section 8.3 can be identiﬁed as Z = {(φ, R) ∈ sp(F) × IGrass(n, F) | φ(F) ⊂ R, φ(R) = 0}. Let R be a tautological subbundle (of dimension n ) on IGrass(n, F). We apply the results of section 5.1 to Z . The vector bundles ξ and η are easily identiﬁed: η = S2 (F/R) and ξ = Ker(S2 F → S2 R). Theorem (5.1.2)(b) implies now that R i q∗ O Z = H i (IGrass(n, F), Sym(S2 (F/R))). Using the Cauchy formula (2.3.8)(a), we see that if we denote by EP(d) the set of partitions of 2d with even parts, then Sd (S2 (F/R)) = K β (F/R). β∈EP(d), β1 ≤n

Theorem (4.3.1) implies that R i q∗ O Z = 0 for i > 0 and that ¯ n ))] = K[C((2 Vβ (F), d≥0 β∈EP(d), β1 ≤n

where we identify the partition with at most n parts with the dominant weight for the group SP(F). We have an exact sequence describing the representation Vβ (F) as a cokernel of a map of Schur functors K β/(12 ) F → K β F → Vβ (F) → 0 (compare Exercise 4 of chapter 6) with the left map being induced by the trace

element tr = 1≤i≤n ei ∧ e¯ i ∈ 2 F.

300

The Nilpotent Orbit Closures

¯ n )) consists of all polynomials This means that the deﬁning ideal of C((2 which can be expressed as (x1 ei )(x2 x3 ) . . . (x2t−1 e¯ i ) − (x1 e¯ i )(x2 x3 ) . . . (x2t−1 ei ) i

i

where x1 , . . . , x2t−1 ∈ F. Since any such polynomial is clearly a product of a polynomial of degree 2 and t − 2 polynomials of degree 1, we see that the deﬁning ideal of C((2n )) is generated by elements of degree 2 of type (x1 ei )(x2 e¯ i ) − (x1 e¯ i )(x2 ei ). i

i

(8.5.5) The Nonnormal Orbit. We give the example of the smallest nonnormal orbit closure for the symplectic group. It is Y(3,3,1,1) . Let us analyze this case. The grid G R(3, 3, 1, 1) looks as follows: X

GC((3, 3, 1, 1)) =

X X

X X O

X X O O

X X O O O

X X O O O O

O O O O O O O

O O O O . O O O O

Let us consider the second symmetric power S2 (η). By the Hinich– Panyushev theorem (8.3.3) we know that the only cohomology group that does not vanish is H 0 (G/P, S2 (η)). To calculate this group, it is enough to calculate the Euler characteristic of the bundle S2 (η). To do this one can replace the bundle S2 (η) with the direct sum of its composition factors of dimension 1. The whole matter becomes an exercise of using Bott’s theorem 55 times. The result is H 0 (G/P, S2 (η)) = V(4,0,0,0) (F) ⊕ V(2,2,0,0) (F) ⊕ V(1,1,0,0) (F) ⊕ V(1,1,1,1) (F). Applying (5.1.3)(b), we see that the group H 0 (G/P, S2 (η)) is the second graded component of the normalization of the coordinate ring of Y(3,3,1,1) . If this variety were normal, H 0 (G/P, S2 (η)) would be a factor of S2 (S2 F). However using the sequences of exercise 4 of chapter 6, we can easily see that S2 (S2 F) = V(4,0,0,0) (F) ⊕ V(2,2,0,0) (F) ⊕ V(1,1,0,0) (F) ⊕ V(0,0,0,0) (F). This proves that our closure is not normal.

8.5. Conjugacy Classes for the Symplectic Group

301

The normal orbit closures for the symplectic Lie algebra were determined by Kraft and Procesi in [KP2]. They used the method of minimal degenerations. We state their result. (8.5.6) Theorem (Kraft–Procesi). Let µ ∈ Ps (2n). The orbit closure Yµ is normal if and only if for every i < j such that µi and µ j are odd, with µi > µ j , at least one of the parts µi+1 , . . . , µ j−1 has to be even. Even though the method of minimal degenerations is more effective in describing normality, calculations such as the one above are still useful, as they allow to describe the decomposition to irreducibles of the normalization of the coordinate ring of an orbit closure. For the remainder of this section we look at the even conjugacy classes, i.e., we assume that the partition µ has only even parts. These classes show certain similarities with the conjugacy classes for the general linear group. In particular, they have another desingularization which is more convenient for explicit calculations. Let µ = (µ1 , . . . , µr ) be a partition of 2n with only even parts, µi = 2νi for i = 1, . . . , r . We denote ν = (ν1 , . . . , νr ). Take V = IFlag(ν1 , ν1 + ν2 , . . . , n; F). The desingularization Zˆ ν of C(µ) is deﬁned as follows: Zˆ ν = {(φ, (Rν1 , Rν1 +ν2 , . . . , Rn )) ∈ sp(F) × IFlag(ν1 , ν1 + ν2 , . . . , n; F) | for i = 2, . . . , r, φ(Rν1 ) = 0, φ(Rν1 +...+νi ) ⊂ Rν1 +...+νi−1

φ(Rν∨ +...+ν ) ⊂ Rν∨ +...+ν for i = 2, . . . , r, φ(F) ⊂ Rν∨ }. 1

i−1

i

1

1

(8.5.7) Example. Let n = 5, µ = (6, 4). The partition ν = (2, 2, 1). The grid corresponding to Zˆ ν is X

& GC((4, 4, 2)) =

X X

X X X

X X X X

X X X X X

X X X X O O

X X O O O O O

X X O O O O O O

O O O O O O O O O

O O O O O . O O O O O

302

The Nilpotent Orbit Closures

Notice that in this grid all entries corresponding to ei e j are marked with X , all entries corresponding to e¯ i e¯ j are marked with O, and in the region corresponding to ei e¯ j we have the same pattern as for the desingularization of the conjugacy class O(ν) for GL(n). This rule is true for a general even partition. The variety Zˆ ν again ﬁts all the assumptions of the setup from chapter 5. Denote the corresponding bundles ξ := Sν , η := Tν . Deﬁne the partition νˆ = (ν1 − 1, . . . , ν j − 1). The idea of the inductive procedure for even conjugacy classes is to look at the bundle corresponding to the weights in our grid in the columns entirely ﬁlled by circles. There are j = ν1 such columns. The corresponding bundle S j lives on IGrass( j, F), and it can be best described by two exact sequences it ﬁts: 0→

2

R → R ⊗ F → S j → 0,

0 → S2 R → S j → R ⊗ (F/R) → 0.

(∗) (∗∗)

Notice that the factor T j := S2 F/S j can be identiﬁed with S2 (F/R). The point now is that we have an exact sequence 0 → Sν1 → Sν → Sνˆ (R∨ /R) → 0, where Sνˆ (R∨ /R) is the bundle Sνˆ for the partition νˆ in the relative situation where the symplectic space of dimension 2(n − j) is replaced by the bundle R∨ /R. Our strategy is again to take a complex F•νˆ in a relative situation and for each term of that complex (which is a symplectic irreducible V(α1 ,...,αn− j ) (R∨ /R)) to estimate the terms resulting in the cohomology of V(α1 ,...,αn− j ) (R∨ /R) ⊗ • S j . Notice that these cohomology groups are the terms of a twisted complex F(V(α1 ,...,αn− j ) (R∨ /R))• supported in the variety Y j which is the image of the incidence variety Z j = {(φ, R) ∈ X × IGrass( j, F) | φ(R) = 0}. We start with the results we need about the twisted complexes of that kind. (8.5.8) Proposition. Assume that α1 ≤ j. (a) The complex F(V(α1 ,...,αn− j ) (R∨ /R))• has terms in nonnegative degree, i.e., m H i IGrass( j, F), V(α1 ,...,αn− j ) (R∨ /R) ⊗ Sj = 0 for i > m.

8.5. Conjugacy Classes for the Symplectic Group

303

(b) The terms of the complex F(V(α1 ,...,αn− j ) (R∨ /R))• contain only the representations V(β1 ,...,βn ) F with β1 ≤ j. Proof. We start with property (a). We use the exact sequence (∗). The m-th exterior power of the two term complex 2 R → R ⊗ F gives by (2.4.7) an acyclic complex 2 2 0 → Sm R → Sm−1 R ⊗ (R ⊗ F) → . . . 2 m−1 m R⊗ (R ⊗ F) → (R ⊗ F) (?) → m S j . The complex (?) ⊗ V(α1 ,...,αn− j ) (R∨ /R) provides a resoluresolving m S j ⊗ V(α1 ,...,αn− j ) (R∨ /R). We split that complex into short exact tion of sequences and use the induced long sequences of cohomology groups. It follows that in order to show (a) it is enough to prove that

H

i

IGrass( j, F),

m−u

(R ⊗ F) ⊗ Su

2

∨ R ⊗ V(α1 ,...,αn− j ) (R /R) = 0

for i > m + u. Decomposing into representations K γ R using the formula (2.3.8)(a), the Cauchy formula (2.3.3), and the Littlewood–Richardson rule (2.3.4), we see that in order to prove part (a) of Proposition (8.5.8) we need to show (8.5.9) Lemma. Let α = (α1 , . . . , αn− j ) be a partition such that α1 ≤ j. Let γ be a partition of m. Then H i (IGrass( j, F), K γ R ⊗ V(α1 ,...,αn− j ) (R∨ /R)) = 0 for all i > m. Proof. We ﬁx α and look at the set of all γ such that the cohomology H l(α,γ ) (IGrass( j, F), K γ R ⊗ V(α1 ,...,αn− j ) (R∨ /R)) = 0 for a unique number l(α, γ ). We deﬁne N (α, γ ) = |γ | − l(α, γ ). We want to show that N (α, γ ) ≥ 0. We recall that by Corollary (4.3.4) the cohomology of the vector bundle K γ R ⊗ V(α1 ,...,αn− j ) (R∨ /R) is calculated by looking at the sequence (−γ j + n, −γ j − 1 + n − 1, . . . , −γ1 + n − j + 1, α1 + n − j, . . . , αn− j + 1)

304

The Nilpotent Orbit Closures

and trying to make it decreasing using the Weyl group action. Since we have −γ j + n > −γ j−1 + n − 1 > . . . > −γ1 + n − j + 1, there exists the smallest s for which −γs + n − j + s < 0. The number l(α, γ ) is calculated as follows. We move −γ1 + n − j + 1 to the end by a sequence of exchanges with numbers following it, change its value to its negative, and do the same with −γ2 + n − j + 2, . . . , −γs + n − j + s. We get a sequence of positive numbers P(α, γ ), which we reorder to get a decreasing sequence R(α, γ ). Each exchange of neighboring elements in this process and changing of the sign of the last number contribute 1 to N (α, γ ). Notice that in the sequence P(α, γ ) the numbers from the positions of γ1 , . . . , γs occupy the last s spots. We also notice that α1 + n − j, . . . , αn− j + 1 as well as −γs+1 + (n − j) + s + 1, . . . , −γ j + n are all numbers ≤ n. Therefore there is a unique γ1 , . . . , γs for which the resulting sequence is (n, n − 1, . . . , 1). We claim that the minimum of N (α, γ ) is achieved for such γ . Indeed, assume that some of the numbers in R(α, γ ) are bigger than n. Let u be the smallest number not occurring in R(α, γ ). We have u ≤ n. Let m be the smallest number ≥ u that occurs in R(α, γ ) and comes from the positions corresponding to γ1 , . . . , γs , say from γt . Let u = m − l. Then there exists a unique partition δ such that δ ⊂ γ , |δ| = |γ | − l, R(α, δ) = R(α, γ ) ∪ {u} \ {m}. Here δ is obtained from γ in at most l steps. Each step consists of decreasing a part of γ by 1 (which forces some of the following parts to also decrease so we again get a partition), starting with γt . At each stage the new repetition in the reordered sequence appears. This repetition has to involve a position coming from γ1 , . . . , γs , so we have to decrease this part of γ next, and so on. We know that |δ| = |γ | − l. Comparing the reordering processes for (α, γ ) and (α, δ), we see that l(α, δ) ≥ l(α, γ ) − l + 1, and the decrease in the number of exchanges is equal to the number of entries {m − 1, . . . , u + 1} in R(α, γ ) that did not come from positions corresponding to −γs + n − j + s, . . . , −γ1 + n − j + 1. Thus N (α, δ) ≤ N (α, γ ) − 1, as desired. Assume that (α, γ ) is such that the reordered sequence is (n, n − 1, . . . , 1). We shall prove that for ﬁxed α and for such γ for which R(α, γ ) = (n, n − 1, . . . , 1), the partition γ with minimal N (α, γ ) is the one with s = 0. Indeed, let us assume that s > 0. Consider the sequence (−γ j + n, −γ j − 1 + n − 1, . . . , −γ1 + n − j + 1, α1 + n − j, . . . , αn− j + 1). We construct a new partition δ as follows. We modify the above sequence by changing the negative entry −γs + n − j + s to its negative x = γs − n + j − s > 0. Then we order the positive part of this sequence and see that it

8.5. Conjugacy Classes for the Symplectic Group

305

can be written as (−δ j + n, −δ j−1 + n − 1, . . . , −δ1 + n − j + 1, α1 + n − j, . . . , αn− j + 1) for a partition δ with |δ| = |γ | − 2x. The partition δ has smaller invariant s than γ has. We count the number l(α, γ ) − l(α, δ). The difference is that the number −γs + n − j + s gets exchanged twice with numbers from the set α1 + n − j, . . . , αn− j + 1 that are in [1, x − 1], then it is reﬂected to its negative, and then it gets exchanged additionally with the numbers from the interval [1, x − 1] that preceed it. Therefore we have l(α, γ ) − l(α, δ) ≤ 2x − 1 and thus N (α, γ ) > N (α, δ). It remains to show that for ﬁxed α and for partitions γ such that R(α, γ ) = (n, n − 1, . . . , 1) and s = 0 we have N (α, γ ) ≥ 0. But now we see that the last reﬂection does not occur in exchanging entries. Therefore we are in the situation of Lemma (8.1.5). We see that the choice of γ minimizing N (α, γ ) is γ = α and that N (α, α ) = 0. This proves Lemma (8.5.9) and therefore part (a) of Proposition (8.5.8). We prove part (b) of Proposition (8.5.8). We use the exact sequence (∗∗). We will prove that the cohomology of •

(S2 R) ⊗

•

(R ⊗ (F/R)) ⊗ Vα (R∨ /R)

does not contain the representations Vβ F with β1 > j. Using the exact sequence 0 → R∨ /R → F/R → F/R∨ → 0, we see that it is enough to show that the cohomology of • • • (S2 R) ⊗ (R ⊗ (R∨ /R)) ⊗ (R ⊗ (F/R∨ )) ⊗ Vα (R∨ /R)

does not contain the representations Vβ F with β1 > j. Let us look at a typical term K β R ⊗ K γ R ⊗ K γ (R∨ /R) ⊗ K δ R ⊗ K δ (F/R∨ ) ⊗ Vα (R∨ /R). After decomposing the Weyl functors to the symplectic irreducibles and calculating tensor products, we see that the term above will decompose to the

306

The Nilpotent Orbit Closures

summands of the form K λ (F/R∨ ) ⊗ Vθ (R∨ /R). Such a bundle has a representation Vβ F with β1 > j in cohomology if a sequence (λ1 + n, λ2 + n − 1, . . . , λ j + n − j + 1, θ1 + n − j, . . . , θn− j + 1) contains either a number > n + j or a number < −n − j. Since θ is an integral dominant weight for Sp(2n − 2 j), i.e. a partition, this can happen if one of the following cases occurs: (a) λ1 + n > n + j, i.e. λ1 > j, (b) θ1 + n − j > n + j, i.e. θ1 > 2 j, (c) λ j + n − j + 1 < −n − j, i.e. −λ j > 2n + 1. Case (a) cannot occur because K λ (F/R∨ ) is a summand in a tensor product K β R ⊗ K γ R ⊗ K δ R ⊗ K δ (F/R∨ ) and the only possible positive weights come from δ . However, δ1 ≤ j, since dim R = j. Case (b) cannot occur because the weight θ comes from a summand in K γ (R∨ /R) ⊗ Vα (R∨ /R), and α1 ≤ j and γ1 ≤ j (because the corresponding functor K γ R is nonzero). Thus K γ (R∨ /R) decomposes to the representations V (R∨ /R), and in the tensor product V (R∨ /R) ⊗ Vα (R∨ /R) all weights have to be ≤ α + by Klimyk’s formula (exercise 12 of chapter 4), and therefore the ﬁrst entry of each occurring weight is ≤ 2 j. Case (c) cannot occur because in the tensor product K β R ⊗ K γ R ⊗ K δ R ⊗ K δ (F/R∨ ) the last entries are −β1 ≥ − j − 1, −γ1 ≥ 2 j − 2n, and −δ1 ≥ − j (because K δ (F/R∨ ) has to be nonzero). This concludes the proof of part (b) of Proposition (8.5.8).•

8.5. Conjugacy Classes for the Symplectic Group

307

(8.5.10) Corollary. (a) The term F• consists of one copy of the trivial representation in homogeneous degree 0. In particular the variety Z j is normal, with rational singularities. (b) The terms F1 contains only the representations Vβ (F) with β1 ≤ 1. In particular the deﬁning ideal of Y j := q(Z j ) is generated by such representations. More precisely, the minimal set of generators of the deﬁning ideal of Y j consists of the vectors in the representation 2(n− j+1)

F ⊂ K (22(n− j+1) ) F ⊂ S2(n− j+1) (S2 F).

Proof. The ﬁrst claim follows from the proof of Lemma (8.5.9). Indeed, the only term contributing to F0 corresponds to γ = 0, which, as seen from the sequences (?), can appear only for m = 0. The proof of the second part is based on the reﬁnement of Lemma (8.5.9). Claim. The only partition γ with |γ | = m such that H m−1 (IGrass( j, F), K γ R) = 0 is γ = (2(n − j + 1)), and the resulting cohomology group is a trivial representation of Sp(F). Proof. We repeat the proof of Lemma (8.5.9) with α = (0). Our partition γ has to lead in one step to the partition γ = 0, which gives the only partition with N (γ , (0)) = 0. The sequence R(γ , (0)) has to be (n, n − 1, . . . , 1), because if not, then s > 0 and the ﬁrst step in the procedure does decrease s. We see now that s = 1; otherwise we need s steps to reduce to γ = 0. Now we can do a case by case analysis, because the number of possibilities for γ corresponds to the possibilities for the number γ1 − n + j − 1 ∈ {n, n − 1, . . . , n − j + 1}. The only viable possibility turns out to be γ1 − n + j − 1 = n − j + 1, which implies γt = 0 for t ≥ 2. Now we see using the exact sequence (?) resolving m S j that the only 2(n− j+1) F in homogeneous degree 2(n − j + possible contribution to F1 is 1). We easily identify this set of equations with the ones given in (8.5.10)(b), because they vanish on Y j . We can ﬁnally state the results of our induction. (8.5.11) Theorem. Let µ be an even partition, µ = 2ν. (a) The orbit closure Yµ is normal, with rational singularities,

308

The Nilpotent Orbit Closures

(b) The irreducible representations occurring in the terms of the complex F•µ have highest weights (β1 , . . . , βn ) with β1 ≤ ν1 . µ (c) The term F1 contains only the representations Vβ (F) with β1 ≤ 1. Therefore the deﬁning ideal of Yµ is generated by the representations of this kind. Proof. We use the induction described above, connecting the relative complex F•µˆ to the complex F•µ . We use the exact sequence 0 → Sν1 → Sν → Sνˆ (R∨ /R) → 0 to estimate the terms in the cohomology of Sν . They come from the terms of the cohomology in Vα (R∨ /R) ⊗ • (Sν1 ). By induction the terms Vα (R∨ /R) satisfy the assumption of Proposition (8.5.8), which shows that the i-th term of the complex F µˆ (R∨ /R)• can produce only the terms in the homological degree ≥ i. Thus, by induction, only the terms in nonnegative homological µ degree occur, and by Corollary (8.5.10) the term F0 consists of one copy of the trivial representation in homogeneous degree 0. Parts (a) and (b) of the theorem follow. µ To prove part (c) let us analyze the term F1 . The terms in this complex come from the term F• , which has the required form by Corollary µ ˆ (8.5.10)(b), and from the terms F1 (R∨ /R), which by induction are the terms of homological degree zero in F(V(α1 ,...,αn− j ) (R∨ /R))• , with weight α for which α1 ≤ 1. By the proof of Lemma (8.5.9) the only such term comes from γ = α , so (by the use of the complexes Vα (R∨ /R) ⊗ (?)) the resulting contribution to the possible terms of F(V(α1 ,...,αn− j ) (R∨ /R))• is K α F ⊗ V(0) F, which decomposes to the representations of Sp(F) of the required form. Let me ﬁnish this section with a general conjecture regarding the equations of the nilpotent orbit closures for the symplectic group. (8.5.12) Conjecture. Let µ ∈ Ps (2n). The deﬁning ideal of Yµ is generated by representations V(β1 ,...,βn ) (F) with β1 ≤ 2. More precisely, the generators of the deﬁning ideal of Yµ can be chosen as subrepresentations of Schur functors K (2i ) F inside Si (S2 F) for 1 ≤ i ≤ 2n. (8.5.13) Remark. Klimek has proved that Conjecture (8.5.12) is true up to a radical.

Exercises for Chapter 8

309

Exercises for Chapter 8 Orbits Corresponding to Special Partitions for Classical Groups Type Bn . We assume that F is a vector space of dimension 2n + 1 and that ( , ) is a nondegenerate symmetric form. We identify F with F ∗ by means of the morphism induced by the form. 1. Consider the adjoint representation X = 2 F. We identify A with 2 F). Show that the space K (22i ) F (1 ≤ i ≤ n) contains a unique Sym( (up to a scalar) SO(F)-invariant v2i . Prove that the ring of invariants ASO(F) is isomorphic to the polynomial ring generated by the invariants v2 , . . . , v2n . This is a special case of Chevalley’s theorem. 2. Let F be a vector space of dimension 2n + 1, and let ( , ) be a nondegene rate symmetric form. We take g = so(2n + 1) = 2 F. The subregular orbit in g is the only orbit of codimension 2 in the nilpotent cone. It corresponds to the partition µ = (2n − 1, 1, 1). Let {e, h, f } be an sl2 -triple where e is an element from the subregular orbit. We denote by g i the component of weight i in g considered as a representation of sl2 . (a) Let ξn be the vector bundle occurring in the resolution (5.1.1). We saw in the section 8.4 that we have p ∗ (ξn ) = g ∗

COHOMOLOGY OF VECTOR BUNDLES AND SYZYGIES The central theme of this book is an exposition of the geometric technique of calculating syzygies. It is written from the point of view of commutative algebra; without assuming any knowledge of representation theory, the calculation of syzygies of determinantal varieties is explained. The starting point is a deﬁnition of Schur functors, and these are discussed from both an algebraic and a geometric point of view. Then a chapter on various versions of Bott’s theorem leads to a careful explanation of the technique itself, based on a description of the direct image of a Koszul complex. Applications to determinantal varieties follow. There are also chapters on applications of the technique to rank varieties for symmetric and skew symmetric tensors of arbitrary degree, closures of conjugacy classes of nilpotent matrices, discriminants, and resultants. Numerous exercises are included to give the reader insight into how to apply this important method.

CAMBRIDGE TRACTS IN MATHEMATICS General Editors

B. BOLLOBAS, W. FULTON, A. KATOK, F. KIRWAN, P. SARNAK

149

Cohomology of Vector Bundles and Syzygies

Jerzy Weyman Northeastern University

Cohomology of Vector Bundles and Syzygies

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge , United Kingdom Published in the United States by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521621977 © Jerzy Weyman 2003 This book is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2003 ISBN-13 ISBN-10

978-0-511-06601-6 eBook (NetLibrary) 0-511-06601-5 eBook (NetLibrary)

ISBN-13 978-0-521-62197-7 hardback ISBN-10 0-521-62197-6 hardback

Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

To Katarzyna

Contents

Preface

page xi

1 Introductory Material 1.1 Multilinear Algebra and Combinatorics 1.2 Homological and Commutative Algebra 1.3 Determinants of Complexes 2 Schur Functors and Schur Complexes 2.1 Schur Functors and Weyl Functors 2.2 Schur Functors and Highest Weight Theory 2.3 Properties of Schur Functors. Cauchy Formulas, Littlewood– Richardson Rule, and Plethysm 2.4 The Schur Complexes Exercises for Chapter 2 3 Grassmannians and Flag Varieties 3.1 The Pl¨ucker Embeddings 3.2 The Standard Open Coverings of Flag Manifolds and the Straightening Law 3.3 The Homogeneous Vector Bundles on Flag Manifolds Exercises for Chapter 3 4 Bott’s Theorem 4.1 The Formulation of Bott’s Theorem for the General Linear Group 4.2 The Proof of Bott’s Theorem for the General Linear Group 4.3 Bott’s Theorem for General Reductive Groups Exercises for Chapter 4 5 The Geometric Technique 5.1 The Formulation of the Basic Theorem 5.2 The Proof of the Basic Theorem 5.3 The Proof of Properties of Complexes F(V)• ix

1 1 12 27 32 32 49 57 66 78 85 85 91 98 104 110 110 117 123 132 136 137 141 146

x

6

7

8

9

Contents

5.4 The G-Equivariant Setup 5.5 The Differentials in Complexes F(V)• . 5.6 Degeneration Sequences Exercises for Chapter 5 The Determinantal Varieties 6.1 The Lascoux Resolution 6.2 The Resolutions of Determinantal Ideals in Positive Characteristic 6.3 The Determinantal Ideals for Symmetric Matrices 6.4 The Determinantal Ideals for Skew Symmetric Matrices 6.5 Modules Supported in Determinantal Varieties 6.6 Modules Supported in Symmetric Determinantal Varieties 6.7 Modules Supported in Skew Symmetric Determinantal Varieties Exercises for Chapter 6 Higher Rank Varieties 7.1 Basic Properties 7.2 Rank Varieties for Symmetric Tensors 7.3 Rank Varieties for Skew Symmetric Tensors Exercises for Chapter 7 The Nilpotent Orbit Closures 8.1 The Closures of Conjugacy Classes of Nilpotent Matrices 8.2 The Equations of the Conjugacy Classes of Nilpotent Matrices 8.3 The Nilpotent Orbits for Other Simple Groups 8.4 Conjugacy Classes for the Orthogonal Group 8.5 Conjugacy Classes for the Symplectic Group Exercises for Chapter 8 Resultants and Discriminants 9.1 The Generalized Resultants 9.2 The Resultants of Multihomogeneous Polynomials 9.3 The Generalized Discriminants 9.4 The Hyperdeterminants Exercises for Chapter 9

References Notation Index Subject Index

149 152 154 156 159 160 168 175 187 195 209 213 218 228 228 234 239 245 251 252 263 278 283 296 309 313 314 318 328 332 355 359 367 369

Preface

This book is devoted to the geometric technique of calculating syzygies. This technique originated with George Kempf and was ﬁrst used successfully by Alain Lascoux for calculating syzygies of determinantal varieties. Since then it has been applied in studying the deﬁning ideals of varieties with symmetries that play a central role in geometry: determinantal varieties, closures of nilpotent orbits, and discriminant and resultant varieties. The character of the method makes it comparable to the symbolic method in classical invariant theory. It works in only a limited number of cases, but when it does, it gives a complete answer to the problem of calculating syzygies, and this answer is hard to get by other means. Even though the basic idea is more than 20 years old, this is the ﬁrst book treating the geometric technique in detail. This happens because authors using the geometric method have usually been interested more in the special cases they studied than in the method itself, and they have used only the aspects of the method they needed. Therefore the basic theorems from chapter 5 stem from the efforts of several mathematicians. The possibilities offered by the geometric method are not exhausted by the examples treated in the book. The method can be fruitfully applied to any representation of a linearly reductive group with ﬁnitely many orbits and with actions such that the orbits can be described explicitly. The varieties treated in chapters 7 and 9 show that the scope of the method is not limited to such actions. The book is written from the point of view of a commutative algebraist. We develop the rudiments of representation theory of general linear group in some detail in chapters 2–4. At the same time we assume some knowledge of commutative algebra and algebraic geometry, including sheaves and their cohomology—for example, the notions covered in chapters II and III of the book [H1] of Hartshorne. The notions of Cohen–Macaulay and Gorenstein

xi

xii

Preface

rings and rational singularities are also used, and their deﬁnitions are brieﬂy recalled in chapter 1. Some parts of the book demand more advanced knowledge. One statement in chapter 5 is an application of Grothendieck duality, whose statement is brieﬂy recalled in chapter 1. We advise less experienced readers to just accept Theorem (5.1.4) and look ﬁrst at its applications. In some sections of chapters 5 and 8 we assume familiarity with highest weight theory and some facts on linear algebraic groups. At the same time, in exercises we use the geometric technique to develop some of that theory for the classical groups. Still, the reader not familiar with these notions should be able to understand all the remaining chapters. Let us describe brieﬂy the contents of the book. The ﬁrst chapter discusses preliminaries. We recall elementary notions from multilinear algebra and combinatorics. The section on commutative and homological algebra covers brieﬂy the deﬁnitions of depth, Koszul complexes, the Auslander– Buchsbaum–Serre theorem, and Cohen–Macaulay and Gorenstein rings. There is also a section on de Jong’s algorithm for explicit calculation of normalization, on the exactness criterion of Buchsbaum and Eisenbud, and on Grothendieck duality. The ﬁnal section is devoted to a brief review of the notion of the determinant of a complex. Chapters 2, 3, and 4 are devoted to developing the representation theory of general linear groups and to the proof of Bott’s theorem on the cohomology of line bundles on homogeneous spaces. This provides the basic tools for the calculations to be performed in later chapters. Our approach is based on Schur and Weyl functors, introduced in the ﬁrst section of chapter 2. They are deﬁned by generators and relations. The relation to highest weight theory and Schur–Young theory is discussed. This is followed by a discussion of Cauchy formulas, the Littlewood–Richardson rule, and plethysm. The ﬁnal section of chapter 2 discusses Schur complexes. In chapter 3 we relate Schur functors to geometry by realizing them as multihomogeneous components of homogeneous coordinate rings of ﬂag varieties. This is followed by a proof of the Cauchy formula based on restriction of the straightening from the Grassmannian to its afﬁne open subset, and by a section on tangent bundles of Grassmannians and ﬂag varieties. In chapter 4 we prove Bott’s theorem on cohomology of line bundles on ﬂag varieties. We follow the approach of Demazure. In the last section we formulate the theorem for arbitrary reductive groups and give explicit interpretations for classical groups.

Preface

xiii

Chapter 5 is devoted to the description of terms and properties of the direct images of Koszul complexes. We study the basic setup, i.e., the diagram Z ⊂ ↓ q Y ⊂

X×V ↓q X

where X is an afﬁne space, V is a nonsingular projective variety, Z is a total space of a vector subbundle S of the trivial bundle X × V , and Y = q(Z ). The variety Z can be described as the vanishing set of a cosection p ∗ (ξ ) → O X ×V , and it is a locally complete intersection. Here ξ is the dual of the factor (X × V )/S. The original idea of Kempf was that the study of a direct image of the resulting Koszul complex can be used to prove results about the deﬁning equations and syzygies of the subvariety Y . We give the basic properties of these direct images in the general case and in important special cases, for example when Z is a desingularization of Y . We also treat the more general case of twisted Koszul complexes. The remaining chapters are devoted to examples and applications. In each of these chapters a different aspect of the method is illustrated. In chapter 6 we study the case of determinantal varieties. Here we show how to handle basic calculations in simple cases when the bundle ξ is a tensor product of tautological bundles. Apart from the proof of Lascoux’s theorem and the calculation of syzygies of determinantal ideals for symmetric and skew symmetric matrices, we also give results on the equivariant modules supported in determinantal varieties. Chapter 7 is devoted to the rank varieties for tensors of degree higher than two. This illustrates that the method can be applied in cases when the variety X or even Y does not have ﬁnitely many orbits with respect to some action of the reductive group. In chapter 8 the study of nilpotent orbit closures allows us to understand how to handle the situation when the cohomology groups needed to get the syzygies cannot be calculated directly, but still partial results can be recovered by estimating the terms in a spectral sequence associated to ﬁltrations on a basic bundle ξ . We also prove the Hinich–Panyushev theorem on rational singularities of normalizations of nilpotent orbits for a general simple group. Chapter 9 illustrates the use of twisted modules supported in resultant and discriminant varieties, which allow one to get natural determinantal expressions for the deﬁning equation. Each chapter is followed by exercises. They should allow readers to learn how to apply the geometric method on their own. At the same time they

xiv

Preface

illustrate further applications of the method. In particular the exercises to chapter 6 deal with the analogues of the determinantal varieties for the symplectic and orthogonal groups. In exercises to chapter 7 we give some calculations of minimal resolutions of Pl¨ucker ideals. The book can be read on several levels. For the reader who is not familiar with representation theory and/or derived categories and Grothendieck duality, we suggest ﬁrst reading chapters 2 through 4. Then one can proceed with the proof of the statement of Theorem 5.1.2 given in section 5.2. At that point most of the following chapters (with the exception of section 8.3) can be understood using only that statement. In such a way the book could be used as the basis of an advanced course in commutative algebra or algebraic geometry. It could also serve as the basis of a seminar. A lot of general notions and theories can be nicely illustrated in the special cases treated in later chapters by the methods given in the book. A reader familiar with representation theory can just skim through chapters 2 through 4 to get familiar with the notation, and then proceed straight to chapter 5 and study the applications. I am indebted to many people who introduced me to the subject, especially to David Buchsbaum, Corrado De Concini, Jack Eagon, David Eisenbud, Tadeusz J´ozeﬁak, Piotr Pragacz, and Joel Roberts. I also beneﬁted from conversations on some aspects of the material with Kaan Akin, Giandomenico Bofﬁ, Michel Brion, Bram Broer, Andrzej Daszkiewicz, Steve Donkin, Toshizumi Fukui, Laura Galindo, Wilberd van der Kallen, Jacek Klimek, Hanspeter Kraft, Witold Kra´skiewicz, Alain Lascoux, Steve Lovett, Olga Porras, Claudio Procesi, Rafael Sanchez, Mark Shimozono, Alex Tchernev, and Andrei Zelevinsky. Throughout my work on the book I was partially supported by grants from the National Science Foundation.

1 Introductory Material

1.1. Multilinear Algebra and Combinatorics 1.1.1. Exterior, Divided, and Symmetric Powers; Multiplication and Diagonal Maps Let K be a commutative ring, and let E be a free K-module with a basis {e1 , . . . , en }. We deﬁne the r -th exterior power r E of E to be the r -th tensor power E ⊗r of E divided by the submodule generated by the elements: u 1 ⊗ . . . ⊗ u r − (−1)sgn σ u σ (1) ⊗ . . . ⊗ u σ (r ) for all σ ∈ r , u 1 , . . . , u r ∈ E. We denote the coset of u 1 ⊗ . . . ⊗ u r by u 1 ∧ . . . ∧ ur . The following basic properties of exterior powers are proved in [L, chapter XIX, section 1]. (1.1.1) Proposition. (a) Let {e1 , . . . , en } be an ordered basis of E. Then the elements ei1 ∧ . . . ∧ eir for 1 ≤ i 1 < . . . < ir ≤ n form abasis of r E. In particular, r E is a free K-module of dimension nr . (b) (Universality property of exterior powers) We have a functorial isomorphism θ M : Alt (E , M) → HomK r

r

r

E, M

where Altr (E r , M) denotes the set of multilinear alternating maps from r ( f )(u 1 ∧ . . . ∧ u r ) = f (u 1 , . . . , u r ). E ×r to M, given by the formula θ M

1

2

Introductory Material

(c) We have natural isomorphisms α : r

r

∗

(E ) →

r

∗ E

sending the exterior product l1 ∧ . . . ∧ lr to the linear function l on e E deﬁned by the formula l(u 1 ∧ . . . ∧ u r ) = (−1)sgn σ lσ (1) (u 1 ) . . . lσ (r ) (u r ). σ ∈ r

The r -th exterior power is an endofunctor on the category of free Kmodules and linear maps. More precisely, for two free K-modules E, F and a linear map φ : E → F we have a well-deﬁned linear map r

r

φ:

r

E→

r

F

φ(u 1 ∧ . . . ∧ u r ) = φ(u 1 ) ∧ . . . ∧ φ(u r ). Let us deﬁned by the formula denote m = dim F. Let {e1 , . . . , en } be a basis of E and let { f 1 , . . . , f m } be a basis of F. In these bases φ correspond to the m × n matrix (φ j,i ) where φ(ei ) =

m

φ j,i f j .

j=1

The map r φ can be written in the corresponding bases of r E, r F as follows: r φ(ei1 ∧ . . . ∧ eir ) = M( j1 , . . . , jr | i 1 , . . . , ir ; φ) f j1 ∧ . . . ∧ f jr , 1≤ j1 0, and satisfying η(1) = 1. deﬁned to be zero on all spaces The following proposition is an elementary calculation.

(1.1.2) Proposition. (a) The maps m, , , η deﬁne on • (E) the structure of commutative, cocommutative bialgebra. (b) The map α : • (E ∗ ) → ( • E)∗ deﬁned in (1.1.1) (c) is an isomorphism of bialgebras. Part (b) of the proposition means that the dual map to the multiplication map m on • (E) is the diagonal map on • (E ∗ ) and vice versa. We deﬁne the r -th symmetric power Sr E of E to be the r -th tensor power E ⊗r of E divided by the submodule generated by the elements u 1 ⊗ . . . ⊗ u r − u σ (1) ⊗ . . . ⊗ u σ (r ) for all σ ∈ r , u 1 , . . . , u r ∈ E. We denote the coset of u 1 ⊗ . . . ⊗ u r by u 1 . . . ur . The following basic properties of symmetric powers are proved in [L, chapter XVI, section 8].

4

Introductory Material

(1.1.3) Proposition. (a) Let {e1 , . . . , en } be an ordered basis of E. Then the elements e1i1 . . . enin for i 1 + . . . + i n = r form a basis of Sr E. In particular Sr E is a free n+r −1 K-module of dimension . r (b) (Universality property of symmetric powers) We have a functorial isomorphism θ M : Symr (E r , M) → HomK (Sr E, M) where Symr (E r , M) denotes the set of multilinear symmetric maps r ( f )(u 1 . . . u r ) = f (u 1 , . . . , u r ). from E ×r to M, given by the formula θ M The r -th symmetric power is an endofunctor on the category of free Kmodules and linear maps. More precisely, for two free K-modules E, F and a linear map φ : E → F we have a well-deﬁned linear map Sr φ : Sr E → Sr F deﬁned by the formula Sr φ(u 1 . . . u r ) = φ(u 1 ) . . . φ(u r ). Let us denote m = dim F. Let {e1 , . . . , en } be a basis of E, and let { f 1 , . . . , f m } be a basis of F. In these bases φ correspond to the m × n matrix (φ j,i ) where φ(ei ) =

m

φ j,i f j .

j=1

The map Sr φ can be written in the corresponding bases of Sr E, Sr F as follows: Sr φ(ei1 . . . eir ) = P( j1 , . . . , jr | i 1 , . . . , ir ; φ) f j1 . . . f jr , 1≤ j1 0, and satisfying η(1) = 1. We have the following analogue of (1.1.2) (a). (1.1.4) Proposition. The maps m, , , η deﬁne on Sym(E) the structure of a commutative, cocommutative bialgebra. Let us investigate the duality. The algebra Sym(E) = r ≥0 Sr E is not ﬁnite dimensional, so instead of the dual we have to work with the graded dual (Sr E)∗ . Sym(E)∗gr := r ≥0

6

Introductory Material

The module map E ∗ = (S1 E)∗ → Sym(E)∗gr induces by universality an algebra map β : Sym(E ∗ ) → Sym(E)∗gr . This map β is an isomorphism only when K contains a ﬁeld of rational numbers. In fact it is given by the formula lσ (1) (u 1 ) . . . lσ (r ) (u r ). β(l1 . . . lr )(u 1 . . . u r ) = σ ∈r

In particular, when l1 = . . . = lr , u 1 = . . . = u r we see that β(l1r ) = r !(u r1 )∗ . In order to describe the graded dual of the symmetric algebra we introduce the divided powers. We deﬁne the r -th divided power Dr (E) as the dual of the symmetric power. Dr (E) := (Sr (E ∗ ))∗ . Its basis is the dual basis to the natural basis of the symmetric power. If {e1 , . . . , en } is a basis of E, we deﬁne e1(i1 ) . . . en(in ) to be the element of the dual basis to the basis {(e1∗ ) j1 . . . (en∗ ) jn }, dual to (e1∗ )i1 . . . (en∗ )in . For every u ∈ E we can deﬁne its r -th divided power u (r ) ∈ Dr E. It is given by the formula (r ) n p (p ) u i ei = u 1 1 . . . u npn e1 1 . . . en( pn ) . i=1

p1 +...+ pn =r

It is easy to check that this deﬁnition does not depend on the choice of basis {e1 , . . . , en }. (1.1.5) Proposition. The divided powers have the following properties: (a) (b) (c) (d) (e)

u (0) = 1, u(1) = u, u (r ) ∈ Dr E, u ( p) u (q) = p+q u ( p+q) , q

p (u + v)( p) = k=0 u (k) v ( p−k) , ( p) ( p) ( p) (uv) = u v , (u ( p) )(q) = [ p, q]u ( pq) for u ∈ E; [ p, q] = [( pq)!]/(q! pq !).

(1.1.6) Remark. In the notation used above, e1(i1 ) . . . en(in ) has a double meaning. It is the element of the dual basis to the basis in the symmetric power,

1.1. Multilinear Algebra and Combinatorics

7

and it is the product of divided powers. It is not difﬁcult to see that the two elements coincide. The r -th divided power is an endofunctor on the category of free K-modules and linear maps. More precisely, for two free K-modules E, F and a linear map φ : E → F we have a well-deﬁned linear map Dr φ : Dr E → Dr F which is best described as the transpose of the map Sr (φ ∗ ) : Sr (F ∗ ) → Sr (E ∗ ). This also gives the description of the matrix coefﬁcients for Dr φ as polynomials in the entries of φ, which we leave to the reader. Dr (E) on E is a commutative, The divided power algebra D(E) := cocommutative algebra because it is a graded dual of the symmetric algebra on E ∗ . Again we denote the components of the multiplication map by m : Dr E ⊗ Ds E → Dr +s E, and the components of the comultiplication by : Dr +s E → Dr E ⊗ Ds E. Let us record the duality statements. (1.1.7) Proposition. (a) The multiplication map m : Dr E ⊗ Ds E → Dr +s E is the dual of the diagonal map : Sr +s E ∗ → Sr E ∗ ⊗ Ss E ∗ . (b) The diagonal map : Dr +s E → Dr E ⊗ Ds E is the dual of the multiplication map m : Sr E ∗ ⊗ Ss E ∗ → Sr +s E ∗ . (c) The diagonal map : Dr +s E → Dr E ⊗ Ds E is given by the formula (e1(i1 ) . . . en(in ) ) =

j1 +...+ jn =r, 0≤ js ≤i s for s=1,...,n

(j )

(i − j1 )

e1 1 . . . en( jn ) ⊗ e1 1

. . . en(in − jn ) .

8

Introductory Material

1.1.2. Partitions, Skew Partitions. Combinatorics of Z2 -Graded Tableaux. Let n be a natural number. A partition λ of n is a sequence λ = (λ1 , . . . , λs ) of natural numbers such that λ1 ≥ λ2 ≥ . . . ≥ λs ≥ 0 and λ1 + λ2 + . . . + λs = n. We identify the partitions (λ1 , . . . , λs ) and (λ1 , . . . , λs , 0). To each partition λ we associate its Young frame (or Ferrers diagram) D(λ). It can be deﬁned as D(λ) = {(i, j) ∈ Z × Z |(1 ≤ i ≤ s, 1 ≤ j ≤ λi }. To represent the Young frames graphically we think of them as contained in the fourth quadrant. A Young frame is a set of boxes with λi boxes in the i-th row from the top. Formally it could be achieved by considering the point ( j, −i) instead of (i, j). (1.1.8) Example. λ = (4, 2, 1): D(λ) =

.

Formally the boxes of D((4, 2, 1)) correspond to the set of points {(1, −1), (2, −1), (3, −1), (4, −1), (1, −2), (2, −2), (1, −3)} in the grid Z × Z. Let λ be a partition. We say that λ has a Durfee square of size r (or rank λ = r ) if λr ≥ r , λr +1 ≤ r , i.e., if the biggest square ﬁtting inside of λ is an r × r square. Let λ be a partition, and let X be a box in λ. The set of boxes to the right of X (including X ) is called an arm of X . The set of boxes below X (including X ) is called the leg of X . The arm length (leg length) of X are deﬁned as the numbers of boxes in the arm (leg) of X . The arm and leg of X form a hook of X . The number of boxes in the hook of X is called the hook length of X . Let λ be a partition of rank r . Let ai (bi ) be the arm length (leg length) of the i-th box on the diagonal of λ. The partition λ is uniquely determined by its rank r and the numbers ai , bi (1 ≤ i ≤ r ). These numbers satisfy the conditions a1 > . . . > ar > 0, b1 > . . . , br > 0.

1.1. Multilinear Algebra and Combinatorics

9

We will sometimes denote by λ = (a1 , . . . , ar |b1 , . . . , br ) the partition with diagonal arm lengths ai and diagonal leg lengths bi . We refer to this as a Frobenius (or hook) notation for λ. (1.1.9) Example. The partition λ = (4, 3, 2) in the hook notation is (4, 2|3, 2). ¯ The boxes in the arm (leg) of the i-th diagonal box are ﬁlled with symbol i (i): X 1¯ 1¯

1 1 1 . X 2 ¯2

Let λ be a partition. The conjugate (or dual) partition λ is deﬁned by setting λi = card{t |λt ≥ i}. The Young frame of λ is obtained from the Young frame of λ by reﬂecting in the line y = −x. (1.1.10) Example. λ = (4, 2, 1), λ = (3, 2, 1, 1):

D(λ ) =

.

Let λ and µ be two partitions. We say that µ is contained in λ (denoted µ ⊂ λ) if for each i we have µi ≤ λi . Let λ and µ be two partitions with µ ⊂ λ. We refer to such a pair as a skew partition λ/µ. We associate to a skew partition λ/µ the skew Young frame D(λ/µ) := D(λ) \ D(µ). Graphically we can represent it as a Young frame of λ with the boxes corresponding to µ missing. (1.1.11) Example. λ = (4, 2, 2, 1, 1), µ = (3, 1):

D(λ/µ) =

.

10

Introductory Material

Let A = (A0 , A1 ) be a Z2 -graded set, i.e. the pair of sets indexed by {0, 1}. Assume that the set A is ordered by a total order . A tableau of shape λ/µ with values in A is a function T : D(λ/µ) → A. (1.1.12) Definition. (a) A tableau T of shape λ/µ with values in A is row standard if for each (u, v) we have T (u, v) T (u, v + 1) with equality possible if T (u, v) ∈ A1 . (b) We say that a tableau T of shape λ/µ with values in A is column standard if T (u, v) T (u + 1, v) with equality possible when T (u, v) ∈ A0 . (c) A tableau T of shape λ/µ with values in A is standard if it is both column standard and row standard. (1.1.13) Notation. We denote by RST(λ/µ, A) (CST(λ/µ, A), ST(λ/µ, A)) the set of row standard (column standard, standard) tableaux of shape λ/µ with values in A. We denote by [1, m] ∪ [1, n] the Z2 -graded set A with A0 = [1, m], A1 = [1 , n ] and with the order deﬁned to be the natural order on A0 and A1 with A0 preceeding A1 . Similarly we deﬁne the Z2 -graded set [1, n] ∪ [1, m] (here A1 preceeds A0 ). (1.1.14) Examples. Let λ = (4, 2, 2, 1, 1), µ = (2, 1). Let A = [1, 2] ∪ [1, 3] . (a) The tableau 1 2

1 T1 = 1 1 2 2 is row standard but not column standard. (b) The tableau 1 1

1 T2 = 1 2 2 3 is column standard but not row standard.

1.1. Multilinear Algebra and Combinatorics

11

(c) The tableau 1 2 1 T3 = 1 2 2 3 is standard. Let λ/µ be a skew partition, and let A = (A0 , A1 ) be a Z2 -graded set ordered by the total order . We deﬁne the orders (relative to ) on the sets of row standard (column standard, standard) tableaux as follows. Consider the set RST(λ/µ, A). Given two tableaux T, U , we have T U if T = U . Assume that T = U . Let us write them as T = (T1 , . . . , Ts ), U = (U1 , . . . , Us ) with Ti (Ui ) being the part of T (U ) from the i-th row of λ/µ. Let j be the minimal i for which Ti = Ui . We have T j = (T ( j, 1), . . . , T ( j, λ j − µ j )), U j = (U ( j, 1), . . . , U ( j, λ j − µ j )). Now let k be the smallest index for which T ( j, k) = U ( j, k) (such a k exists by the choice of j). We say that T U if and only if T ( j, k) U ( j, k). The order on ST(λ/µ, A) is deﬁned to be the restriction of from RST(λ/µ, A). Finally we deﬁne the order on CST(λ/µ, A). Given two tableaux T, U from CST(λ/µ, A), then T = U implies T ≤ U . Assume T = U . We write T = (T 1 , . . . , T s ), U = (U 1 , . . . , U s ) with T i (U i ) being the part of T (U ) from the i-th column of λ/µ. Let j be the minimal i for which T i = U i . We have T j = (T (1, j), . . . , T (λj − µj , j)), U j = (U (1, j), . . . , U (λj − µj , j)). Now let k be the smallest index for which T (k, j) = U (k, j) (such k exists by the choice of j). We say that T U if and only if T (k, j) U (k, j). Note. The order on ST(λ/µ, A) is the restriction of the order on RST(λ/µ, A). It is different from the restriction of on CST(λ/µ, A). (1.1.15) Examples. Let λ = (2, 1), µ = (0). Set A = [1, 2] ∪ [1, 2] . In RST (λ/µ, A) we have 1 2 1 1 2 1 . 1 2 1

12

Introductory Material

In CST(λ/µ, A) we have 1 1 1 2 2 1 . 2 1 1 In ST(λ/µ, A) we have 1 2 1 1 . 1 2

1.2. Homological and Commutative Algebra 1.2.1. Regular Sequences, Koszul Complexes, Depth Let R be a commutative Noetherian ring. Let M be an R-module. The dimension dim M of M is deﬁned to be the Krull dimension of R/Ann(M), where Ann(M) = {x ∈ R | x M = 0 } is the annihilator of M. Let I be an ideal in R. If I M = M, we deﬁne the I -depth of M as depth R (I, M) = min {i | ExtiR (I, M) = 0 }. In the case I M = M we deﬁne depth R (I, M) = ∞. For a ﬁnitely generated R-module M we have I M = M if and only if depth R (I, M) < ∞ if and only if depth R (I, M) ≤ dim M. A sequence a = (a1 , . . . , an ) of elements from R is an M-sequence (or a regular sequence on M) if M = (a1 , . . . , an )M and the multiplication ai : Mi−1 → Mi−1 is injective for i = 0, 1, . . . , n − 1, where Mi := M/ (a1 , . . . , ai )M. The connection between these notions is expressed in (1.2.1) Theorem. Let R be a Noetherian ring, and M a ﬁnitely generated R-module. Let I be an ideal in R. The following conditions are equivalent: (a) depth R (I, M) ≥ n. (b) ExtiR (R/I, M) = 0 for i < n. (c) There exists an M-sequence a = (a1 , . . . , an ) of length n with ai ∈ I for i = 1, . . . , n. A regular sequence (a1 , . . . , an ) is a maximal regular M-sequence if there is no b such that (a1 , . . . , an , b) is an M-sequence. In particular the theorem

1.2. Homological and Commutative Algebra

13

implies that two maximal regular M-sequences with terms from I must have the same length, equal to depth(I, M). Let M be an R-module, and let a = (a1 , . . . , an ) be a sequence of elements from R. We deﬁne the Koszul complex K (a, M)• as follows. For an n-dimensional free R-module E = R n with a basis e1 , . . . , en we set K (a, M)i = i E ⊗ R M, and the differential d:

i

E ⊗R M →

i−1

E ⊗R M

is deﬁned by the formula d(e j1 ∧ . . . ∧ e ji ⊗ m) =

i

(−1)u+1 e j1 ∧ . . . ∧ eˆ ju ∧ . . . ∧ e ji ⊗ a ju m.

u=1

Let M be a ﬁnitely generated R-module. We deﬁne the codimension of M, codim R (M) := ht Ann(M), where ht denotes the height of an ideal. We also deﬁne the grade of M, grade R (M) = depth R (Ann(M), R). The homological properties of Koszul complex include the information about the depth. (1.2.2) Theorem. Let M be a ﬁnitely generated R-module, and let a = (a1 , . . . , an ) be a sequence of elements from R. Denote I = (a1 , . . . , an ). Then depth R (I, M) = n − max{ i |Hi (K (a, M)) = 0 }. (1.2.3) Corollary. Let R be a commutative ring. Assume that I = (a1 , . . . , an ) is an ideal generated by a regular sequence. Then the Koszul complex K (a, R)• is a free resolution of the R-module R/I . The ideal I is a complete intersection ideal of codimension n if there exists a regular sequence (a1 , . . . , an ) such that I = (a1 , . . . , an ). Thus the ﬁnite free resolutions of complete intersection ideals are provided by Koszul complexes. The projective dimension, codimension, and grade of an R-module are related.

14

Introductory Material

(1.2.4) Theorem. For an R-module M = 0 we have pd R (M) ≥ codim(M) ≥ grade R (M). A ﬁnitely generated R-module N is perfect if pd R (N ) = grade(N ). In that case the inequalities in (1.2.4) become equalities. We call codim(N ) the codimension of N . Sometimes by abuse of notation we call the grade of R/I the grade of the ideal I . Let us note the following consequence of Theorem (1.2.4) applied to N = R. (1.2.5) Proposition. Let I be an ideal of codimension n. The functor N → ExtnR (N , R) is an exact contravariant involution on the category of perfect modules N with Ann(N ) = I up to radical. If R/I is a perfect module, we call I a perfect ideal. An ideal I is Gorenstein if I is perfect and Extn (R/I, R) ∼ = R/I for n = codim(R/I ). 1.2.2. Cohen–Macaulay Rings and Modules, Gorenstein Rings Let (R, m) be a local ring. The depth of a module M is deﬁned as depth R (M) := depth R (m, M). We have the following inequalities: (1.2.6) Proposition. Let M be a ﬁnitely generated module over a local ring R. Then depth(M) ≤ dim M ≤ dim R. An R-module M is Cohen–Macaulay if depth(M) = dim M. If depth (M) = dim R, we say that M is maximal Cohen–Macaulay. The zero module is by deﬁnition maximal Cohen–Macaulay. The ring R is Cohen–Macaulay if it is Cohen–Macaulay as a module over itself. The projective dimension and depth of a module over a local ring are complementary to each other. (1.2.7) Theorem (Auslander–Buchsbaum Formula). Let R be a Noetherian local ring. Assume that pd R (M) < ∞. Then we have pd R (M) + depth(M) = depth(R).

1.2. Homological and Commutative Algebra

15

It follows that if R is Cohen–Macaulay and M is a maximal Cohen– Maculay module of ﬁnite projective dimension over R, then M is R-free. If R is a Cohen–Macaulay local ring and M is a ﬁnitely generated R-module of ﬁnite projective dimension, then M is Cohen–Macaulay if and only if it is perfect. If R is a Cohen–Macaulay local ring, then dim R = dim R/P for every associated prime P of R. This means that R is equidimensional. A local ring (R, m) is Gorenstein if an only if R has a ﬁnite injective dimension as an R-module. (1.2.8) Theorem. Let (R, m) be a local ring of dimension d. The following conditions are equivalent: (a) (b) (c) (d) (e)

R is Gorenstein, for i = d we have ExtiR (K , R) = 0, Extd (K , R) = K , there exists i > d such that ExtiR (K , R) = 0, ExtiR (K , R) = 0 for i < d, ExtdR (K , R) = K , R is Cohen–Macaulay and ExtdR (K , R) = K .

Recall that the embedding dimension of a local ring is emdim (R) = dim K m/m 2 . A local ring R is regular if emdim(R) = dim R. We denote by gl dim R the global dimension of R. (1.2.9) Theorem (Auslander and Buchsbaum, Serre). Let (R, m) be a local ring of dimension d. Then the following are equivalent: (a) (b) (c) (d) (e)

R is a regular local ring, gl.dim R < ∞, gl.dim R = d, pd R K = d, m is generated by a regular sequence of length d.

The connection between the notions of Cohen–Macaulay (Gorenstein) ring and perfect (Gorenstein) ideal is stated in the next proposition. (1.2.10) Proposition. (a) Let R be a Cohen–Macaulay local ring. Then the ring R/I is Cohen– Macaulay if and only if I is perfect, (b) Let R be a Gorenstein local ring. Then R/I is Gorenstein if and only if I is a Gorenstein ideal.

16

Introductory Material

The theory outlined above for local rings has an analogue for graded rings and graded modules. Let us state the corresponding statements. Let R be a graded ring R = i≥0 Ri where R0 = K is a ﬁeld and Ri are ﬁnite dimensional vector spaces over K . We assume that R is generated as a K -algebra by elements of degree 1, which implies that R is Noetherian. We denote by m the maximal ideal m = R+ = i>0 Ri . For a graded R-module M we denote depth R (M) := depth R (m, M). Then the following statements hold. (1.2.6) Proposition. Let M be a ﬁnitely generated graded module over a graded ring R. Then depth R (M) ≤ dim M ≤ dim R. (1.2.7) Theorem (Auslander–Buchsbaum formula). Let R be a graded ring, and let M be a graded R-module. Assume that pd R (M) < ∞. Then we have pd R (M) + depth R (M) = depth R (R). (1.2.8) Theorem. Let R be a graded ring of dimension d with the maximal ideal m = R + . The following conditions are equivalent: (a) (b) (c) (d) (e)

R is Gorenstein, for i = d we have ExtiR (K , R) = 0, Extd (K , R) = K , there exists i > d such that ExtiR (K , R) = 0, ExtiR (K , R) = 0 for i < d, ExtdR (K , R) = K , R is Cohen–Macaulay and ExtdR (K , R) = K .

The theorem characterizing the regular rings differs because the only graded regular ring is a polynomial ring. The embedding dimension of a graded ring R is emdim(R) = dim K m/m 2 . A graded ring R is regular if emdim(R) = dim R. (1.2.9) Theorem. Let R be a graded ring of dimension d with the maximal ideal m = R + . Then the following are equivalent: (a) (b) (c) (d) (e) (f)

R is a regular graded ring, gl.dim R < ∞, gl.dim R = d, pd R K = d, m is generated by a regular sequence of length d, R is a polynomial ring over K in d variables.

1.2. Homological and Commutative Algebra

17

The deﬁnitions of Cohen–Macaulay, Gorenstein, and regular rings generalize to the global case. We deﬁne a commutative ring R to be regular (Cohen–Macaulay, Gorenstein) if for every prime ideal P ∈ P in R(R) the localization R P is regular (Cohen–Macaulay, Gorenstein). 1.2.3. Minimal Resolutions We will be working throughout the book with complexes dn

d1

F• : . . . → Fn → . . . → F1 → F0 . . . of free R-modules. We use the following notation. The rank of the free module Fi will be denoted by f i . Sometimes we will choose bases {e(i) j }1≤ j≤ f i in free modules Fi . Then the homomorphism di can be identiﬁed with f i−1 × f i (i) matrix (φk, j )1≤k≤ f i−1 ,1≤ j≤ f i where di (e(i) j )=

f i−1

(i) (i−1) φk, . j ek

k=1

We deﬁne the rank ri of di to be the biggest integer r such that there exists a nonzero r × r minor of φ (i) . We denote by Iri (di ) the ideal of ri × ri minors (i) of the matrix φk, j. Assume that R is a local (resp. graded) ring, and let m denote the maximal ideal (resp. m = R+ ). A complex dn

d1

F• : . . . → Fn → . . . → F1 → F0 of free R-modules is minimal if for each i, 1 ≤ i ≤ n, we have di (Fi ) ⊂ m Fi−1 . Equivalently, after choosing bases in Fi we can identify the differential di with a matrix with entries in R. The minimality condition says that all entries of matrices of differentials di are in the maximal ideal m. (1.2.11) Proposition. Let (R, m) be a local ring. Denote K = R/m. Let M be an R-module. (a) The module M has a minimal free resolution F• . The resolution F• is unique up to isomorphism. (b) Fi ⊗ R R/m = ToriR (R/m, M), so rank Fi = dim K ToriR (R/m, M) . Let us notice that one can construct the minimal free resolution of a ﬁnitely generated module M from short exact sequences πi

0 → i+1 (M) → Fi → i (M) → 0

18

Introductory Material

where the modules i (M) and maps π1 are constructed inductively as follows. We take 0 (M) = M and consider the vector space V (M) = M/m M. We choose a basis {v1(0) , . . . v (0) f 0 } of the vector space V (M) and take as F0 a (0) (0) free R-module with a basis {e1(0) , . . . e(0) f 0 }. We deﬁne π0 (e j ) := m j where (0) m (0) j ∈ 0 (M) is an element whose representative modulo m M is v j . The homomorphism π0 is onto by Nakayama’s lemma. Next we deﬁne 1 (M) = Ker π0 and continue the procedure with 1 (M). The module i (M) is the i-th syzygy module of M. It is unique up to isomorphism. The uniqueness of minimal free resolution occurs also in the case of graded rings and graded modules. Let R = i≥0 Ri be a graded ring, as deﬁned in the previous section. We denote by m = R+ the maximal ideal of elements of positive degree. Let M = ⊕i≥0 Mi be a graded R-module. Recall that we deﬁne the shifted module M(n) by setting M(n)i := Mn+i . The analogue of (1.2.11) is true in the graded case. Then one can construct the minimal free resolution as sketched above, as the analogue of Nakayama’s lemma holds, and the analogue of (1.2.11) is true. Let F• be a minimal resolution of M. We will write (i, j) R(− j) f . Fi = j≥i

The numbers f (i, j) are the dimensions of the graded pieces of graded K -vector spaces ToriR (K , M). Sometimes they are called the graded Betti numbers of M. We have a very useful exactness criterion for acyclicity of a ﬁnite complex of free R-modules. (1.2.12) Theorem (Buchsbaum–Eisenbud acyclicity criterion, [BE1]). Let R be a Noetherian ring, and let dn

d1

F• : 0 → Fn → . . . → F1 → F0 be a complex of free ﬁnitely generated R-modules. Then F• is acyclic (i.e. Hi (F• ) = 0 for i > 0) if and only if the following two conditions hold: (a) ri + ri−1 = f i for 1 ≤ i ≤ n + 1 (with the convention rn+1 = 0), (b) depth(I (di )) ≥ i for 1 ≤ i ≤ n. (1.2.13) Proposition. Let F• be a complex from (1.2.12). √ √ (a) We have I (di ) ⊂ I (di+1 ) for 1 ≤ i ≤ n − 1,

1.2. Homological and Commutative Algebra

19

(b) Assume that F• is a resolution of a perfect module M, and let I be the deﬁning ideal of the support of M. Then for every i, 1 ≤ i ≤ n, we √ have I (di ) = I . Proof. Assume that I (di ) = R. Then the map di splits and it follows that all maps d j have to split for j > i. This means that I (d j ) = R for j > i. Applying localization, we get (a). To prove (b) we notice that if F• is a resolution of a perfect module M, then F ∗ is also a resolution of the perfect module with the same support by (1.2.5). Applying (a), we see that all radicals of ideals I (di ) are equal and therefore have to be equal to I . This statement implies the following result of Eagon and Northcott. (1.2.14) Theorem (Generic Perfection Theorem, [EN2]). Let K be a commutative ring, and let R = K[X 1 , . . . , X n ] be a polynomial ring over K. Let F• : 0 → Fm → Fm−1 → . . . → F1 → F0 be a free resolution of an R-module M. Assume that the complex F• is perfect, i.e., m = depth Ann R (M). Assume that M is free as a K-module. Then for every ring homomorphism φ : R → S such that m = depth Ann S (M ⊗ R S), the complex F• ⊗ R S is a free resolution over S of an S-module M ⊗ R S. Proof. Applying Proposition (1.2.13) (b) to the complex F• , we see that all ideals I (di ) for this complex are equal up to a radical to Ann R (M). We deduce that the same is true for the complex F ⊗ R S, and the depth assumption implies that the complex F ⊗ R S is acyclic by the Buchsbaum–Eisenbud criterion.

1.2.4. Effective Calculation of Normalization We describe a very useful algorithm due to Grauert and Remmert and to de Jong ([GRe], [dJ]) for calculating the normalization of a reduced afﬁne ring. The algorithm is based on the following criterion of normality. Consider a radical ideal J containing a nonzero divisor, whose zero set contains the nonnormal locus of R. Then we have canonical inclusions R ⊂ Hom R (J, J ) ⊂ R¯

20

Introductory Material

given by the maps r → ψr , ψ →

ψ(x) , x

where ψr is a multiplication by r and x ∈ J is a nonzero divisor. (1.2.15) Proposition ([DGJP]). Let R be a reduced Noetherian ring. Let J be a radical ideal in R containing a nonzero divisor, whose zero set V (J ) contains the nonnormality locus of R. Then R is normal if and only if R = Hom R (J, J ). The proposition implies that if R is not normal, then Hom R (J, J ) is an ¯ different from R. The algorithm consists intermediate ring between R and R, of ﬁnding J and then replacing R by the bigger ring Hom R (J, J ). The authors ¯ prove that after ﬁnitely many steps we reach R. One also has an interesting presentation of Hom R (J, J ) as a ring. One starts with the R-module generators u 0 = x, u 1 , . . . , u s of Hom R (J, J ). Since Hom R (J, J ) is an algebra, we have the quadratic relations s ui u j i, j u k ak = . x x x k=0

We also have the linear relations between u 0 , . . . , u s . Let us assume that the

j relations sk=0 bk u k ( j = 1, . . . , m) are the generators of the ﬁrst syzygies between u 0 , . . . , u s . We deﬁne an epimorphism of commutative R-algebras θ : R[T1 , . . . , Ts ] → Hom R (J, J ). (1.2.16) Proposition ([DGJP]). The kernel of the homomorphism θ is gen

j erated by the linear relations sk=1 bk T j ( j = 1, . . . , m) and the quadratic

s i, j relations Ti T j − k=0 ak Tk (1 ≤ i, j ≤ s). The above algorithm and presentation allow to write down the normalization quite explicitly.

1.2.5. Duality for Proper Morphisms and Rational Singularities In this subsection we use the notions related to derived categories. Our principal references are [H2], [GM]. Apart from the notion of rational singularities, these results will be used only in the proof of the duality statement for complexes F• (V) (Theorem (5.1.4)) and in the proof of the Hinich–Panyushev theorem in section 8.3. Thus this subsection can skipped in the ﬁrst reading.

1.2. Homological and Commutative Algebra

21

Let X be a locally Noetherian scheme. We denote by D ∗ (X ) (where ∗ = ∅, +, −, b) the derived category D ∗ (A), where A is the category of ∗ ∗ O X -modules. By DQco X (X ) we denote the thick subcategory DA (A) where A is the category of O X -modules and A = Qco X is a category of quasicoherent O X -modules. Note that by [H2, Proposition I.4.8] the natural embedding ∗ D ∗ (Qco X ) → DQco X (X ) is an equivalence of categories, for a quasicompact ∗ (X ). scheme X and for ∗ = +, ∅. We will use the abbreviation Dqc We start with the discussion of dualizing complexes. Let X be a Noetherian scheme. A complex of quasicoherent O X -modules I • is a dualizing complex of X if I • is bounded, each term of I • is an injective module, each cohomology group is coherent, and the canonical map O X → Hom•O X (I • , I • ) is a quasiisomorphism. A dualizing complex is treated as an object in the derived category, so any complex isomorphic to a dualizing complex in + DQco X (X ) is also called a dualizing complex. If I • is a dualizing complex of X , then for any complex F • of O X -modules with coherent cohomology groups, the canonical map F • → Hom•O X (Hom•O X (F • , I • ), I • ) is a quasiisomorphism. Dualizing complexes are unique in the following sense. (1.2.17) Theorem ([H2, Theorem V.3.1]). Let I • be a dualizing complex on X , and I • a complex of O X -modules bounded above with coherent cohomology groups. Then I • is dualizing if and only if there exists an invertible sheaf + L and an integer n such that I • is isomorphic to I • ⊗O X L[n] in DQco X (X ). In this case L and n are determined by L[n] = R Hom•O X (I • , I • ). Let X be a Noetherian scheme with a dualizing complex I X• . Let r := min{i ∈ Z | H i (I X• ) = 0}. We deﬁne the canonical sheaf ω X := H r (I X• ). The coherent sheaf ω X is deﬁned up to tensoring with an invertible sheaf. If the scheme X is afﬁne, this means that the canonical module ω X is deﬁned uniquely up to isomorphism as an O X -module. (1.2.18) Proposition. Let X be a Noetherian scheme. (a) If X is a Cohen–Macaulay scheme, then the dualizing complex I X• has only one cohomology, so ω X = I X• . (b) If X is a Gorenstein scheme, then ω X = O X .

22

Introductory Material

It follows that in the case of afﬁne Cohen–Macaulay schemes we recover the theory of canonical modules as described in [HKu]. Let X, Y be Noetherian schemes. The following theorem was proved by Nagata [N]. (1.2.19) Theorem. Let f : X → Y be a morphism of ﬁnite type between Noetherian schemes. Then f is compactiﬁable, i.e., there exists a scheme X˜ , a proper morphism p : X˜ → Y , and an open immersion i : X → X˜ such that pi = f . The factorization f = pi is called a compactiﬁcation of f . The following theorem is known as a global duality theorem for proper morphisms. (1.2.20) Theorem. Let p : Y → X be a proper morphism. Then the de+ + + (Y ) → Dqc (X ) has the right adjoint p ! : Dqc (X ) → rived functor R + p∗ : Dqc + (Y ). Dqc Let Fin(X ) denote the category of X -schemes of ﬁnite type. A morphism f : Y → Y in Fin(X ) has a compactiﬁcation f = pi by (1.2.19). We deﬁne + + f ! := i ∗ ◦ p ! : Dqc (Y ) → Dqc (Y ),

where p ! is the right adjoint of R + p∗ . The functors f ! have the following properties. (1.2.21) Proposition. With the above notation, the following hold: (a) The deﬁnition of f ! is independent of a compactiﬁcation f = pi. (b) For any two morphisms f, g in Fin(X ) we have (g ◦ f )! = f ! ◦ g ! , provided g ◦ f is deﬁned. (c) If h : Y → Y is a smooth morphism in Fin(X ), of relative dimension d, then h ! is isomorphic to the functor h , where L h (F) = h ∗ F ⊗O ωY /Y [d]. Y

(d) If g : Y → Y is a ﬁnite morphism in Fin(X ), then g ! is isomorphic to g , where g (F) = g¯ ∗ R Hom•OY (g∗ OY , F), where g¯ : (Y, OY ) → (Y , g∗ OY ) is the canonical morphism of ringed spaces associated to g.

1.2. Homological and Commutative Algebra

23

(e) Let f : Y → Y be a morphism from Fin(X ), and let g : Z → Y be a ﬂat morphism of Noetherian schemes. Let Z = Y ×Y Z with the commutative square f

Z ↓g

→

Y

→

f

Z ↓g . Y

Then we have a canonical isomorphism (g )∗ ◦ f ! = ( f )! ◦ g ∗ . (f) Let f : X → Y be a morphism of ﬁnite type. Let IY• be a dualizing complex on Y . Then I X• := f ! (IY• ) is a dualizing complex on X . (1.2.22) Theorem (Duality for Proper Morphisms). Let p : Y → X be a − + (Y ), G • ∈ Dqc (X ). proper morphism between Noetherian schemes, F• ∈ Dqc Then there is an isomorphism θ p : R p∗ R Hom•OY (F• , p ! G • ) ∼ = R Hom•O X (R p∗ F• , G • ). (1.2.23) Remark. The proof of Theorem (1.2.22) is the main subject of Hartshorne’s lecture notes [H2]. The existence of the adjoint p ! is proven in the appendix by Deligne. In our application we will use only the case when p is projective, and therefore one needs only the contents of ﬁrst three chapters of [H2]. The proofs in [H2] are rather complicated, and the signs are not always correct. These questions are addressed in recent lecture notes of Brian Conrad [C], where the fully rigorous version of the duality theorem is developed. An alternative approach based on techniques from algebraic topology was developed by Neeman [Ne]. Still, this approach depends on unbounded derived functors and techniques of Thomason [TT]. The reader should also compare the notes [Ha7] of Hashimoto for fuller treatment of this material. One of the applications of duality for proper morphisms is the notion of rational singularities. We follow the approach of Kempf from [KKMSD]. (1.2.24) Corollary. Let p : Y → X be a proper morphism of smooth varieties. Denote by m, n the dimensions of X, Y respectively. Let L be an invertible sheaf on Y . Assume that R i f ∗ (ωY ⊗ L−1 ) = 0 for i > 0. Then there are natural isomorphisms of sheaves on X given by R i p∗ L = Extn−m+i ( p∗ (ωY ⊗ L−1 ), ω X ). OX Proof. We apply Theorem (1.2.22) to the morphism f , and to the sheafs F = ωY ⊗ L−1 , G = ω X . Then we take the cohomology sheaves of complexes on both sides.

24

Introductory Material

Let X be a smooth scheme of dimension m. A quasicoherent sheaf F on m− j X is Cohen–Macaulay of pure dimension k if ExtO X (F, ω X ) is zero unless j = k is the dimension of the support of F. (1.2.25) Remark. The above deﬁnition is related to the deﬁnition of Cohen– Macaulay modules. Assume that X is an afﬁne smooth variety. Let A be the ˜ For a sheaf F = M ˜ the coordinate ring of X . Then we have ω X = O X = A. m− j condition on vanishing of ExtO X (F, ω X ) is equivalent to the vanishing of m− j modules Ext A (M, A). This translates to the depth condition by Theorem (1.2.1), which, by Proposition (1.2.6) and the following deﬁnition, is equivalent to the Cohen–Macaulayness of the module M. (1.2.26) Corollary. Let p : Y → X be a proper map of smooth varieties. Assume that R i f ∗ (ωY ⊗ L−1 ) = 0 for i > 0 and additionally R i p∗ L = 0 for i > 0. Then the sheaves p∗ (ωY ⊗ L−1 ), p∗ L are Cohen–Macaulay of pure dimension y. In fact the duality F → Extm−n (−, ω X ) interchanges them. We refer to the duality F → Extm−n (−, ω X ) as the Ext duality. It is an exact involution on the category of Cohen–Macaulay sheaves supported in p(Y ). (1.2.27) Definition. Let f : Z → Y be a proper birational morphism, with Z smooth. We call such f a resolution of singularities. The resolution f is a rational resolution if the following conditions are satisﬁed: (a) Y is normal, i.e., the natural map OY → f ∗ O Z is an isomorphism, (b) R i f ∗ O Z = 0 for i > 0, (c) R i f ∗ ω Z = 0 for i > 0. Condition (c) is not needed over a ﬁeld of characteristic 0 because of the following relative version of the Kodaira vanishing theorem. (1.2.28) Theorem (Grauert–Riemenschneider, [GR], [Ke4]). Let K be a map of characteristic 0. Let f : Z → Y be a proper map of smooth varieties deﬁned over K, with k = dim Z , n = dim Y . Then Ri f∗ω Z = 0 for i > k − n.

1.2. Homological and Commutative Algebra

25

(1.2.29) Proposition. Let f : Z → Y be a resolution of singularities. Assume that condition (c) from (1.2.27) holds. Assume that Y can be embedded in a smooth variety X . Then the resolution f is a rational resolution if and only if the following conditions are satisﬁed: (d) OY is Cohen–Macaulay. (e) The natural morphism f ∗ ω Z → ωY (where ωY = Extm−n O X (OY , O X ) is a dualizing sheaf on Y ) is an isomorphism. Proof. Let j : Y → X be an embedding. Let g = j ◦ f . Condition (c) is satisﬁed, so by (1.2.24) we get the isomorphisms j∗ R i f ∗ O Z = Extm−n+i (g∗ ω Z , ω X ). OX Let us assume that conditions (a) and (b) are satisﬁed. Condition (b) implies that g∗ ω Z or f ∗ ω Z are Cohen–Macaulay. If conditions (a) and (b) are satisﬁed, we have that OY = f ∗ O X is the Ext-dual of the Cohen–Macaulay sheaf g∗ ω Z . Thus OY is Cohen–Macaulay. The homomorphism in (e) is the Ext-dual of the isomorphism OY → f ∗ O Z . Thus (d) and (e) hold. Assume that (d) and (e) are true. Then OY is Cohen–Macaulay with Extdual Cohen–Macaulay sheaf ωY by (d). Using (e), we see that f ∗ ω Z is Cohen– Macaulay. The isomorphisms j∗ R i f ∗ O Z = Extm−n+i (g∗ ω Z , ω X ) OX imply condition (b) and that f ∗ O Z is the Ext-dual of f ∗ ω Z . Further, the Extdual of the homomorphism OY → f ∗ O Z of Cohen–Macaulay sheaves is an isomorphism by (e). This implies that the homomorphism OY → f ∗ O Z is an isomorphism. This implies (a), and the proposition is proven. (1.2.30) Remark. Let us assume that we work over a ﬁeld K of characteristic zero. By [GR] the sheaf f ∗ ω Z does not depend on the choice of desingularization Z , only on the variety Y . Therefore conditions (d) and (e) are just conditions on the variety Y . The assumption of embeddability in a smooth variety is locally satisﬁed for any variety Y . (1.2.31) Definition. The variety Y deﬁned over a ﬁeld K of characteristic zero has rational singularities if one of the equivalent conditions holds: (a) Conditions (d) and (e) are (locally) satisﬁed. (b) There exists a desingularization f : Z → Y which is a rational resolution. (c) Every desingularization f : Z → Y is a rational resolution.

26

Introductory Material

We ﬁnish this section with a nice criterion, due to Hinich [Hi], for a scheme to have rational singularities and be Gorenstein. (1.2.32) Proposition. Let f : X → Y be a proper birational morphism of ﬁnite type schemes over a ﬁeld K, with X smooth and Y normal. Suppose that there exists a morphism of sheaves φ : O X → ω X such that f ∗ (φ) : f ∗ (O X ) → f ∗ (ω X ) is an isomorphism. Then Y is Gorenstein and it has rational singularities. Proof. Consider the diagram f

−→

X p

Y !q

.

Spec K By (1.2.21) (b), p (K) = f q (K). Applying (1.2.21) (c) to p : X → Spec (K), one obtains p ! (K) = L p ∗ (K) ⊗ L ω X [n] = ω X [n], where n = dim X = dim Y . Denoting ωY := q ! (K)[−n], we have f ! (ωY ) = ω X . By (1.2.21) (f), ωY is the dualizing complex on Y . Applying the duality theorem (1.2.22) to the morphism f and complexes ω X , ωY , we get !

! !

R f ∗ R Hom X (ω X , ω X ) = R HomY (R f ∗ (ω X ), ωY ).

(∗)

Since H i (R Hom X (ω X , ω X )) = O X for i = 0 and 0 for i > 0, the left hand side can be identiﬁed with R f ∗ O X . By the Grauert–Riemenschneider theorem (1.2.28), R i f ∗ (ω X ) = 0 for i > 0. Since f is proper, birational, X is smooth, and Y is normal, we have f ∗ O X = OY . Therefore by the assumption of the proposition we have the isomorphism α : R f ∗ (ω X ) → OY , α = f (φ)−1 , and (∗) gives the isomorphism β : R f ∗ (O X ) → ωY . Recall that we have a morphism φ : O X → ω X . It induces the morphism ψ = αφβ −1 : ωY → OY . By assumption of the proposition we know that ψ induces an isomorphism H 0 (ψ) in zero cohomology, since R 0 f ∗ (φ) = f ∗ (φ). Let i : U → Y be an open immersion. Then by (1.2.21) (f), the complex i ∗ (ωY ) is a dualizing complex on U , and i ∗ (ψ) induces an isomorphism in zero cohomology. Since being Gorenstein is a local property, it is enough

1.3. Determinants of Complexes

27

to prove the assertion in the case when Y = Spec(R) for some commutative Noetherian ring R. We can therefore work in the category of R-modules. We are given a morphism ψ : ω R → R in D + (R) which induces an isomorphism H 0 (ψ) : H 0 (ω R ) → R. By deﬁnition ψ is represented by a diagram

s

ω R ← C • → R, where s is a quasiisomorphism. The morphism is given by a diagram ... ...

→ C −1 ↓ → 0

d −1

→

C0 ↓ 0 → R

d0

→ C1 ↓ → 0

→

...

→

...

with 0 d −1 = 0. Since H 0 () : H 0 (C • ) → R is an isomorphism, there exists a cycle z ∈ C 0 such that 0 (z) = 1. Therefore the R-module map a → az from R to C 0 extends to a morphism σ : R → C • of complexes. One obviously has σ = 1 R . Thus C • = R ⊕ Ker(). Since C • has a ﬁnite injective dimension (it is a dualizing complex), R also has a ﬁnite injective dimension. Therefore R is Gorenstein. Now, since Y is Gorenstein, ωY has only one nonzero cohomology, so ψ : ωY → OY is an isomorphism. We conclude that R i f ∗ (O X ) = H i (ωY ) = 0 for i > 0.

1.3. Determinants of Complexes In this section we collect the facts we need about determinants of complexes of vector spaces and modules. Let us start with the complex of vector spaces dn

dn−1

dm+1

V• : 0 → Vn →Vn−1 → . . . → Vm+1 → Vm over a ﬁeld K. For a vector space of dimension n we deﬁne its determinant to be the one dimensional vector space det(V ) := n V . Similarly we deﬁne the inverse of the determinant of V by setting det(V )−1 := n (V ∗ ). We deﬁne the determinant of a complex V• to be a one dimensional vector space det(V• ) =

n i=m

i

det(Vi )(−1) .

28

Introductory Material

(1.3.1) Proposition. Let 0 → V → V → V → 0 be an exact sequence of complexes. Then we have a canonical isomorphism det(V ) ⊗ det(V ) → det(V ). Proof. Let {u 1 , . . . , u m } be a basis of V . Let {v1 , . . . , vn } be a basis of V . Denote by g the linear map from V to V , and let f denote the linear map from V to V . We choose the elements w1 , . . . , wn in V so f (wi ) = vi for i = 1, . . . , n. We deﬁne the isomorphism j : det(V ) ⊗ det(V ) → det(V ) by setting j(u 1 ∧ . . . ∧ u m ⊗ v1 ∧ . . . ∧ vn ) := g(u 1 ) ∧ . . . ∧ g(u m ) ∧ w1 ∧ . . . ∧ wn . One checks directly that this isomorphism does not depend on the choice of vectors w1 , . . . , wn or on the choice of bases {u 1 , . . . , u m }, {v1 , . . . , vn }. (1.3.2) Proposition. Let V• be a complex of vector spaces. Let H (V• ) be a complex of vector spaces whose i-th term is Hi (V• ), with zero differential. Then we have a canonical isomorphism Eud : det(V• ) = det(H (V• )). Proof. Decompose the complex V• to short exact sequences 0 → Ker(di ) → Vi → Im(di ) → 0, 0 → Im(di+1 ) → Ker(di ) → Hi (V• ) → 0, and use isomorphisms from Proposition (1.3.1) We have the following properties of determinants of complexes of vector spaces. They easily follow from deﬁnitions. (1.3.3) Proposition. (a) Let V [i]• be a complex V• shifted to the left (i.e. V [i] j := Vi+ j ). Then i det(V [i]• ) = det(V• )(−1) . (b) Let 0 → V• → V• → V• → 0

1.3. Determinants of Complexes

29

be an exact sequence of complexes of vector spaces. Then det(V• ) = det(V• ) ⊗ det(V• ). (c) If the complex V• is exact, we have the isomorphism Eud : det(V• ) → K. Assume that (V• , v) is a based exact complex, i.e., V• is exact and v denotes the choice of bases {v1(i) , . . . , vn(i)i } of vector spaces Vi . Then we have a natural basis element (denoted also v) in one dimensional vector space (−1)i given by the tensor product of volume elements (v1(i) ∧ vn(i)i ) i det(Vi ) and their duals. We deﬁne a determinant of the based complex (V• , v) to be a nonzero scalar det(V• , v) = Eud (v). Let R be a commutative domain, and let dm+1

dn

F• : Fn →Fn−1 → . . . → Fm be a complex of free R-modules. Assume that F• is generically exact. This means that, tensoring with the ﬁeld of fractions K := R(0) , we get an exact sequence of vector spaces. Let us ﬁx bases { f 1(i) , . . . , f n(i) } in the R-modules i Fi . Then we can talk about the element det(F• , f ) ∈ K∗ , where f is the corresponding volume element. For another choice of bases (over R) in Fi corresponding to a volume element f we see that det(F• , f ) = det(F• , f ) u where u is a unit in R. Thus a determinant of a generically exact complex of free modules is well deﬁned as an element of K∗ /R ∗ . We want to investigate the numerator and denominator of this function. Notice that the construction of the determinant of a complex of free modules commutes with the localization. The properties (1.3.3) of determinants of complexes are also true for generically exact complexes of free modules over R. Let us assume that R is a unique factorization domain. We can write

det(F• ) = f ord f (det(F• )) . f ∈Irr(R)

where Irr(R) denotes the set of irreducible elements in R. In order to understand the determinant of F• it is enough to understand the numbers ord f (det(F• )) Let us ﬁx the irreducible element f . We recall that the ideal ( f ) is prime and that the localization R( f ) is a discrete valuation ring. For each ﬁnitely generated R-module M we deﬁne its f -multiplicity mult f (M) as the length of the localization M ⊗ R( f ) . The multiplicity of a ﬁnitely generated R-module

30

Introductory Material

M is ﬁnite if and only if the localization M( f ) is annihilated by some power of f . (1.3.4) Theorem. Let us assume that R is a unique factorization domain. Let F• be a complex of free R-modules that is generically exact. Then the order ord f (det(F• )) of the irreducible element f in the determinant of F• is given by the formula (−1)i mult f (Hi (F• )). ord f (det(F• )) = i

Proof. By localizing we may assume that the ring R is a discrete valuation ring, so we need just to calculate the determinant of a free complex over such ring. However, each homology module Hi (F• ) is torsion, because F• is generically exact. Therefore each Hi (F• ) is a direct sum of cyclic modules (i) R/( f j ). Let G (i) : 0 → G (i) 1 → G 0 be a free resolution of Hi (F• ) (which is f

j

a direct sum of complexes 0 → R →R). We use the decreasing induction on j, starting with j = n + 1 to deﬁne complexes F•( j) such that Hi (F• ) if i < j , Hi (F•( j) ) = 0 otherwise. The complex F•(n+1) = F• . Suppose we constructed the complex F•( j+1) . Its top nonzero homology is H j (F•( j+1) ) = H j (F• ). Therefore we can construct a map ψ j : G ( j) [ j]• → F•( j+1) lifting the identity map on homology in degree j. We deﬁne F•( j) to be the cone of the map ψ j . The complex F•(m−1) will be exact, and thus its determinant will be a unit in R. Using multiplicativity of the determinant with respect to short exact sequences and using exact sequences associated to the cone construction, we see that it is enough to prove the statement of the theorem for the complexes G (•j) . jThis means it is enough to f check the statement for each summand 0 → R →R. Here both sides of the equality give j, so Theorem (1.3.4) is proved. (1.3.5) Corollary. Assume that m = 0 and the complex F• is acyclic in codimension 2, i.e., all modules Hi (F• ) are supported in codimension 2. Then the determinant of F• is the greatest common divisor of the maximal minors of the map d1 . Proof. Using the statement (1.3.4), we see that the only factors that can occur in det(F• ) are the equations of codimension 1 components of the support of

1.3. Determinants of Complexes

31

H0 (F• ). This means after localizing at ( f ) the complex will be acyclic. This again reduces the statement to the case of complexes of length 1 resolving a torsion module, in which case we can check it directly. (1.3.6) Remark. The last statement is closely related to the so-called ﬁrst structure theorem of Buchsbaum and Eisenbud [BE3]. It states that if rank (di ) = ri , then there exists a sequence of maps ai : R → ri Fi−1 such that ri ∗ di = ai ai+1 . In fact one knows (Theorem 3, Chapter 7.) that a complex satisfying the assumptions of (1.3.5) always satisﬁes this theorem. One knows also that for i > 1 the ideal generated by entries of ai has depth ≥ 2. This means that the determinant of F• is the entry of a1 . But the ideal of maximal minors of d1 equals a1 I (a2 ), where I (a2 ) is the ideal of entries of a2 , which has depth ≥ 2. Thus in this case a1 is the greatest common divisor of maximal minors of d1 .

2 Schur Functors and Schur Complexes

In this chapter we develop the representation theory of general linear groups. We follow the approach from [ABW2] based on the explicit characteristic free deﬁnition of Schur and Weyl functors. This approach is sufﬁcient for our goals, and it seems to be easier to grasp for the reader not familiar with representation theory. In section 2.1 we deﬁne Schur and Weyl functors and prove the standard basis theorems. In section 2.2 we discuss the connection of Schur functors with highest weight theory, and provide the alternate deﬁnition using Young symmetrizers in characteristic 0. In section 2.3 we derive various formulas from representation theory, including the Littlewood–Richardson rule and Cauchy formulas. Finally in section 2.4 we give the deﬁnition and basic properties of Schur complexes. 2.1. Schur Functors and Weyl Functors Let us ﬁx a free module E of dimension n over a commutative ring K. Let λ = (λ1 , . . . , λs ) be a partition of a number m. We consider the module Lλ E =

λ1

E⊗

λ2

E ⊗ ... ⊗

λs

E/R(λ, E),

where the submodule R(λ, E) is the sum of submodules: λ1

E ⊗ ... ⊗

λ a−1

E ⊗ Ra,a+1 (E) ⊗

λ a+2

E ⊗ ... ⊗

λs

E

for 1 ≤ a ≤ s − 1, where Ra,a+1 (E) is the submodule spanned by the images of the following maps θ(λ, a, u, v; E) with u + v < λa+1 : u E ⊗ λa −u+λa+1 −v E ⊗ v E ↓ 1⊗⊗1 u E ⊗ λa −u E ⊗ λa+1 −v E ⊗ v E ↓ m 12 ⊗m 34 λa E ⊗ λa+1 E.

32

2.1. Schur Functors and Weyl Functors

33

Let e1 , . . . , en will be a ﬁxed ordered basis of E. We introduce the ordered set [1, n] = {1, 2, . . . , n}, which is the set indexing our basis. We refer to section 1.1.2. for notions related to tableaux used in this section. Let T be a tableau of shape λ with the entries in [1, n]. We associate to T the element in L λ E which is a coset of the tensor eT (1,1) ∧ . . . ∧ eT (1,λ1 ) ⊗ eT (2,1) ∧ . . . ∧ eT (2,λ2 ) ⊗ . . . ⊗ eT (s,1) ∧ . . . ∧ eT (s,λs ) in L λ E. In the sequel we will identify these two objects and we will call both of them the tableaux of shape λ corresponding to the basis {e1 , e2 , . . . , en }. (2.1.1) Remark. It is convenient to think about the relations R(λ, E) in graphical terms using the Young frames. Let us deﬁne the Young scheme of shape λ to be the Young frame of shape λ with some boxes empty and some ﬁlled. We associate to the map θ (λ, a, u, v; E) its Young scheme which is empty in all rows except the a-th and (a + 1)-st, and its restriction to these rows is ... • ... • • ... • • ... • • • ... • • ... • ... with u empty boxes, followed by λa − u ﬁlled boxes in the a-th row, and λa+1 − v ﬁlled boxes, followed by v empty boxes in the (a + 1)-st row. Notice that the condition u + v < λa+1 assures that there will be at least one column in this frame with two boxes ﬁlled. The image of typical element U1 ⊗ . . . ⊗ Ua−1 ⊗ V1 ⊗ V2 ⊗ V3 ⊗ Ua+2 ⊗ . . . ⊗ Us , where U j = eT ( j,1) ∧ . . . ∧ eT ( j,λ j ) , V1 = ex1 ∧ . . . ∧ exu , V2 = e y1 ∧ . . . ∧ e yλa +λa+1 −u−v , V3 = ez1 ∧ . . . ∧ ezv , is a sum of tableaux, where we put in each tableau element T ( j, t) in the t-th box in the j-th row for j = a, a + 1. In the a-th and (a + 1)st row we put x 1 , . . . , xu in the empty u boxes in the a-th row, put z 1 , . . . , z v in the empty v boxes in the (a + 1)st row, and shufﬂe the elements y1 , . . . , yλa +λa+1 −u−v between the ﬁlled boxes in the a-th and (a + 1)st rows, with the appropriate signs coming from exterior diagonal. (2.1.2) Example. Take λ = (3, 3), u = v = 1. The corresponding Young scheme is • • . • •

34

Schur Functors and Schur Complexes

Take x1 = 1, z 1 = 6, {y1 , y2 , y3 , y4 } = {2, 3, 4, 5}. The image of the corresponding vector by θ(λ, 1, u, v; E) is 1 2 3 1 2 4 1 2 5 1 3 4 1 3 5 1 4 5 − + + − + . 4 5 6 3 5 6 3 4 6 2 5 6 2 4 6 2 3 6 (2.1.3) Example. (a) If λ = (t) then L λ E = t E. Indeed, by deﬁnition L λ E = t E/ R((t), E), but R((t), E) = 0, since the partition (t) has only one part. (b) If λ = (1t ) then L λ E = St E. Indeed, the relations Ra,a+1 (E) express the symmetry between the a-th and (a + 1)st row. (c) Let λ = (2, 1). In that case there is only one choice of u, v, namely u = v = 0. The corresponding Young scheme is • • . • The image of θ(1, 0, 0; E) on the typical element e y1 ∧ e y2 ∧ e y3 is y1 y2 y y y y − 1 3 + 2 3. y3 y2 y1 We conclude that L 2,1 E is the cokernel of the diagonal map :

3

E→

2

E ⊗ E.

(d) Let λ = (2, 2). There are three types of relations θ(1, u, v). The choices of u, v are (u, v) = (0, 0), (1, 0), (0, 1). The corresponding Young schemes are • • , • •

• , • •

• • . •

It is easy to see that the relations coming from θ(1, 1, 0; E) are the consequences of two other types of relations. Therefore we have two types of relations. For the map θ (1, 0, 0; E) the image of the typical element e y1 ∧ e y2 ∧ e y3 ∧ e y4 is y1 y2 y y y y y y y y y y − 1 3 + 1 4 + 2 3 − 2 4 + 3 4. y3 y4 y2 y4 y2 y3 y1 y4 y1 y3 y1 y2

2.1. Schur Functors and Weyl Functors

35

For the relation θ (1, 0, 1; E) the image of the typical element e y1 ∧ e y2 ∧ e y3 ⊗ ez is y1 y2 y1 y3 y2 y3 − + . y3 z y2 z y1 z We conclude that L 2,2 E is the factor of 2 E ⊗ 2 E by the images of two maps 3 E ⊗ E → 2 E ⊗ 2 E (corresponding to u = 0, 4 2 v = 1) and the diagonal E→ E ⊗ 2 E (corresponding to u = v = 0). (e) Let λ = (3, 2). There are three choices of pairs u, v: (u, v) = (0, 0), (1, 0), (0, 1). The Young schemes are • • • , • •

• • , • •

• • • . •

It follows that L 3,2 E is a factor of the module 3 E ⊗ 2 E by the images of three maps: 4 E ⊗ E → 3 E ⊗ 2 E (corresponding to 4 3 2 u = 0, v = 1), E ⊗ E→ E⊗ E (corresponding to u=1, v = 0), and 5 E → 3 E ⊗ 2 E (corresponding to u = v = 0). (f) We will show below that for two rowed partitions there are two ways of choosing smaller set of relations θ (1, u, v; E) that still sufﬁce to deﬁne the Schur functor. One choice is to take all pairs (u, v) with u = 0. The other choice is to take all pairs (u, v) with one overlap, i.e. all pairs (u, v) for which the Young scheme has exactly one column with two ﬁlled boxes (equivalently, u + v = λ2 − 1). (g) Let λ = (2, 2, 1). We have two types of relations corresponding to the ﬁrst pair of rows (described in example (d)), and one type corresponding to the second and third rows (described in example (c)). Choosing the relations with one overlap the Young schemes are • • , •

• • • ,

• • . •

(h) Let λ be a hook i.e. a partition of the form λ = ( p, 1q−1 ). Graphically, ... .. λ= .

,

36

Schur Functors and Schur Complexes

with p boxes in the ﬁrst row and q boxes in the ﬁrst column. The relations between two rows of length 1 express the symmetry (cf. example (b)). There is only one type of relations corresponding to the ﬁrst two rows, for the pair u = v = 0. It follows that the Schur functor L ( p,1q−1 ) E is the cokernel of the composition map p+1

⊗1

E ⊗ Sq−2 E −→

p

1⊗m

E ⊗ E ⊗ Sq−2 E −→

p

E ⊗ Sq−1 E.

We recall from section 1.1.2 that a tableau T is standard if the numbers in each row of T form an increasing sequence and the numbers in each column of T form a nondecreasing sequence. This notion plays a key role in representation theory thanks to the following (2.1.4) Proposition. Let E be a free K-module of dimension n. Let e1 , . . . , en be a basis of E. The set ST(λ, [1, n]) of standard tableaux of shape λ with entries from [1, n] form a basis of L λ E. In particular, L λ E is also a free module. Proof of Proposition (2.1.4). First we prove that the standard tableaux generate L λ E. It is clear that the set RST(λ, [1, n]) of row standard tableaux with entries from [1, n] generate L λ E. Let us order the set of such tableaux by the order deﬁned in section 1.1.2. We will prove that if the tableau T is not standard then we can express it modulo R(λ, E) as a combination of earlier tableaux. Let us assume ﬁrst that T has two rows. Since T is not standard, we can ﬁnd w for which T (1, w) > T (2, w). We consider the map θ (λ, 1, u, v; E) for u = w − 1 and v = λ2 − w. The key observation is that image of the tensor V1 ⊗ V2 ⊗ V3 , where V1 = eT (1,1) ∧ eT (1,2) ∧ . . . ∧ eT (1,w−1) , V2 = eT (1,w) ∧ . . . ∧ eT (1,λ1 ) ∧ eT (2,1) ∧ . . . ∧ eT (2,w) , V3 = eT (2,w+1) ∧ . . . ∧ eT (2,λ2 ), contains the tableau T with the coefﬁcient 1, and all the other tableaux occurring in this image are earlier than T in the order . Indeed, in all summands other than T we shufﬂe the smaller numbers from the second row to the ﬁrst one, replacing bigger numbers. Therefore T can be expressed modulo R(λ, E) as a combination of earlier tableaux. Let us consider the general case. If T is not standard, then we can ﬁnd such a and w that T (a, w) > T (a + 1, w). Now we apply the previous argument

2.1. Schur Functors and Weyl Functors

37

to the tableau S which consists of the a-th and (a+1)st rows of T . Notice that the relations R(λ, E) we are using do not do anything to the other rows of T , so we can express T as a sum of earlier tableaux in the order . It remains to prove that the standard tableaux are linearly independent in L λ E. Consider a map φλ :

λ1

E⊗

λ2

E ⊗ ... ⊗

λs

α

E −→ ⊗(i, j)∈λ E(i, j)

β

−→ Sλ1 E ⊗ Sλ2 E ⊗ . . . ⊗ Sλt E, where α is the tensor product of exterior diagonals :

λj

E → E( j, 1) ⊗ E( j, 2) ⊗ . . . ⊗ E( j, λ j )

and β is the tensor product of multiplications m : E(1, i) ⊗ E(2, i) ⊗ . . . ⊗ E(λi , i) → Sλi E. If we imagine the copies of E correspond to the boxes of λ with λ j E corresponding to the boxes in the ﬁrst row and Sλi E corresponding to boxes in the i-th column of λ, we can think of the image φλ (T ) of the tableau T as ﬁrst shufﬂing (with signs) the terms of T in each row, and then multiplying the terms in each column of the tableaux we obtain. The map φλ is called the Schur map associated to the partition λ. (2.1.5) Example. (a) Let λ = (2, 2). Then the map φλ :

2

E⊗

2

E −→ S2 E ⊗ S2 E

is given by the formula φλ (x ∧ y ⊗ u ∧ v) = xu ⊗ yv−yu ⊗ xv−xv ⊗ yu+yv ⊗ xu. (b) It is useful to think about the map φλ in graphical terms as follows. We consider the case λ = (3, 2), but the general case will become clear. The tensor product 3 E ⊗ 2 E has a basis corresponding to the set RST(λ, [1, n]). It can be thought of as a set of standard tableaux of shapes (3) and (2), corresponding to rows of λ. The tensor product S2 E ⊗ S2 E ⊗ E has a basis consisting of triples of costandard

38

Schur Functors and Schur Complexes

tableaux of shapes (2), (2), (1), corresponding to columns of λ. The map φλ acts according to the scheme

↓ , and the image of a tableau T is the sum (with signs) of tableaux obtained from T by shufﬂing each of its rows. The role of the map φλ is explained in the next statement. (2.1.6) Proposition. The image φλ (R(λ, E)) equals 0. Proof. We want to show that the spaces λ1

E ⊗ ... ⊗

λ a−1

E ⊗ Ra,a+1 (E) ⊗

λ a+2

E ⊗ ... ⊗

λs

E

are in the kernel of φλ . Since Ra,a+1 is the span of images of the maps of type θ (λ, a, u, v; E), we choose one such map (i.e., we choose a and u, v such that u + v < λa+1 ). Let us consider the element U = U1 ⊗ U2 ⊗ . . . ⊗ Ua−1 ⊗ U ⊗ Ua+2 ⊗ . . . ⊗ Us , where Ui = xi,1 ∧ xi,2 ∧ . . . ∧ xi,λi ∈ λi E and U is the image under θ(λ, a, u, v; E) of the element x1 ∧ . . . ∧ xu ⊗ y1 ∧ . . . ∧ yλa +λa+1 −u−v ⊗ z 1 ∧ . . . ∧ z v . Let us consider the tensor φλ U . It is formally a sum of tensors in Sλ1 E ⊗ Sλ2 E ⊗ . . . ⊗ Sλt E each of which is the tensor product of products of xi, j ’s, x j ’s, ym ’s, and z p ’s shufﬂed in some way. Let us write our image formally as such a sum by writing products in each Sλj E in the order they get multiplied by β. Let us consider a summand T in our sum. Since u + v < λa+1 , two of the ym ’s have to occur in the same symmetric power Sλj E. Let us choose the smallest such j. Let yb and yc occur in Sλj E. Let us consider another summand T in our sum φλ U which differs from T by changing places of yb and yc when applying the map 1 ⊗ ⊗ 1 from θ(λ, a, u, v; E). We can easily check that the correspondence T → T deﬁnes an involution on the summands in φλ . Moreover, each pair of such summands cancels out in

2.1. Schur Functors and Weyl Functors

39

Sλ1 E ⊗ Sλ2 E ⊗ . . . ⊗ Sλt E, because they come with different signs and the product yb yc is symmetric. This means φλ U = 0. (2.1.7) Example. Let us take λ = (3, 3), u = v = 1. Consider U = x ⊗ y1 ∧ y2 ∧ y3 ∧ y4 ⊗ z. Then if T = x y1 ⊗ y3 y4 ⊗ y2 z then T = x y1 ⊗ y4 y3 ⊗ y2 z. Now T occurs as a summand in φλ (x ∧ y3 ∧ y2 ⊗ y1 ∧ y4 ∧ z), and T occurs as a summand in φλ (x ∧ y4 ∧ y2 ⊗ y1 ∧ y3 ∧ z). One checks easily that T and T cancel out. Proposition (2.1.6) means that φλ induces a surjective map from L λ E to Im φλ . Next we will show that the map φλ maps standard tableaux to linearly independent elements of Sλ1 E ⊗ Sλ2 E ⊗ . . . ⊗ Sλt E. This will prove (2.1.4), and at the same time it will prove that L λ E = Im φλ . In order to see that the images of standard tableaux are linearly independent, we notice that the module Sλ1 E ⊗ Sλ2 E ⊗ . . . ⊗ Sλt E has a basis corresponding naturally to the set RST(λ , [1, n] ) of row costandard tableaux of shape λ . We order this set by the order deﬁned in section 1.1.2. If T is a standard tableau of shape λ, then the smallest element (with respect to the order ) occurring in φλ (T ) is eT (1,1) . . . eT (λ1 ,1) ⊗ eT (1,2) . . . eT (λ2 ,2) ⊗ . . . ⊗ eT (1,t) . . . eT (λt ,t) . Indeed, if in applying the map α we make an exchange of elements in some row, we put bigger elements to the earlier columns, so we get the later (with respect to ) elements. Moreover, one sees instantly that eT (1,1) . . . eT (λ1 ,1) ⊗ eT (1,2) . . . eT (λ2 ,2) ⊗ . . . ⊗ eT (1,t) . . . eT (λt ,t) occurs in α(T ) with coefﬁcient 1. It is also obvious that the initial elements eT (1,1) . . . eT (λ1 ,1) ⊗ eT (1,2) . . . eT (λ2 ,2) ⊗ . . . ⊗ eT (1,t) . . . eT (λt ,t) are different for different standard tableaux T . This proves that the images φλ T of standard tableaux T are linearly independent.• Since the exterior and symmetric powers are GL(E)-modules, and the diagonal and multiplication maps are GL(E)-equivariant, it is clear that the group GL(E) acts on L λ E in a natural way. The space L λ E becomes a GL(E)module, which is called the Schur module corresponding to the partition λ. The notion of a Schur module can be generalized to skew partitions. Let λ/µ be a skew partition. We deﬁne the Schur map φλ/µ :

λ1 −µ1

E⊗

λ2 −µ2

E ⊗ ... ⊗

→ Sλ1 −µ1 E ⊗ . . . ⊗ Sλt −µt E

λ s −µs

E

40

Schur Functors and Schur Complexes

as a composition φλ/µ :

λ1 −µ1

E⊗

λ2 −µ2

E ⊗ ... ⊗

λ s −µs

α

E −→ ⊗(i, j)∈λ/µ E(i, j)

β

−→ Sλ1 −µ1 E ⊗ Sλ2 −µ2 E ⊗ . . . ⊗ Sλt −µt E, where α is the tensor product of exterior diagonals :

λ j −µ j

E → E( j, µ j + 1) ⊗ E( j, µ j + 2) ⊗ . . . ⊗ E( j, λ j )

and β is the tensor product of multiplications m : E(µi + 1, i) ⊗ E(µi + 2, i) ⊗ . . . ⊗ E(λi , i) → Sλi E. If we imagine the copies of E correspond to the boxes in a skew Young frame λ/µ, with λ j −µ j E corresponding to the boxes in the j-th row, and Sλi −µi E corresponding to boxes in the i-th column, we can think of the image φλ/µ (T ) of a tableau T as ﬁrst shufﬂing (with signs) the terms of T in each row, and then multiplying the terms of each column of each summand. (2.1.8) Example. Let λ = (3, 2), µ = (1). The tensor product 2 E ⊗ 2 E has a basis corresponding to the set RST(λ/µ, [1, n]). It can be thought of as a pair of standard tableaux of shapes (2) and (2) (corresponding to the rows of (3, 2)/(1)). The tensor product E ⊗ S2 E ⊗ E has a basis consisting of triples of costandard tableaux of shapes (1), (2), (1) (corresponding to columns of (3, 2)/(1)). The map φ(3,2)/(1) acts according to the scheme

↓

↓

2.1. Schur Functors and Weyl Functors

41

We deﬁne the skew Schur module L λ/µ E to be the image of φλ/µ . The description of the relations and of the standard basis of skew Schur modules is the same as for the Schur modules. (2.1.9) Proposition. (a) L λ/µ E = λ1 −µ1 E ⊗ λ2 −µ2 E ⊗ . . . ⊗ λs −µs E/R(λ/µ, E) where R(λ/µ, E) is spanned by the subspaces: λ1 −µ1

E ⊗ ... ⊗

⊗ ... ⊗

λ s −µs

λa−1 −µa−1

E ⊗ Ra,a+1 (E) ⊗

λa+2 −µa+2

E

E

for 1 ≤ a ≤ s − 1, where Ra,a+1 (E) is the vector space spanned by the images of the following maps θ (λ/µ, a, u, v; E) with u + v < λa+1 − µa : u E ⊗ λa −µa −u+λa+1 −µa+1 −v E ⊗ v E ↓ 1⊗⊗1 λa −µa −u u E⊗ E ⊗ λa+1 −µa+1 −v E ⊗ v E ↓ m 12 ⊗m 34 λa −µa E ⊗ λa+1 −µa+1 E. (b) The standard tableaux of shape λ/µ form a basis of L λ/µ . Proof. First of all one can check by direct calculation that the map φλ/µ sends the elements from R(λ/µ, E) to zero. This is done in the same way as in the case µ = ∅, so we leave it to the reader. Then it is enough to check two things: (1) The standard tableaux of shape λ/µ generate the factor λ1 −µ1

E⊗

λ2 −µ2

E ⊗ ... ⊗

λ s −µs

E/R(λ/µ, E).

(2) The images φλ/µ (T ) for the standard tableaux T of shape λ/µ are linearly independent. Facts (1) and (2) allow us to identify the image of φλ/µ with the factor λ1 −µ1

E⊗

λ2 −µ2

E ⊗ ... ⊗

λ s −µs

E/R(λ/µ, E).

The proof of (1) and (2) is exactly the same as for Schur functors, so we leave it to the reader as an exercise.

42

Schur Functors and Schur Complexes

We can associate to each of the relations θ(λ/µ, a, u, v; E) its Young scheme, as we did in the case µ = ∅. The only difference is that there are missing boxes. Again the condition u + v < λa+1 − µa assures at least one overlap in the Young scheme of the relation. (2.1.10) Example. Let us take λ = (4, 3), µ = (1, 0). The Young scheme of the relation θ((4, 3)/(1, 0), 1, 1, 0; E) is • • . • • • (2.1.11) Remark. We can interpret the relations R(λ/µ, E) in terms of tableaux as we did in the case µ = ∅. Again we start with the case of two rows. The image of typical element V1 ⊗ V2 ⊗ V3 with V1 = ex1 ∧ . . . ∧ exu , V2 = e y1 ∧ . . . ∧ e yλa −µa +λa+1 −µa+1 −u−v , V3 = ez1 ∧ . . . ∧ ezv is a sum of tableaux, where we put in each tableau x 1 , . . . , xu in the u empty boxes in the ﬁrst row, put z 1 , . . . , z v in the v empty boxes in the second row, and shufﬂe the elements y1 , . . . , yλa +λa+1 −u−v between the ﬁlled boxes in the ﬁrst and second rows, with the appropriate signs coming from exterior diagonal. If the number of parts of λ is bigger than 2, the relations R(λ/µ, E) can be interpreted in terms of tableaux as follows. Fix a, u, v. The Young scheme of the map θ (λ/µ, a, u, v; E) has all the boxes empty except the a-th and (a + 1)st, where the scheme is the same as in the case of two rows. The image of the typical element U1 ⊗ . . . ⊗ Ua−1 ⊗ V1 ⊗ V2 ⊗ V3 ⊗ Ua+2 ⊗ . . . ⊗ Us , where U j = ei( j,1) ∧ . . . ∧ ei( j,λ j −µ j ) , V1 = ex1 ∧ . . . ∧ exu , V2 = e y1 ∧ . . . ∧ e yλa −µa +λa+1 −µa+1 −u−v , and V3 = ez1 ∧ . . . ∧ ezv , is the same as in the case of two rows, except that in each summand we put element i ( j, t) in the t-th box in the j-th row for j = a, a + 1. (2.1.12) Example. Let λ = (3, 2), µ = (1, 0), u = v = 0. The corresponding Young scheme is • • . • • Take {y1 , y2 , y3 , y4 } = {2, 3, 4, 5}. The image of the corresponding vector e2 ∧ e3 ∧ e4 ∧ e5 by θ((3, 2)/(1, 0), 1, 0, 0; E) is 2 3 2 4 2 5 3 4 3 5 4 5 − + + − + . 4 5 3 5 3 4 2 5 2 4 2 3

2.1. Schur Functors and Weyl Functors

43

(2.1.13) Example. (a) Consider the skew shape λ = (3, 2), µ = (1, 0). The only possible relation in this case is θ ((3, 2)/(1, 0), 1, 0, 0; E). Its Young scheme is • • . • • The Schur module L (3,2)/(1,0) E is a factor of 2 E ⊗ 2 E by the image 4 E embedded by the exterior diagonal. of (b) Take λ = (4, 2), µ = (2, 0). In this case there are no possible relations, and L (4,2)/(2,0) E is just the tensor product 2 E ⊗ 2 E. (c) The previous example generalizes as follows. Assume that the skew Young frame λ/µ is disconnected, i.e., it can be written as a union of two skew Young frames λ(1)/µ(1) and λ(2)/µ(2) with no boxes in the same row or column. Then L λ/µ E = L λ(1)/µ(1) E ⊗ L λ(2)/µ(2) E. Proof of (c). The relations of type θ (λ/µ, a, u, v; E) between two rows of λ(i)/µ(i) (i = 1, 2) are the same as the relations between corresponding rows of λ/µ. Since λ/µ is disconnected, there are no relations between rows of λ(1)/µ(1) and of λ(2)/µ(2). We conclude this section with the deﬁnition of the skew Weyl modules K λ/µ . They are the duals of the skew Schur modules. To deﬁne the modules K λ/µ E we take the deﬁnition of L λ/µ E, but instead of exterior powers we use divided powers, and instead of symmetric powers we use exterior powers. Thus we deﬁne the Weyl map ψλ/µ : Dλ1 −µ1 E ⊗ Dλ2 −µ2 E ⊗ . . . ⊗ Dλs −µs E →

λ1 −µ1

E ⊗ ... ⊗

λ t −µt

E

as a composition α

ψλ/µ : Dλ1 −µ1 E ⊗ . . . ⊗ Dλs −µs E −→

E(i, j)

(i, j)∈λ−µ β

−→

λ1 −µ1

E ⊗ ... ⊗

λ t −µt

E,

where α is the tensor product of divided diagonals : Dλ j −µ j E → E( j, µ j + 1) ⊗ E( j, µ j + 2) ⊗ . . . ⊗ E( j, λ j ),

44

Schur Functors and Schur Complexes

and β is the tensor product of multiplications m:

E(µi

+ 1, i) ⊗

E(µi

+ 2, i) ⊗ . . . ⊗

E(λi , i)

→

λ i −µi

E.

We deﬁne the skew Weyl module K λ/µ E to be the image of ψλ/µ . The description of the relations and of the standard basis of skew Weyl modules is analogous to that for the skew Schur modules. Before we state it, let us deﬁne the tableaux. As before we ﬁx an ordered basis e1 , . . . , en of E. Let r ( j,1) r ( j,n) . . . en , where of course us choose the tensors U j ∈ Dλ j −µ j , U j = e1 r ( j, 1) + . . . + r ( j, n) = λ j − µ j . Then the image ψλ/µ (U1 ⊗ . . . ⊗ Us ) will be denoted by a tableau T of shape λ/µ which in the j-th row has r ( j, 1) 1’s, r ( j, 2) 2’s, . . . , r ( j, n) n’s. The order of these elements will not matter, because we will assume each tableau to be symmetric in the symbols in every row. We will identify the tableau T with the tensor ψλ/µ (U1 ⊗ . . . ⊗ Us ). (2.1.14) Example. Take λ = (3, 2), µ = ∅. The tableau T =

1 1 2 2 2

is identiﬁed with the image ψ(3,2) (e1(2) e2 ⊗ e2(2) ). (2.1.15) Proposition. (a) K λ/µ E = Dλ1 −µ1 E ⊗ Dλ2 −µ2 E ⊗ . . . ⊗ Dλs −µs E/U (λ/µ, E), where U (λ/µ, E) is the sum of subspaces Dλ1 −µ1 E ⊗ . . . ⊗ Dλa−1 −µa−1 E ⊗ Ua,a+1 (E) ⊗ Dλa+2 −µa+2 E ⊗ . . . ⊗ Dλs −µs E for 1 ≤ a ≤ s − 1, where Ua,a+1 (E) is the module spanned by the images of the following maps θ (λ/µ, a, u, v; E) with u + v < λa+1 − µa : Du E ⊗ Dλa −µa −u+λa+1 −µa+1 −v E ⊗ Dv E ↓ 1⊗⊗1 Du E ⊗ Dλa −µa −u E ⊗ Dλa+1 −µa+1 −v E ⊗ Dv E ↓ m 12 ⊗m 34 Dλa −µa E ⊗ Dλa+1 −µa+1 E. (b) The costandard tableaux ST(λ, µ, [1, n] ) of shape λ/µ (cf. section 1.1.2) with the entries from [1 , n ] form a basis of K λ/µ E.

2.1. Schur Functors and Weyl Functors

45

Proof. For the remainder of the section we write j instead of j for j [1, n]. First of all, one can check by direct calculation that the map ψλ/µ sends the elements from U (λ/µ, E) to zero. This is done in the same way as in the case of Schur modules. Then it is enough to check two things: (1) The costandard tableaux of shape λ/µ generate the factor Dλ1 −µ1 E ⊗ Dλ2 −µ2 E ⊗ . . . ⊗ Dλs −µs E/U (λ/µ, E). (2) The images ψλ/µ (T ) for the costandard tableaux T of shape λ/µ are linearly independent. Facts (1) and (2) allow us to identify the image of ψλ/µ with the factor Dλ1 −µ1 E ⊗ Dλ2 −µ2 E ⊗ . . . ⊗ Dλs −µs E/U (λ/µ, E). The proof of (2) is exactly the same as for Schur modules, so we leave it to the reader. However, we prove fact (1) because the proof in the case of Weyl modules is slightly different. The reason is that the map θ (λ/µ, a, u, v; E) involves the multiplication in the divided powers algebra, which involves some integer coefﬁcients. It is clear that the row costandard tableaux RST(λ/µ, [1, n] ) with entries from [1, n] generate K λ/µ E. Let us order the set of such tableaux by the order deﬁned in section 1.1.2. We will prove that if the tableau T is not costandard, then we can express it modulo U (λ/µ, E) as a combination of earlier tableaux. Let us assume ﬁrst that T has two rows. Since T is not costandard, we can ﬁnd w for which T (1, w) ≥ T (2, w). Let us ﬁnd the smallest w with this property. Let w be the biggest number for which T (2, w ) = T (2, w). We consider the map θ (λ/µ, a, u, v, E) for u = w − µ1 − 1 and v = λ2 − w . The key observation is that image of the tensor V1 ⊗ V2 ⊗ V3 where V1 = eT (1,µ1 +1) ∪ eT (1,µ1 +2) ∪ . . . ∪ eT (1,w−1) , V2 = eT (1,w) ∪ . . . ∪ eT (1,λ1 ) ∪ eT (2,µ2 +1) ∪ . . . ∪ eT (2,w ), V3 = eT (2,w +1) ∪ . . . ∪ eT (2,λ2 ), where ∪ indicates we take the corresponding tensor in the divided power, contains the tableau T with the coefﬁcient 1, and all the other tableaux occurring in this image are earlier than T . The coefﬁcients of multiplication in the divided algebra do not spoil anything, because by choice of w and w we have T (1, w − 1) < T (1, w) and T (2, w ) < T (2, w + 1). Therefore T can be expressed modulo U (λ/µ, E) as a combination of earlier tableaux.

46

Schur Functors and Schur Complexes

Let us consider the general case. If T is not costandard, then we can ﬁnd a and w such that T (a, w) ≥ T (a + 1, w). Now we apply the previous argument to the tableau S which consists of the a-th and (a+1)st rows of T . Notice that the relations U (λ/µ, E) we are using do nothing to the other rows of T , so in the same way we can express T as a sum of earlier tableaux. (2.1.16) Example. Let us standardize the tableau 1 4 5 2 3 3 in K (3,3) E. We have to use the relation θ (λ, a, u, v; E) with u = 1, v = 0: 1 4 5 1 2 3 1 2 4 1 2 5 1 3 3 =− − − − 2 3 3 3 4 5 3 3 5 3 3 4 2 4 5 −

1 3 4 1 3 5 − . 2 3 5 2 3 4

All the tableaux on the right hand side except the third one and the last one are standard, and we have 1 2 5 1 2 3 1 2 4 =− − . 3 3 4 3 4 5 3 3 5 Similarly, 1 3 5 1 2 3 1 3 3 1 3 4 =− −2 − . 2 3 4 3 4 5 2 4 5 2 3 5 Putting all these expressions together, we get 1 4 5 1 2 3 1 3 3 = + . 2 3 3 3 4 5 2 4 5 In the case when µ = 0, i.e. in the case of the partition, we denote K λ/µ by K λ and we call it a Weyl module. (2.1.17) Example. (a) For λ = (t), K λ E = Dt E. Indeed, by deﬁnition K λ E = Dt E/U ((t), E), but U ((t), E) = 0, since the partition (t) has only one part. (b) For λ = (1t ), K λ E = t E. Indeed, the map ψλ is onto in this case. The relations θ (a, u, v; E) between two rows of length 1 correspond to u = v = 0, and they express antisymmetry in the rows.

2.1. Schur Functors and Weyl Functors

47

(c) Let λ = (2, 1). The only relation that occurs in K (2,1) E is θ (1, 0, 0; E). The corresponding Young scheme is • • . • The image by θ (1, 0, 0; E) of the typical element e y1 ∪ e y2 ∪ e y3 is y1 y2 y y y y + 1 3 + 2 3 y3 y2 y1 if y1 , y2 , y3 are all different, y1 y1 y1 y3 + y3 y1 if y1 = y2 = y3 , and y1 y1 y1 if y1 = y2 = y3 . (d) Let λ = (2, 2). There are three types of relations θ (1, u, v; E). The possible pairs (u, v) are (0, 0), (1, 0), (0, 1). The corresponding Young schemes are • • , • •

• , • •

• • . •

Therefore we have three types of relations. For the map θ (1, 0, 0; E) the image of typical element e y1 ∪ e y2 ∪ e y3 ∪ e y4 is y1 y2 y1 y3 y1 y4 y2 y3 y2 y4 y3 y4 + + + + + y3 y4 y2 y4 y2 y3 y1 y4 y1 y3 y1 y2 when all numbers yi are different, with easy adjustments when repetitions occur. For the map θ (1, 0, 1; E) the image of the typical element e y1 ∪ e y2 ∪ e y3 ⊗ ez is y1 y2 y1 y3 y2 y3 + + y3 z y2 z y1 z if all numbers are different, with easy adjustments when repetitions occur. Finally, for the map θ (1, 1, 0; E) the image of typical element

48

Schur Functors and Schur Complexes

ex ⊗ e y1 ∪ e y2 ∪ e y3 is x y1 x y2 x y3 + + y2 y3 y1 y3 y1 y2 if all numbers are different, with easy adjustments when repetitions occur. We note a slight difference between the cases of Schur and Weyl modules. For Schur modules we could eliminate the relation θ (1, 0, 0; E). This is impossible for Weyl modules. Indeed, let us consider the case when all four numbers are the same and they equal y. The relations coming from θ (1, 1, 0; E) and θ (1, 0, 1; E) give the relation 2

y y =0 y y

in K 2,2 E. To get the relation y y =0 y y we need the relation θ (1, 0, 0; E). (e) Let λ = (3, 2). There are three choices of pairs u, v. (u, v) = (0, 0), (1, 0), (0, 1). The Young schemes are • • • , • •

• • , • •

• • • . •

It follows that K 3,2 E is a factor of the module D3 E ⊗ D2 E by the images of three maps: D4 E ⊗ E → D3 E ⊗ D2 E (corresponding to u = 0, v = 1), E ⊗ D4 E → D3 E ⊗ D2 E (corresponding to u = 1, v = 0), and D5 E → D3 E ⊗ D2 E (corresponding to u = v = 0). (f) We will show below that for two rowed partitions one can choose a smaller set of relations θ (1, u, v; E) with u = 0 that still sufﬁce to deﬁne the Weyl functor. Example (c) above shows that the other choice that worked for Schur functors (choosing relations with one overlap) does not deﬁne a Weyl functor. (g) Let λ = (2, 2, 1). We have three types of relations corresponding to the ﬁrst pair of rows (described in example (d)), and one type corresponding to the second and third row (described in example (c)). The Young schemes are • • • • ,

• • , •

• • • ,

• • . •

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49

(h) Let λ be a hook, i.e. a partition of the form λ = ( p, 1q−1 ). Graphically ... .. λ= .

,

with p boxes in the ﬁrst row and q boxes in the ﬁrst column. The relations between two rows of length 1 express the antisymmetry (cf. example (b)). There is only one type of relations corresponding to the ﬁrst two rows, for the pair u = v = 0. It follows that the Weyl functor K ( p,1q−1 ) E is the cokernel of the map D p+1 E ⊗

q−2

⊗1

E −→ D p E ⊗ E ⊗

q−2

1⊗m

E −→ D p E ⊗

q−1

E.

We ﬁnish this section by stating the obvious functoriality property of all above constructions. (2.1.18) Proposition. (a) The constructions of modules L λ/µ E, K λ/µ E are functorial with respect to the free module E. They deﬁne the endofunctors of the category of free K-modules. We refer to these functors as Schur functors and Weyl functors respectively. (b) The functors L λ/µ , K λ/µ are polynomial, homogeneous of degree |λ/µ|. (c) We have the functorial isomorphisms K λ/µ E = (L λ /µ E ∗ )∗ . Proof. We start with the proof of (a). The modules L λ/µ E, K λ/µ E are deﬁned as images of natural transformations φλ/µ , ψλ/µ of endofunctors of a category of free K-modules. They are therefore functors themselves. Their values are again in the category of free K-modules by the standard basis theorem. Part (b) follows because the exterior, symmetric, and divided powers are homogeneous polynomial functors. Statement (c) is a consequence of (1.1.7), because that statement implies that the map ψλ /µ is the dual of the map φλ/µ for E ∗ . 2.2. Schur Functors and Highest Weight Theory The modules L λ E play a crucial role in the representation theory of the general linear group. In this section we describe this connection. We assume that the commutative ring K is an inﬁnite ﬁeld of arbitrary characteristic.

50

Schur Functors and Schur Complexes

Let us denote by T ( by U ) the subgroup of GL(E) of all diagonal matrices (all upper triangular matrices with 1’s on the diagonal) with respect to a ﬁxed basis e1 , . . . , en of E. We recall that a rational representation of GL(E) is a vector space V together with a homomorphism of algebraic groups ρ : GL(E) → GL(V ). In terms of coordinate rings this means that we have a homomorphism ρˆ : K[GL(V )] −→ K[GL(E)] satisfying the conditions dual to the conditions for homomorphism. A rational representation V is called polynomial if ρ extends to the algebraic map ρ : EndK (E) → EndK (V ). (2.2.1) Proposition. Let V be a rational representation of GL(E), and let n E be a determinant representation of GL(E). Then for m >> 0 the rep resentation V ⊗K ( n E)⊗m is polynomial. Proof. Indeed, for each k, l (1 ≤ k, l ≤ dim V ) the image ρ(Y ˆ k,l ) is an element of K[GL(E)] = K[{X i, j }1≤i, j≤n , T ]/(T det(X i, j ) − 1), where X i, j is the (i, j)-th entry function on GL(E). Multiplying the representation ρ by the ˆ k,l ) will be muldeterminant n E means that the corresponding image ρ(Y tiplied by det(X i, j ). In this way, by multiplying by a sufﬁciently high power of det(X i, j ) we can clear the denominators of all elements ρ(Yi, j ). The following fact is well known in representation theory (cf. [B, section 8]). (2.2.2) Proposition. (a) Every character χ : T → GL(1) = K∗ is of the form (t1 , . . . , tn ) → χ χ χ t1 1 t2 2 . . . tn n for some integers χ1 , χ2 , . . . , χn . Here we denote by (t1 , . . . , tn ) the diagonal n × n matrix with the entries t1 , . . . , tn . (b) Every rational representation V of GL(E) has a decomposition Vχ , V = χ∈char(T)

where Vχ = {v ∈ V |ρ(t)v = χ (t)v} for all t ∈ T. The characters of T are called weights. The subspace Vχ of V is called the weight space of V corresponding to the weight χ . We denote by i the weight i (t1 , . . . , tn ) = ti .

2.2. Schur Functors and Highest Weight Theory

51

The next step is to investigate how the elements of U change weights of vectors from V . We denote by Ai, j (x) the elementary endomorphism Ai, j (x)(es ) = es + xδs j ei . (2.2.3) Proposition. (a) Let V be a polynomial representation of GL(E), and W a subrepresentation. Let v ∈ Vχ . Then there exists a natural number r and elements v0 , . . . , vr in V such that for every x ∈ K Ai, j (x)v = v0 + xv1 + ... + x r vr . Moreover, the vector vs is a weight vector of weight χ + s(i − j ) and if v ∈ W , then v0 , . . . , vr ∈ W . (b) Every nonzero polynomial representation V of GL(E) contains a nonzero U-invariant weight vector. Proof. The existence of the vectors vi follows at once from the fact that the representation V is a polynomial representation. To calculate the weight of vi we notice that if t = (t1 , . . . , tn ) is a diagonal matrix, then t Ai, j (x) = Ai, j (xti t −1 j )t. Applying both sides to v yields that the weight of vs is χ + s(i − j ). To prove the last statement we notice that we can use the assumption that K is inﬁnite. Then we can ﬁnd r + 1 values x1 , . . . , xr +1 for which the corresponding Vandermonde determinant is nonzero. This means that if W is a subrepresentation and the vectors Ai, j (x1 )v, . . . , Ai, j (xr +1 )v are in W , then v0 , . . . , vr +1 are also in W . This completes the proof of the ﬁrst part. To prove the second part we order the weights χ = (χ1 , χ2 , . . . , χn ) lexicographically with respect to the sequence χ1 , χ2 , . . . , χn . If V is a ﬁnite dimensional rational representation of GL(E), then there exists the earliest weight χ in this order for which Vχ = 0. By part (a) we see that every element of Vχ is a U-invariant. Indeed, if v ∈ Vχ and i < j, then the weights of elements vs from part (a) for s > 0 are earlier than χ. This means that vs = 0 for s > 0, so Ai, j (x)v = v. Since the elements Ai, j (x) (i < j, x ∈ k) generate U, v is a U-invariant. (2.2.4) Example. The canonical tableau cλ (i, j) = j for all (i, j) ∈ λ is a U-invariant in L λ E.

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Schur Functors and Schur Complexes

The following fact is crucial for the whole theory. (2.2.5) Lemma. The vector space (L λ E)U of U-invariants in L λ E is one dimensional (i.e., it is spanned by cλ ). Proof. Let U− be the subgroup of lower triangular matrices with 1’s on the diagonal in GL(E). We will actually prove the opposite result stating that the space of U− -invariants in L λ E is one dimensional, spanned by the anticanonical tableau cˆ λ deﬁned by cˆ λ (i, j) = n − λi + j. We start with two combinatorial results. Let T be a standard tableau, and let us take 1 ≤ i < j ≤ n. Deﬁne the j tableau Si (T ) to be the tableau obtained from T by replacing i by j in every j row in T containing i but not j. Let h i (T ) denote the number of rows in T j containing i but not j. The tableau Si (T ) does not have to be standard. In j some situations it turns out Si (T ) is standard. (2.2.6) Lemma. Let us ﬁx 1 ≤ i < j ≤ n. Assume that T is a standard tableau of shape λ, with entries from [1, n]. Assume that every row of T containing j an integer ≤ i contains also all integers i, i + 1, . . . , j − 1. Then Si (T ) is j j standard, and T is determined by h i (T ) and Si (T ). Proof. From our assumptions it follows that the rows of T not containing i j must follow the rows of T containing i. Thus T is obtained from Si (T ) by j replacing j by i in the ﬁrst h i (T ) rows that contain j but not i. j To show that Si (T ) is standard it is enough to do it in the case when T has j two rows. The action of Si on a row satisfying our assumption can at most replace the entry p with p + 1 for i ≤ p < j. Therefore the only possible j violation of standardness in Si (T ) arises when i occurs in the ﬁrst row of T and, for some p with i ≤ p < j, p occurs in the same positions in the ﬁrst and second row of T . Since the ﬁrst row of T must contain i, i + 1, . . . , p and since the entries in the second row are strictly increasing and T is standard, i must occur in the second row of T in the same position as in the ﬁrst row. If j does not occur in the second row of T , then p in the second row will be replaced by p + 1 and standardness will be preserved. But if j occurs in the second row, then it must occur exactly j − i positions after i, since all integers i + 1, . . . , j − 1 must occur there by assumption. However the ﬁrst row of T

2.2. Schur Functors and Highest Weight Theory

53

must have an element > j that is j − i positions after i, because it contains i + 1, . . . j − 1 but not j. This contradicts the standardness of T . We can pass from any standard tableau T to the anticanonical tableau by applying the composite operator n−1 n n Sn−2 Sn−2 . . . S13 S12 . Sn−1 j

Denote by h i the number of substitutions of j for i made by the application j of Si in that sequence. (2.2.7) Corollary. The standard tableau T of shape λ is determined by the j numbers h i deﬁned above. Proof. The corollary follows from Lemma (2.2.6) by induction. Now we conclude the proof of (2.2.5). For 1 ≤ i < j ≤ n and for x ∈ K we denote by A j,i (x) the matrix from U− which has entries on the diagonal equal to 1, the entry in the ( j, i)th position equal to x, and all other entries equal to 0. Consider the expansion from Proposition (2.2.3). It is clear that j j for v = T we have vr = Si (T ) and that r = h i (T ). Consider a nonzero linear combination m as Ts y= s=1

of standard, distinct tableaux Ts with all as = 0. To prove (2.2.5) it is enough to show that the anticanonical tableau cˆ λ is contained in the span of U− y. Consider A j,i (x)(y). This can be expanded as a polynomial in x, as in (2.2.3). The expansion gives j as Si (Ts ) + {terms of lower degree in x}. A j,i (x)(y) = x h j

h i (Ts )=h

The element

y =

j

as Si (Ts )

j

h i (Ts )=h

is nonzero by (2.2.6) and by the assumption as = 0. The element y is contained in the linear span of U− y by the Vandermonde determinant argument used in the proof of Proposition (2.2.3).• The statement (2.2.5) has important consequences.

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Schur Functors and Schur Complexes

(2.2.8) Proposition. The submodule Mλ E of L λ E generated by the canonical tableau cλ is an irreducible GL(E)-module. Every irreducible polynomial representation of GL(E) is isomorphic to Mλ E for some λ. Proof. A nonzero submodule W of Mλ E contains a U-invariant element, so it contains cλ . This means that W = Mλ E. This proves the ﬁrst statement. To prove the second part of the proposition let us consider the irreducible polynomial representation V of GL(E). By the second part of (2.2.3) we see that V contains for some weight λ a nonzero U-invariant vector vλ of weight λ. We will show that V and Mλ are isomorphic. Let us consider the submodule W in V ⊕ Mλ generated by (vλ , cλ ). We consider two projections p : W → V and q : W → Mλ . Both maps p, q are nonzero, so they are epimorphisms. Let us assume that one of those maps, say p, has nonzero kernel. This means that the module Ker p is isomorphic to Mλ , so W = V ⊕ Mλ . This is however impossible, because the weight space Wλ is one dimensional. Indeed, if U− denotes the group of lower triangular matrices with 1’s on the diagonal, then the subset U− TU is a Zariski dense subset of GL(E). Therefore W is spanned by U− TU(vλ , cλ ), which is the span of U− (vλ , cλ ). By part (b) of (2.2.3) we see that the only weight vector in the last span is (vλ , cλ ). The same reasoning works for the projection q. Thus p and q are isomorphisms and we are done. If K is a ﬁeld of characteristic zero, then it is well known (see [Hu1] for the proof) that GL(E) is linearly reductive, i.e., all ﬁnite dimensional representations are direct sums of simple ones. Then it follows instantly from (2.2.7) that L λ E is irreducible. Let us state these facts. (2.2.9) Theorem. All irreducible rational representations of GL(E) are iso morphic to Mλ E ⊗K ( n E)⊗m for some partition λ = (λ1 , . . . , λn−1 ) and m ∈ Z. This correspondence is bijective. (2.2.10) Theorem. Assume that K is a ﬁeld of characteristic zero. (a) We have Mλ E = L λ E. (b) Every rational representation of GL(E) is a direct sum of irreducibles. Theorem (2.2.9) is the main statement of the highest weight theory. The irreducible representations are parametrized by the sequences (λ1 , . . . , λn ) with λi ∈ Z, λ1 ≥ . . . ≥ λn . These are dominant integral weights for the group GL(E). They can be deﬁned as the weights whose values on the roots of GL(E)

2.2. Schur Functors and Highest Weight Theory

55

are integral, and whose values on positive roots are nonnegative. The integral weights are the weights whose values on the roots of GL(E) are integral. Let V be a rational representation of GL(E). We deﬁne the character of V χ char(V ) = dimVχ x1 1 . . . xnχn , where x1 , . . . , xn are indeterminates. Obviously we have char(V ⊕ V ) = char(V ) + char(V ), char(V ⊗ V ) = char(V )char(V ). (2.2.11) Remark. (a) When K is a ﬁeld of characteristic 0, then every representation is determined by its character. This follows easily from linear reductivity (2.2.10) (b) and from (2.2.5). (b) The function char(V ) is a symmetric function of x 1 , . . . , xn , because for each χ1 , . . . , χn and for each permutation σ ∈ n we have dim Vχ1 ,...,χn = dim Vχσ (1) ,...,χσ (n) . Indeed, the permutation matrix in GL(N ) corresponding to σ carries one space isomorphically into another. (c) The function char(L λ E) is called the Schur function (cf. [MD, chapter I]). The Schur functions play an important role in combinatorics. (d) The correspondence E → L λ E gives rise to a functor from the category VectK to itself. We will refer to it as a Schur functor. We ﬁnish our discussion with the simple example showing that (2.2.10) is false in positive characteristics. (2.2.12) Example. Let char K = p > 0. Let λ = (1 p ). Then L λ E = S p E, p but Mλ E is the span of the elements ei for 1 ≤ i ≤ n. This subspace is GL(E)-invariant, because in characteristic p we have (x + y) p = x p + y p . For the remainder of this section we assume that the commutative ring K has characteristic 0, i.e., it is a Q-algebra. We give an alternate description of Schur modules using Young idempotents (cf. [DC]). Consider the natural action of the symmetric group m on the tensor product E ⊗m given by σ (v1 ⊗ . . . ⊗ vm ) = vσ −1 (1) ⊗ . . . ⊗ vσ −1 (m) . Let λ be a partition of m and let D be a tableau of shape λ, with entries from [1, m] of weight (1m ) (i.e. with distinct entries). We deﬁne the Young

56

Schur Functors and Schur Complexes

symmetrizer e(D) ∈ K[m ] as follows. Denote by R(D) (by C(D)) the subgroup of m of permutations preserving the rows (columns) of D. We set e(D) = sgn(σ ) τ σ. τ ∈R(D) σ ∈C(D)

We deﬁne the representation of GL(E) depending on the tableau D by L D (E) = e(D)E ⊗m . (2.2.13) Lemma. (a) If D, D are the tableaux of the same shape λ, then L D (E) and L D (E) are isomorphic as GL(E)-modules. (b) If D is a tableau of shape λ, then L D (E) is isomorphic to the Schur module L λ (E). Proof. We start with part (a). Let σ be a permutation such that σ (D) = D , i.e., for every (i, j) ∈ D(λ) we have D (i, j) = σ (D(i, j)). Then we have R(D ) = σ R(D)σ −1 , C(D ) = σ C(D)σ −1 , and therefore e(D ) = σ e(D)σ −1 , which implies L D (E) = σ (L D (E)). Indeed, the isomorphism is given by acting by σ on a tensor. To prove (b) let us choose D which is a row standard tableau, minimal with respect to the order deﬁned in section 1.1.2 or, more precisely, D(i, j) = λ1 + . . . + λi−1 + j. We can identify representations L λ E and L D (E) as follows. L λ E can be interpreted as Im φλ . We embed Im φλ into the tensor product ⊗(i, j)∈D(λ) E by symmetrizing in each column. This can be done, because in characteristic 0 the symmetric power can be identiﬁed with the set of symmetric tensors (which in a characteristic free way is isomorphic to the divided power). Call the symmetrization map η. Then the image ηφλ (v) can be identiﬁed with e(D)(α(v)), where α is the product of exterior diagonals used to deﬁne φλ . The approach based on the use of the action of m on E ⊗m , due to Schur, can also give the irreducibility of Schur modules in characteristic 0. We sketch the basic steps in the proof. The interested reader may consult [DC]. The actions of m and of GL(E) on E ⊗m commute. Let us denote the span of all endomorphisms of type g ⊗m (g ∈ GL(E)) in EndK (E ⊗m ) by S(m, E). We also denote by (m, E) the endomorphisms of E ⊗m that are induced by the elements of the group ring K[m ].

2.3. Properties of Schur Functors

57

(2.2.14) Lemma (Schur Commutation Lemma). The algebras S(m, E) and (m, E) are their own commutants in EndK (E ⊗m ). More precisely, S(m, E) = {h ∈ EndK (E ⊗m ) | gh = hg for all g ∈ (m, E) }, (m, E) = {h ∈ EndK (E ⊗m ) | gh = hg for all g ∈ S(m, E) }. The algebra (m, E) is semisimple by Maschke’s theorem. This means the action of S(m, E) on E ⊗m is also semisimple and the Schur modules are precisely the irreducible representations. This follows from the following facts, proven in [DC], in the appendix on non-commutative algebra, part IV. (2.2.15) Proposition. Let B be a semisimple subalgebra in the matrix algebra Mn (K). Let C be the commutant of B, i.e., C = {x ∈ M N (K) | x y = yx for all y ∈ B }. Then the subalgebra C is also semisimple. Denote by VC the vector space V = K N with the structure of a C-module. Every simple module in VC is isomorphic to a module bV where b is an element of a minimal left ideal in B. If b, b generate the same left ideal in B, then bVC and b VC are isomorphic as C-modules. (2.2.16) Proposition. The isomorphism classes of minimal left ideals in K [m ] are in one to one correspondence with partitions of m. The correspondence is given by associating to λ the left ideal generated by e(D), where D is an arbitrary tableau of shape λ with entries from [1, m], with distinct entries.

2.3. Properties of Schur Functors. Cauchy Formulas, Littlewood–Richardson Rule, and Plethysm In this section we discuss some formulas from the representation theory of general linear groups. They will be used in the calculations involving vector bundles in chapters 6 through 9. We try to give both characteristic 0 and characteristic free statements. We start with the direct sum decompositions. Let E and F be two free modules over a commutative ring K. Our formulas express the modules L λ/µ (E ⊕ F) and K λ/µ (E ⊕ F) through the corresponding modules for E and F.

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Schur Functors and Schur Complexes

(2.3.1) Proposition. (a) There is a GL(E) × GL(F)-equivariant ﬁltration on L λ/µ (E ⊕ F) with the associated graded object L λ/ν E ⊗ L ν/µ F. ν|µ⊂ν⊂λ

If K is a commutative ring of characteristic 0, then we have a GL(E) × GL(F)-equivariant isomorphism L λ/µ (E ⊕ F) = L λ/ν E ⊗ L ν/µ F. ν|µ⊂ν⊂λ

(b) There is a GL(E) × GL(F)-equivariant ﬁltration on K λ/µ (E ⊕ F) with the associated graded object K λ/ν E ⊗ K ν/µ F. ν|µ⊂ν⊂λ

If K is a commutative ring of characteristic 0, then we have a GL(E) × GL(F)-equivariant isomorphism K λ/µ (E ⊕ F) = K λ/ν E ⊗ K ν/µ F. ν|µ⊂ν⊂λ

Proof. Since the proofs of both parts of the proposition are the same, we will just prove part (a). We observe that it is enough to prove the ﬁrst statement, since from it one deduces that the characters of the left and right hand sides of the second formula are the same. Let us denote dim E = n, dim F = m. Let us choose the bases e1 , . . . , en of E, f 1 , . . . , f m of F. Then e1 , . . . , en , f 1 , . . . f m is the basis of E ⊕ F, and we order it so f 1 < . . . < f m < e1 < . . . < en . We can consider the tableaux corresponding to this basis. For each such tableau T we deﬁne its F-part f (T ) to be the sequence a1 , . . . , as , where ai is the number of elements f j in the i-th row of T . We order sequences f (T ) lexicographically and denote this order by . Now for each ν such that µ ⊂ ν ⊂ λ we deﬁne the subspace Fν to be the span of all tableaux T such that (ν1 − µ1 , . . . , νs − µs ) f (T ). It is clear that Fν is a GL(E) × GL(F)-submodule. We also order all possible partitions ν by saying that ν ξ if (ξ1 , . . . , ξs ) (ν1 , . . . , νs ). It is clear by deﬁnition that if ξ ν then Fξ ⊂ Fν . Claim. Fν /

ξ ≺ν

Fξ is a factor of L ν/µ F ⊗ L λ/ν E.

It is obvious from the deﬁnition that the factor Fν / ξ ≺ν Fξ is spanned by the tableaux T such that f (T ) = (ν1 − µ1 , . . . , νs − µs ). Each tableau T such

2.3. Properties of Schur Functors

59

that f (T ) = (ν1 − µ1 , . . . , νs − µs ) can be considered as a pair of tableaux: the tableau T (1) of shape ν/µ with the entries from the set { f 1 , . . . , f m }, and the tableau T (2) of shape λ/ν with the entries from the set {e1 , . . . en }. We will denote such T by (T (1), T (2)). We show that the standard relations deﬁning L ν/µ F ⊗ L λ/ν E are satisﬁed

in Fν / ξ ≺ν Fξ .

Let us consider the element in Fν / ξ ≺ν Fξ which is the relation of type θ (λ/ν, a, u, v; E) (cf. section 2.1) on the tableau T (2). More precisely, let us denote by T (1) j the j-th row of T (1), and let us choose the rows T (2) j for j different than a, a + 1, and ﬁnally let us choose V1 ∈ u E, λa −νa +λa+1 −νa+1 −u−v v V2 ∈ E and V3 ∈ E. We consider the relation which shufﬂes the entries of V2 into the last λa − νa − u spots in the a-th row of λ/ν and the ﬁrst λa+1 − νa+1 − v spots in the (a + 1)st row of λ/ν and leaves all other entries in their spots. Let us call this relation R1 . We want to show that

this relation is zero in Fν / ξ ≺ν Fξ . Let us consider the relation R2 of type θ (λ/µ, a, νa − µa + u, v; E ⊕ F) which shufﬂes the entries of V2 ∧ T (1)a . This relation, being the deﬁning relation of L λ/µ (E ⊕ F), is identically zero

in Fν / ξ ≺ν Fξ . The relation R2 has more summands than R1 . However, all the summands occurring in R2 and not in R1 involve shufﬂing some basis elements from F into the earlier rows of λ/µ. Such elements are automati

cally contained in ξ ≺ν Fξ , so they are automatically zero in Fν / ξ ≺ν Fξ . Similarly we deal with the relations on the F-side. This proves our claim. Now the statement of the proposition follows, because by the standard basis theorem the left and right hand sides of the second formula have the same dimension, so in fact Fξ = L ν/µ F ⊗ L λ/ν E. Fν / ξ ≺ν

This completes the proof of (2.3.1). Next we discuss the Cauchy formulas. Let E and F be two free modules. We are interested in the modules Sm (E ⊗ F) and m (E ⊗ F). We want to express these modules in terms of Schur and Weyl modules. (2.3.2) Theorem. (a) There is a natural ﬁltration on Sm (E ⊗ F) whose associated graded object is L λ E ⊗ L λ F. |λ|=m

60

Schur Functors and Schur Complexes

(b) There is a natural ﬁltration on object is

m

(E ⊗ F) whose associated graded

L λ E ⊗ K λ F.

|λ|=m

We will prove part (a) of this theorem in chapter 3. We also give there the description of the ﬁltration giving part (b). For the proof we refer to [ABW2], or exercises 4, 5, 6 in chapter 3. Now we state the characteristic zero consequence. (2.3.3) Corollary. Let K be a commutative ring of characteristic 0. We have natural isomorphisms L λ E ⊗ L λ F, Sm (E ⊗ F) = |λ|=m m L λ E ⊗ L λ F. (E ⊗ F) = |λ|=m

Proof. Without loss of generality we can assume that K is a ﬁeld. Indeed, if the theorem is true over the ﬁeld Q of rational numbers, then the result extends to any commutative ring of characteristic 0 by base change. By Theorem (2.3.2) the characters of both sides of our formulas are the same (we notice that since char K = 0, K λ F = L λ F). Using the remark (2.2.11) (a), we get our statement. The formulas from Corollary (2.3.3) are special cases of the problem of outer plethysm. The general problem is to ﬁnd the multiplicities v(µ, ν, λ) in the decomposition L λ (E ⊗ F) = v(µ, ν, λ) L µ E ⊗ L ν F. |µ|=|ν|=|λ|

This is a very difﬁcult problem, solved in very few cases. Notice that substituting in (2.3.3) (a) F ⊗ G for F, we see that Sm (E ⊗ F ⊗ G) = v(µ, ν, λ) L λ E ⊗ L µ F ⊗ L ν G, |λ|=|µ|=|ν|=m

so the multiplicities v(µ, ν, λ) are symmetric in λ, µ, and ν.

2.3. Properties of Schur Functors

61

This interpretation explains why the problem of outer plethysm is so complicated. Let us consider the action of the group SL (E) × SL(F) × SL(G) on E ⊗ F ⊗ G. The dimension of E ⊗ F ⊗ G is in general bigger than the dimension of the acting group SL(E) × SL(F) × SL(G). This means that the structure of the ring of invariants is very complicated. Finding the expression for the Hilbert function of the ring of invariants is, however, equivalent to ﬁnding some of the multiplicities v(µ, ν, λ). This does not exclude the existence of a combinatorial formula for v(µ, ν, λ), but it means that such a formula will not lead to an easy calculation of our multiplicities. Next we state the Littlewood–Richardson rule. It describes a decomposition of the tensor product of Schur functors into Schur functors. Let K be a commutative ring of characteristic 0. Let λ, µ be two partitions. In the case where K is a ﬁeld we have by (2.2.10) u(λ, µ; ν)L ν E, Lλ E ⊗ Lµ E = |ν|=|λ|+|µ|

where u(λ, µ; ν) are some multiplicities. This decomposition carries over to the case of an arbitrary ring K of characteristic 0. Indeed, the explicit isomorphism over Q remains an isomorphism when tensored with K. The Littlewood–Richardson rule gives a beautiful combinatorial description of these multiplicities. In order to state the rule, we need one combinatorial notion. A word w = w1 . . . wt , with w1 , . . . , wt being positive integers, is a lattice permutation if for each s(1 ≤ s ≤ t) and each positive integer i, the number of occurrences of i in w1 , . . . , ws is not smaller than the number of occurrences of i + 1. Let T be a tableau of skew shape ν/λ. From such T we form a word w(T ) by reading T column by column, starting in each column with the lowest entry. In other words, w(T ) = (T (ν1 , 1), T (ν1 − 1, 1), . . . , T (ν1 − λ1 + 1, 1), T (ν2 , 2), . . . , T (νs − λs + 1, s)). We say that the tableau T satisﬁes the condition LP if the word w(T ) is a lattice permutation. Let us denote by P(λ, µ; ν) the set of all standard tableaux of shape ν/λ of weight µ satisfying the condition LP. Then we have (2.3.4) Theorem (Littlewood–Richardson Rule). u(λ, µ; ν) = card P(λ, µ; ν).

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Schur Functors and Schur Complexes

Again there is a characteristic free statement of the Littlewood–Richardson rule involving ﬁltrations (cf. [Bo2]), but we will not need it in the applications. Since we will use the rule sporadically, we give a combinatorial proof as a series of exercises at the end of this chapter. The reader might also look for the proof in [MD, chapter I]. MacDonald proves the statement about the symmetric functions, but we notice that his statement means that the representations L λ E ⊗ L µ E and |ν|=|λ|+|µ| u(λ, µ; ν)L ν E, with u(λ, µ; ν) deﬁned by (2.3.4), have the same characters. Let us state two important special cases of the Littlewood–Richardson rule, known as Pieri’s formulas. We recall that a skew partition λ/µ is called a vertical strip if it contains at most one box in each row, i.e., λi ≤ µi + 1 for all i. We denote the set of all vertical strips by VS. Similarly, the skew partition λ/µ is a horizontal strip if it contains at most one box in each column, i.e. when λ /µ is a vertical strip. We denote the set of all horizontal strips by HS. (2.3.5) Corollary (Pieri’s Formulas). Let K be a commutative ring of characteristic 0. Then we have the natural isomorphisms L ν E, Lλ E ⊗ Sj E = {ν|λ⊂ν,|ν/λ|= j,ν/λ∈VS}

Lλ E ⊗

j

E=

L ν E.

{ν|λ⊂ν,|ν/λ|= j,ν/λ∈HS}

The Littlewood–Richardson rule has an analogue for skew Schur functors. Assuming that K has characteristic 0, we can write L ν/µ E = w(ν/µ; λ)L λ E |λ|=|ν/µ|

Then we have (2.3.6) Theorem (Littlewood–Richardson Rule for Skew Schur Functors). w(ν/µ; λ) = u(λ, µ; ν) = card P(λ, µ; ν). Again we refer for the proof of this fact to [MD, chapter I]. The characteristic free statement involving ﬁltrations is also true (cf. [Bo2]). We state the analogues of Pieri’s formulas for skew shapes.

2.3. Properties of Schur Functors

63

(2.3.7) Corollary. Let K be a commutative ring of characteristic 0. (a)

L ν/(1 j ) E =

L λ E.

{λ | λ⊂ν, |ν/λ|= j, ν/λ∈VS }

(b)

L ν/( j) E =

L λ E.

{λ | λ⊂ν, |ν/λ|= j, ν/λ∈HS }

We conclude this section with a brief discussion of the problem of inner plethysm. The general problem is to decompose the functor L λ (L µ E) into Schur functors. This problem is probably even more difﬁcult than the outer plethysm. To see why, let us look at the situation from the point of view of invariant theory. We look at the special case λ = (1m ), µ = (1n ). Decomposing Sm (Sn E) into Schur functors involves the formula for the dimension of the homogeneous components of the ring of invariants of SL(E) acting on Sn E. Such rings are extremely complicated, at least according to nineteenth century invariant theorists. In view of this remark it is logical to expect nice formulas for Sn (S2 E) and Sn ( 2 E), because the action of GL(E) on S2 E or 2 E has ﬁnitely many orbits, so the rings of SL(E)-invariants are very simple. Such formulas are given in the next proposition. (2.3.8) Proposition. Let K be a commutative ring of characteristic 0. (a)

Sm (S2 E) =

L λ E,

|λ|=2m, λi even for all i

(b) 2 Sm E =

L λ E.

|λ|=2m, λi even for all i

Proof. It is enough to prove the proposition when K is a ﬁeld. The corresponding formulas for characters are given in [MD, chapter I]. However, it is convenient for later applications to give a proof based on U-invariants. Let us ﬁx an ordered basis e1 , . . . , en of the vector space E. We ﬁrst prove the formula (a). We identify the symmetric algebra S = Sym(S2 E) with the polynomial ring K[X i, j ], where X i, j = ei e j . We consider the generic n × n symmetric matrix X = (X i, j )1≤i, j≤n over S. For each r , 1 ≤ r ≤ n, we choose

64

Schur Functors and Schur Complexes

the r × r minor of X by deleting rows and columns with numbers > r . Let gr be the determinant of this minor. Then an easy calculation shows that gr is an U-invariant of weight (2r , 0n−r ) in Sr (S2 E). Let λ be a partition of 2m such that λj = 2µ j for 1 ≤ j ≤ n. This means that for each i > 0, λ2i−1 = λ2i . Therefore the product gλ = i>0 gλ2i is a nonzero U-invariant of weight λ in Sm (S2 E). This means that the left hand side of (a) contains the right hand side of (a). Now we prove (a) by induction on n = dim E. For n = 1 the formula is obvious. Let us assume that the formula (a) is true for dim E = n. We will prove that the left and right hand sides have the same dimension for dim E = n + 1. In view of the above construction, it is enough to prove (a). To prove the statement about the dimensions, let us consider the space E ⊕ K of dimension n + 1 over K. We will actually prove that the left and right hand sides of (a) for E ⊕ K are isomorphic as GL(E)-modules. The left hand side decomposes in the following way: S2 (E ⊕ K) = S2 E ⊕ E ⊕ K. Therefore Sm (S2 (E ⊕ K)) = Si (S2 E) ⊗ S j E ⊗ Sl K i+ j+l=m

=

Si (S2 E) ⊗ S j E.

i+ j≤m

Applying the formula (a) for i ≤ n, we get Sm (S2 (E ⊕ K)) =

L ν E ⊗ S j E.

i+ j≤m |ν|=2i, νl even for all l

On the other hand, for the right hand side we get L λ (E ⊕ K) = |λ|=2m, λi even for all i

|λ|=2m, λi even for all i

L µ E.

µ⊂λ, λ/µ∈VS

It remains to show that for any partition α, L α E occurs in the right hand sides of both formulas with the same multiplicity. Let us ﬁx m and α. The multiplicity of L α E in the ﬁrst formula is equal to the cardinality of the set Aα1 = {(ν, i, j) | |ν| = 2i, νl is even for all l, ν ⊂ α, α/ν ∈ VS, |α/ν| = j, i + j ≤ m}. The multiplicity of L α E in the second formula is equal to the cardinality of the set Aα2 = {λ | |λ| = 2m, λl even for all l, α ⊂ λ, λ/α ∈ VS}.

2.3. Properties of Schur Functors

65

To ﬁnish the proof of (a) it is enough to construct a bijection h from Aα1 to Aα2 . Let (ν, i, j) ∈ Aα1 . We construct λ ∈ Aα2 as follows. Let us deﬁne the numbers a j (0 ≤ j). If α j is even, then α j − ν j = 2a j . If α j is odd, then α j − ν j = 2a j + 1. We also deﬁne a−1 = 0. Let us construct the partition β by adding to the j−th column of α 2a j−1 boxes if α j is even and 1 + 2a j−1 boxes if α j is odd. Then β is a partition of 2i + 2 j (we added j boxes to α) with each β j even. We construct λ = h(ν, i, j) by adding to the ﬁrst column of β 2(m − i − j) boxes. The reader will check easily that the map h deﬁnes a bijection from Aα1 to Aα2 . This proves the statement (a). We prove (b) using the same technique. We identify S = Sym( 2 E) with the polynomial ring in the variables X i, j = ei ∧ e j . Then we consider the n × n skew symmetric generic matrix X = (X i, j )1≤i, j≤n over S. For each even number 2r, 0 ≤ 2r ≤ n, we deﬁne the element g2r to be the Pfafﬁan of the skew symmetric 2r × 2r matrix obtained from X by deleting rows and columns with numbers > 2r . We see easily that for each 2r the element g2r is a U-invariant of the weight (12r , 0n−2r ). Now for a partition λ with all λi even we ﬁnd that gλ = i gλi is a nonzero U-invariant of weight λ . This shows that the right hand side of (b) is contained in the left hand side. Then we can ﬁnish the proof of (b) with a similar (but easier) reasoning to the one in the proof of (a). It turns out that the companion formulas for m (S2 E) and m ( 2 E) also can be easily obtained. Let us recall that every partition λ can be written in the hook notation as λ = (a1 , . . . ar |b1 , . . . , br ) (cf. section 1.1.2). Let us denote by Q 1 (m) the set of partitions λ of m for which ai = bi + 1 for each i. Similarly we denote by Q −1 (m) the set of partitions λ of m for which bi = ai + 1 for each i. Then we have (2.3.9) Proposition. Let K be a commutative ring of characteristic 0. (a)

m

(S2 E) =

L λ E.

λ∈Q −1 (2m)

(b) m 2 E =

L λ E.

λ∈Q 1 (2m)

Proof. The corresponding formulas for characters are given in [MD, chapter I]. Let us just indicate the nonzero U-invariants in m (S2 E) and in m ( 2 E).

66

Schur Functors and Schur Complexes

The proof of (2.3.9) can be ﬁnished by an argument similar to the one in the proof of (2.3.8). Let λ be a partition from Q −1 (2m) which can be written in the hook notation as λ = (a1 , . . . ar |a1 + 1, . . . , ar + 1). Let us denote by X i, j the element ei e j of S2 E. Then we deﬁne g j = X j, j ∧ X j, j+1 ∧ . . . ∧ X j, j+ai −1 and gλ = g1 ∧ g2 ∧ . . . ∧ gr . Then one checks easily that gλ is a U-invariant of weight λ . Similarly for m ( 2 E). We ﬁx λ ∈ Q 1 (2m) such that λ can be written in the hook notation λ = (a1 + 1, . . . ar + 1|a1 , . . . , ar ). We denote by X i, j the element ei ∧ e j from 2 E. Then we deﬁne g j = X j, j+1 ∧ X j, j+2 ∧ . . . ∧ X j, j+ai and gλ = g1 ∧ g2 ∧ . . . ∧ gr . It is easy to check that gλ is a nonzero U-invariant of weight λ . Finally let us mention that all the formulas proven in this section are functorial, so they extend to vector bundles. 2.4. The Schur Complexes In this section we review the theory of Schur complexes. First we deal with the case of a general Z2 -graded module and deﬁne Z2 -graded Schur modules, which are common generalizations of Schur and Weyl modules. We apply this deﬁnition to complexes. Over a ﬁeld of characteristic zero the Schur complexes obtained in this way have many nice acyclicity properties. We review the main properties of these complexes. Then we discuss the special case of complexes of length 1. It turns out that in this case the acyclicity properties are true in characteristic free settings. The Schur complexes will be used in section 6.2 when proving the properties of determinantal ideals in positive characteristic and in the discussion of the differentials in resolutions of determinantal ideals. We work in the category of Z2 -graded free modules over a commutative ring K. The objects of our category are Z2 -graded modules = F0 ⊕ F1 where both F0 , F1 are free K-modules. The maps are all K-linear maps of degree 0. Our theory associates to the Z2 -graded module the family of Z2 -graded modules L λ . They are a common generalization of Schur and Weyl modules. For F1 = 0 we have L λ = L λ F0 , and for F0 = 0 we have L λ = K λ F1 [|λ|], where the bracket denotes shift in homological degree. The strategy of our approach is similar when deﬁning Schur functors. First we deﬁne the exterior and symmetric powers of and then we imitate the deﬁnition from section 2.1. The i-th exterior power i is a Z2 -graded module deﬁned in the following way.

2.4. The Schur Complexes

67

Consider the i-fold tensor product ⊗i . The permutation σ ∈ i acts of ⊗i in the following way: σ (v1 ⊗ . . . ⊗ vi ) = ±vσ −1 (1) ⊗ . . . ⊗ vσ −1 (i) , where vi are homogeneous elements from and the sign ± is determined by the rule that exchanging the elements v and w contributes the sign

(−1)deg(v)deg(w) . This means that ± = (−1) N , where N = (i, j)∈Inv(σ ) deg(vi )deg(v j ), where we sum over inversions of σ . This formula deﬁnes a (Z2 -graded) action of i on ⊗i . We deﬁne the i-th exterior power of as the subset of antisymmetric elements in ⊗i with respect to this action of i . For elements of degree 0 this means antisymmetry of elements, but for degree 1 elements this means symmetry. More precisely, i is a Z-graded vector space whose t-th graded piece is i i−t = Dt F1 ⊗ F0 . t

The i-th symmetric power Si is a Z-graded module deﬁned as a factor of ⊗i by the span of all elements v − σ (v) (v ∈ ⊗i , σ ∈ i ). The t-th graded piece of St is (Si )t =

t

F1 ⊗ Si−t F0 .

(2.4.1) Proposition. (a) There exist natural maps of Z-graded modules :

i+ j

→

i

⊗

j

whose components are given by the products of exterior and divided diagonals. (b) There exist natural maps of Z-graded modules m : Si ⊗ S j → Si+ j whose components are given by the products of exterior and symmetric multiplications. (c) There exist natural maps of Z-graded modules m:

i

⊗

j

→

i+ j

whose components are given by the products of exterior and divided multiplications.

68

Schur Functors and Schur Complexes

In the case when K is a ﬁeld, all above maps are GL(F0 ) × GL(F1 )equivariant. Proof. We will just deﬁne the maps from parts (a), (b), (c) of the proposition. We start with part (a). Choose the pair of indices a, b such that 0 ≤ a ≤ i, 0 ≤ b ≤ j. We deﬁne the component i+ j j i ⊗ a,b : → a+b

as the following map: Da+b F1 ⊗

a

i+ j−a−b

b

F0

↓ ⊗ i−a F0 ⊗ j−b F0 ↓ t23 F0 ⊗ Db F1 ⊗ j−b F0 ,

Da F1 ⊗ Db F1 ⊗ Da F1 ⊗

i−a

where t23 is the map exchanging the second and third positions in the tensor product. Now we proceed with part (b) of the proposition. The component m a,b : (Si )a ⊗ (S j )b → (Si+ j )a+b is deﬁned as a composition a F1 ⊗ Si−a F0 ⊗ b F1 ⊗ S j−b F0 ↓ t23 b a F1 ⊗ F1 ⊗ Si−a F0 ⊗ S j−b F0 ↓ m⊗m a+b F1 ⊗ Si+ j−a−b F0 . Finally, the component j i+ j i ⊗ → m a,b : a

is deﬁned as a composition

b

F0 ⊗ Db F1 ⊗ j−b F0 ↓ t23 i−a Da F1 ⊗ Db F1 ⊗ F0 ⊗ j−b F0 ↓ m⊗m Da+b F1 ⊗ i+ j−a−a F0 . Da F1 ⊗

i−a

This completes the proof of the proposition.

a+b

2.4. The Schur Complexes

69

We call the maps and m from Proposition (2.4.1) the diagonal and multiplication maps. We could also deﬁne the diagonal maps on symmetric powers, and the multiplication maps on exterior powers, but they will not be needed in our application. Now we are ready to deﬁne the Z2 -graded Schur modules. Let = F0 ⊕ F1 be as above, and let λ be a partition. For two partitions µ ⊂ λ we deﬁne L λ/µ =

λ1 −µ1

⊗

λ2 −µ2

⊗ ... ⊗

λ s −µs

/R(λ/µ, ),

where R(λ/µ, ) is the sum of submodules: λ1 −µ1

⊗ ... ⊗

⊗ ... ⊗

λ s −µs

λa−1 −µa−1

⊗ Ra,a+1 () ⊗

λa+2 −µa+2

for 1 ≤ a ≤ s − 1, where Ra,a+1 () is the submodule spanned by the images of the following maps (a, u, v, ): u ⊗ λa −µa −u+λa+1 −µa+1 −v ⊗ v ↓ 1⊗⊗1 u λa −µa −u ⊗ ⊗ λa+1 −µa+1 −v ⊗ v ↓ m 12 ⊗m 34 λa −µa ⊗ λa+1 −µa+1 for u + v < λa+1 − µa . The next step is the description of the standard basis in L λ/µ . Let f 1 , . . . , f m and g1 , . . . , gn be ﬁxed bases in F0 and F1 respectively. We consider the Z2 -graded set A = (A0 , A1 ) where A0 = [1, m], A1 = [1, n]. Let us recall from section 1.1.2 that a Z2 -graded tableau of shape λ/µ with values in A is a map T : D(λ/µ) → A. We deﬁne the map ϕ : A → F0 ⊕ F1 by setting ϕ(i) = f i for i ∈ A0 and ϕ( j) = g j for j ∈ A1 . Let T be a Z2 -graded tableau of shape λ with the values in A. We associate to T the element in L λ which is a coset of the tensor ϕ(T (1, µ1 + 1)) ∧ . . . ∧ ϕ(T (1, λ1 )) ⊗ ϕ(T (2, µ2 + 1)) ∧ . . . ∧ϕ(T (2, λ2 )) ⊗ . . . ⊗ ϕ(T (s, µs + 1)) ∧ . . . ∧ ϕ(T (s, λs )). In the sequel we will identify these two objects and we will call both of them the (Z2 -graded) tableaux of shape λ/µ.

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Let us order the set A by an arbitrary order . Let us recall that in section 1.1 we deﬁned a standard Z2 -graded tableau relative to the order as a tableau satisfying the conditions (1) T (u, v) T (u, v + 1) with equality possible when T (u, v) ∈ A1 , (2) T (u, v) T (u + 1, v) with equality possible when T (u, v) ∈ A0 . We have the following generalization of Proposition (2.1.9) (b). (2.4.2) Proposition. Let us ﬁx the order on A. The standard Z2 -graded tableaux of shape λ/µ with values in A form a basis of the module L λ/µ . Proof of proposition (2.4.2). First we prove that the standard Z2 -graded tableaux generate L λ/µ . It is clear that the row standard Z2 -graded tableaux generate L λ/µ . Let us order the set of such tableaux by the order deﬁned in section 1.1.1. We will prove that if the Z2 -graded tableau T is not standard, then we can express it modulo R(λ/µ, ) as a combination of earlier Z2 -graded tableaux. Since T is not standard, we can ﬁnd a and w for which T (a, w) ( T (a + 1, w) with possible equality if T (a, w) ∈ A1 . Let w be such an index that T (a + 1, w) = T (a + 1, w + 1) = · · · = T (a + 1, w ) T (a + 1, w + 1). We consider the map (a, u, v; ) for u = w − µa − 1 and v = λa+1 − w . The key observation is that the image of the tensor U1 ⊗ . . . ⊗ Ua−1 ⊗ V1 ⊗ V2 ⊗ V3 ⊗ Ua+2 ⊗ . . . ⊗ Us , where U j = ϕ(T ( j, µ j + 1)) ∧ . . . ∧ ϕ(T ( j, λ j − µ j )) for j = a, a + 1, and where V1 = ϕ(T (a, µa + 1)) ∧ ϕ(T (a, µa + 2)) ∧ . . . ∧ ϕ(T (a, w − 1)), V2 = ϕ(T (a, w)) ∧ . . . ∧ ϕ(T (a, λa − µa )) ∧ ϕ(T (a + 1, µa+1 + 1)) ∧ . . . ∧ ϕ(T (a + 1, w )), V3 = ϕ(T (a + 1, w + 1)) ∧ . . . ∧ ϕ(T (a + 1, λa+1 − µa+1 )), contains the tableau T with the coefﬁcient 1, and all the other tableaux occurring in this image are earlier than T . It remains to prove that the standard tableaux are linearly independent in L λ/µ . Let us consider a map Hλ/µ :

λ1 −µ1

⊗

λ2 −µ2

⊗ ... ⊗ β

λ s −µs

α

−→ ⊗(i, j)∈D(λ/µ) (i, j)

−→ Sλ1 −µ1 ⊗ Sλ2 −µ2 ⊗ . . . ⊗ Sλt −µt ,

2.4. The Schur Complexes

71

where α is the tensor product of exterior diagonals δ:

λ j −µ j

→ ( j, µ j + 1) ⊗ ( j, µ j + 2) ⊗ . . . ⊗ ( j, λ j − µ j )

and β is the tensor product of multiplications m : (µi + 1, i) ⊗ (µi + 2, i) ⊗ . . . ⊗ (λi , i) → Sλi −µ1 . The map Hλ/µ is called the Z2 -graded Schur map associated to the partition λ/µ. The straightforward calculation (compare [ ABW2, II.2]) shows that (2.4.3) Proposition. The image Hλ (R(λ/µ, )) equals 0. This means that Hλ/µ induces a surjective map from L λ/µ to Im Hλ/µ . Now we will show that the map Hλ/µ maps standard tableaux to linearly independent elements of Sλ1 −µ1 ⊗ Sλ2 −µ2 ⊗ . . . ⊗ Sλt −µt . This will show (2.4.2), and at the same time it will prove that L λ/µ = Im Hλ/µ . A typical basis element of St is ϕ(s1 ) . . . ϕ(st ) where ϕ(s1 ) ϕ(s2 ) . . . ϕ(st ) with equality ϕ(si ) = ϕ(si+1 ) allowed only when si ∈ A0 . We order these elements lexicographically with respect to the sequence (s1 , . . . , st ). We denote this order by &. If w = w1 ⊗ w2 ⊗ . . . ⊗ wt and w = w1 ⊗ w2 ⊗ . . . ⊗ wt are two tensor products of such elements in Sλ1 −µ1 E ⊗ Sλ2 µ2 E ⊗ . . . ⊗ Sλt −µt E, we say that w & w iff w j & wj for the smallest j for which w j = wj . If T is a standard Z2 -graded tableau of shape λ/µ, then the smallest element (with respect to the order we just deﬁned) occurring in Hλ/µ (T ) is ϕ(T (µ1 + 1, 1)) . . . ϕ(T (λ1 , 1)) ⊗ . . . ⊗ ϕ(T (µt + 1, t)) . . . ϕ(T (λt , t)). Indeed, if, in applying the map α, we make the exchange of elements in some row, then we put bigger elements in earlier columns, so we get the earlier (with respect to &) elements. Moreover, it follows easily from the deﬁnitions that ϕ(T (µ1 + 1, 1)) . . . ϕ(T (λ1 , 1)) ⊗ . . . ⊗ ϕ(T (µt + 1, t)) . . . ϕ(T (λt , t)) occurs in α(T ) with coefﬁcient 1. It is also obvious that the elements ϕ(T (µ1 + 1, 1)) . . . ϕ(T (λ1 , 1)) ⊗ . . . ⊗ ϕ(T (µt + 1, t)) . . . ϕ(T (λt , t))

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are different for different standard tableaux T . This proves that the images Hλ/µ T of standard Z2 -graded tableaux T are linearly independent.• (2.4.4) Example. (a) The module L 2,1 is by deﬁnition the cokernel of the diagonal map : 3 → 2 ⊗ . It is a direct sum (L 2,1 )3 ⊕ (L 2,1 )2 ⊕ (L 2,1 )1 ⊕ (L 2,1 )0 . There are two basic descriptions of the graded components (L 2,1 )i . If we choose the order so F0 F1 , then (L 2,1 )3 = K 2,1 F1 , (L 2,1 )2 = F1 ⊗ F1 ⊗ F0 , (L 2,1 )1 has a ﬁltration with associated graded object F1 ⊗ 2 F0 ⊕ F1 ⊗ S2 F0 , and (L 2,1 )0 = L 2,1 F0 . If we set F1 F0 , we get (L 2,1 )3 = K 2,1 F1 , (L 2,1 )2 has a ﬁltration with associated graded object D2 F1 ⊗ F0 ⊕ 2 F1 ⊗ F0 , (L 2,1 )1 = F1 ⊗ F0 ⊗ F0 , and (L 2,1 )0 = L 2,1 F0 . (b) The module L 2,2 is by deﬁnition the factor of 2 ⊗ 2 di3 ⊗→ vided by the following images of maps (1, u, v; ): 2 ⊗ 2 (corresponding to u = 0, v = 1) and 4 → 2 ⊗ 2 (corresponding to u = v = 0). It is a direct sum (L 2,2 )4 ⊕ (L 2,2 )3 ⊕ (L 2,2 )2 ⊕ (L 2,2 )1 ⊕ (L 2,2 )0 . The graded components have the following descriptions, the same for both possible orders: (L 2,2 )4 = K 2,2 F1 , (L 2,2 )3 = K 2,1 F1 ⊗ F0 , (L 2,2 )1 = L 2,1 F0 ⊗ F1 (L 2,2 )0 = L 2,2 F0 . Here we use the isomorphisms K 2,2/1 F1 ∼ = K 2,1 F1 , L 2,2/1 F0 ∼ = L 2,1 F0 . The middle component has similar description whether we use the order F0 F1 or F1 F0 . The module (L 2,2 )2 has a ﬁltration with the associated graded object D2 F1 ⊗ 2 F0 ⊕ 2 F1 ⊗ S2 F0 . As a consequence of (2.4.2) we prove the following properties of Z2 -graded Schur modules. (2.4.5) Theorem. The Schur modules L λ/µ have the following properties: (a) The t-th term (L λ/µ )t has a natural ﬁltration with the associated graded object K λ/ν F1 ⊗ L ν/µ F0 . |ν|=|λ|−t

2.4. The Schur Complexes

73

(b) The t-th term (L λ/µ )t has a natural ﬁltration with the associated graded object K ν/µ F1 ⊗ L λ/ν F0 . |ν|=t

Proof. We start with (a). Let us choose the order by setting A0 A1 and i j if and only if i < j for i, j ∈ As for s = 0, 1. Let us order all sequences (u 1 , . . . , u s ) by saying that (u 1 , . . . , u s ) (v1 , . . . , vs ) if u j > v j for the smallest j for which u j = v j . For each ν we deﬁne (F≤ν )s as the span of the images of λ λ1 −µ1 s −µs , ⊗ ... ⊗ u1

us

where the sequence (u 1 , . . . , u s ) is than (ν1 , . . . , νs ). We notice that the proof of (2.4.2) implies that if we take the tableau T from (F≤ν )s and we standardize it, we express it as a linear combination of earlier standard tableaux from (F≤ν )s . Let us order all partitions ν by the order , and let us consider the factor (F≤η )s . (F≤ν )s / η λ2 , µ1 > µ2 . Deﬁne λ(1) = (λ1 − 1, λ2 ), µ(1) = (µ1 − 1, µ2 ). The shape λ(1)/µ(1) has rows of the same length as λ/µ, but the ﬁrst row is shifted by one place to the left, so we have t+1 overlaps. Show that there is a natural epimorphism π (λ, µ) : L λ/µ E → L λ(1)/µ(1) E. Show that the kernel of π (λ, µ) is isomorphic to L λ1 −µ2 ,λ2 −µ1 E (with the convention that Ker π (λ, µ) = 0 if λ2 < µ1 ). Formulate and prove the analogous result for skew Weyl modules.

Exercises for Chapter 2

79

Schur and Weyl Modules in Positive Characteristic 4. Deﬁne two morphisms jd : Sd E → Dd E, i d : Dd E → Sd E by formulas i d (e1a1 . . . enan ) =

n! e(a1 ) . . . en(an ) , (a1 )! . . . (an )! 1

jd (e1(a1 ) . . . en(an ) ) = (a1 )! . . . (an )!e1a1 . . . enan . Prove that i d , jd deﬁne GL(E)-equivariant maps and that the compositions i d jd = n!(id Sd E ), jd i d = n!(id Dd E ). 5. Let λ = (λ1 , . . . , λs ) be a partition. (a) Deﬁne a morphism ˜jλ :

λ1

E ⊗ ... ⊗

λs

φλ

E −→ Sλ1 E ⊗ . . . ⊗ Sλt E

jλ ⊗...⊗ jλs 1

−→

Dλ1 E ⊗ . . . ⊗ Dλt E.

Prove that ˜jλ factors to give an equivariant map jλ : L λ E → K λ E. (b) Deﬁne a morphism

ψλ i˜λ : Dλ1 E ⊗ . . . ⊗ Dλs E −→

λ1

E ⊗ ... ⊗

λt

E.

Prove that i˜λ factors to give an equivariant map i λ : K λ E → L λ E. (c) Prove that i λ jλ = h λ (id L λ E ), jλ i λ = h λ (id K λ E ), where h λ = (x,y)∈λ h λ (x, y), where h λ (x, y) = λ y − x + λx − y + 1 is the hook length of a hook in λ with the corner at (x, y). (d) Deduce that over a ﬁeld of characteristic p > 0 the module L λ E is irreducible as long as λ does not contain a box (x, y) such that h λ (x, y) is divisible by p. 6. We call a partition λ p-regular if it does not contain p rows of the same length. Otherwise we call λ p-singular. (a) For a partition λ deﬁne pi λ = ( pi λ1 , . . . , pi λs ). Prove that every

partition can be written uniquely as λ = j≥0 p j λ( j) where λ( j) are p-regular partitions. We call this decomposition a p-adic decomposition of λ.

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Schur Functors and Schur Complexes

(b) Let |λ| = n = dim E. Prove that the module Mλ E contains a nonzero element of weight (1n ) if λ is p-regular. (c) Let K be an inﬁnite ﬁeld of characteristic p > 0. We deﬁne the Frobenius functor Fr : V ectK → V ectK by setting Fr (E) = E and p Fr (φ) = φ ( p) , where φ ( p) is a matrix (φi, j )1≤i≤m,1≤ j≤n . For a functor (i) H : V ectK → V ectK we denote H = H ◦ Fr i . This means that we take “the same” functor as H on object, but when evaluating H ◦ Fr i on a linear map η we raise every entry of the matrix H (η) to the power pi . (d) Deﬁne a U − p-invariant in L λ E to be a vector v ∈ L λ E such that

for the generic matrix t: = id + i< j ti, j E i, j from U tv = v +

α

pα

ti, j i, j vα .

i< j

Prove that if v is a U − p-invariant, then all vectors vα are also U − pinvariants. Prove that if λ is p-regular, then the only U − p-invariant in L λ E is the canonical tableau.

(e) (Steinberg theorem) Let λ be a partition, λ = j≥0 p j λ( j) its p-adic decomposition. Prove that ( j) Mλ( j) E. Mλ E = j≥0

Deduce that the reverse implication in (b) is also true. 7. Let K be an inﬁnite ﬁeld of characteristic p > 0. (a) Prove that the exterior power i E is an irreducible representation of GL(E). (b) Consider the symmetric power Sd E. Assume pi ≤ d < pi+1 . The sequence d = (d0 , . . . , di ) is a p-adic representation of d if d = d0 + d1 p + . . . + di pi . Assume that n = dim E ≥ d. For any p-adic representation d of d we denote by Nd the GL(E)-submodule of Sd E generated by the weight vector of weight (1d0 , p d1 , . . . , ( pi )di ). Describe Nd , and prove that these are the only equivariant subspaces in Sd E. (c) Let d, e be two p-adic representations of the same number d. We say that e is a reﬁnement of d if it can be obtained from d by several steps, each of which involves decreasing d j by 1 and simultaneously increasing d j−1 by p (for some j = 1, . . . , i). This deﬁnes a partial order on the set of p-adic representations of d, denoted d ⊂ e. Prove that d ⊂ e if and only if Nd ⊂ Ne .

Exercises for Chapter 2

81

(d) Let d be a p-adic representation of d. We deﬁne the partition λ( p, d)

where λv = ij=0 m v (d j ) p j , where the numbers m v (d) are deﬁned as follows p−1 if d ≥ (v + 1)( p − 1), m v (d) = d − v( p − 1) if v( p − 1) < d < (v + 1)( p − 1), 0 otherwise.

Prove that for each p-adic representation d of d the module Nd / e⊂d Ne is an irreducible representation of GL(E) of highest weight λ( p, d). 8. Let K be a ﬁeld. Consider a vector space E of dimension n. For a partition λ of m we denote by S λ the weight space of L λ E of weight (1m ). We can think of S λ as a span of tableaux of shape λ of weight (1m ) modulo the usual standard relations. The module S λ is called a Specht module corresponding to the partition λ. (a) Prove that S λ has the natural structure of a m -module. (b) Let K be a ﬁeld of characteristic 0. Prove that the modules S λ give a complete set of nonisomorphic irreducible m -modules. (c) Let K be a ﬁeld of characteristic p > 0. Let λ be a partition of m. Let M λ be the weight space of Mλ E of weight (1m ). The module M λ is = 0 if and only if λ is p-regular. It is proven in [ jm] that the representations M λ give a complete set of isomorphism classes of irreducible representations of Sm . 9. Let K be a ﬁeld of characteristic 3. Let E be a vector space of dimension n. Consider the Schur functors of degree 5. (a) Prove that L (3,1,1) E, L (4,1) E, L (5) E are irreducible, (b) Prove the following exact sequences, which imply the composition series of the remaining Schur functors: 0 → M(5) E → L (15 ) E → M(22 ,1) E → 0, 0 → M(4,1) E → L (2,13 ) E → M(3,2) E → 0, 0 → M(3,2) E → L (2,2,1) E → M(15 ) E → 0, 0 → M(2,2,1) E → L (3,2) E → M(2,13 ) E → 0.

Littlewood–Richardson Rule 10. Use the Littlewood–Richardson rule to ﬁnd the multiplicities of irreducible representations L λ E in the tensor products L 2,1 E ⊗ L 2,1 E, L 3,1 E ⊗ L 2,1 E .

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Schur Functors and Schur Complexes

11. Let λ, µ be two rectangular partitions, i.e. λ = (l s ), µ = (m t ). Prove that the tensor product L λ E ⊗ L µ E is multiplicity free, i.e. that for each ν the multiplicity u(λ, µ; ν) equals 0 or 1. Characterize the partitions ν such that L ν E occurs in L λ E ⊗ L µ E. 12. Let ν = (m t ) be a rectangular partition. Show that L ν E occurs in the tensor product L λ E ⊗ L µ E if and only if λ and µ can be ﬁtted together to ﬁll the rectangle ν, i.e. when for λ = (λ1 , . . . , λt ), µ = (µ1 , . . . , µt ) we have λi + µt+1−i = m for 1 ≤ i ≤ t. Show that if λ and µ can be ﬁtted together to ﬁll the rectangle ν, then the multiplicity of L ν E in L λ E ⊗ L µ E is equal to 1. 13. Let us ﬁx ﬁve numbers a, b, c, d, e with a ≥ b ≥ c, d ≥ e, a + b + c = d + e. Denote by t(m) the multiplicity of L d m ,em E in L a m E ⊗ L bm E ⊗ t(1)+m−2 m L c E. Prove that t(m) = m−1 . 14. Let us ﬁx ﬁve numbers a, b, c, d, e with a ≥ b ≥ c ≥ d, a + b + c + d = 2e. Denote by t(m) the multiplicity of L e2m E in L a m E ⊗ L bm E ⊗ . L cm E ⊗ L d m E. Prove that t(m) = t(1)+m−2 m−1 15. Let us ﬁx seven numbers : a1 , a2 , b1 , b2 , c1 , c2 , d. Assume a1 + b1 + c1 + a2 + b2 + c2 = 3d. Denote by t(m) the multiplicity L d 3m E in t(1)+m−2of m m m m m m L a1 ,a2 E ⊗ L b1 ,b2 E ⊗ L c1 ,c2 E. Prove that t(m) = m−1 . 16. Let λ = (λ1 , . . . , λs ), µ = (µ1 , . . . , µt ) be two partitions. Let ν = (ν1 , . . . , νs+t ) be a partition resulting from permuting the sequence (λ1 , . . . , λs , µ1 , . . . , µt ) to be nonincreasing. Let π ∈ s+t be the resulting permutation, i.e. λi = νπ(i) for 1 ≤ i ≤ s, µ j = νs+ j for 1 ≤ j ≤ t. Prove that the morphism λ1

ˆ m(λ, µ) : →

ν1

E ⊗ ... ⊗

E ⊗ ... ⊗

νs+t

λs

E⊗

µ1

E ⊗ ... ⊗

µt

E

E

given by permuting the factors according to the permutation π factors to give an equivariant epimorphism m(λ, µ) : L λ E ⊗ L µ E → L ν E. Use the Littlewood–Richardson rule to show that the multiplicity of L ν E in L λ E ⊗ L µ E is equal to 1. Let us order the partitions lexicographically. Prove that for all partitions η such that L η E occurs in L λ E ⊗ L µ E we have η ≥ ν. Sometimes the factor L ν E is called the Cartan piece of the tensor product L λ E ⊗ L µ E.

Exercises for Chapter 2

83

17. Let λ = (λ1 , . . . , λs ) and µ = (µ1 , . . . , µs ) be two partitions. Let ν be a partition ν = (λ1 + µ1 , . . . , λs + µs ). Deﬁne the map ˆ (λ, µ) :

λ1 +µ1

⊗ ... ⊗

µs

E ⊗ ... ⊗

λ s +µs

E→

λ1

E ⊗ ... ⊗

λs

E⊗

µ1

E

E

to be the tensor product of the diagonals followed by a permutation of ˆ factors. Prove that (λ, µ) factors to give an equivariant map (λ, µ) : L ν E → L λ E ⊗ L µ E. Use the Littlewood–Richardson rule to show that the multiplicity of L ν E in L λ E ⊗ L µ E is equal to 1. Let us order the partitions lexicographically. Prove that for all partitions η such that L η E occurs in L λ E ⊗ L µ E we have η ≤ ν.

Schur Functors and Duality 18. Let E be a vector space of dimension n. (a) Prove the canonical isomorphisms ∗

L λ1 ,...,λs E = L n−λs ,...,n−λ1 E ⊗

n

E

∗

⊗s

,

(b) Prove the canonical isomorphism K λ1 ,...,λn E ∗ = K −λn ,...,−λ1 E. 19. Let E be a vector space of dimension n. Use duality and the Littlewood– Richardson rule to decompose i E ⊗ j E ∗ to the irreducible highest weight representations as a GL(E)-module.

Acyclicity Properties of Schur Complexes 20. Let R be a commutative ring, let M be an R-module, and let F1 → F0 → M → 0 be a presentation of M. Let be a complex F1 → F0 . We deﬁne the module L λ M to be L λ M = H0 (L λ ).

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Schur Functors and Schur Complexes

Prove that this deﬁnition does not depend on the choice of the presentation . 21. Let M be an R-module of projective dimension 1 with a free resolution 0 → F1 → F0 → M → 0. Let be the complex F1 → F0 . Prove that if λ is a partition of d and M is a (d–1)st syzygy, then L λ () gives a free resolution of L λ M. 22. Let : F1 → F0 be a linear map of free K-modules. For each partition ν the complex L λ has a subcomplex X u 1 , vt > . . . > v1 , u s− j ≥ vt− j for 1 ≤ j ≤ t. This means that when we reverse the order of numbers in this tableau, we will get the standardeness condition for both tableaux. Recalling the deﬁnition of the standard tableau from section 1.A. we see that it is natural to call the double tableau us ... . . . u 1 i1 i2 . . . it . . . is vt . . . v1 j1 j2 . . . jt a standard double tableau. (3.2.2) Proposition. The standard double tableaux form a basis of the coordinate ring K[U I0 ] (which by (3.2.1) is the polynomial ring in the variables z i j for 1 ≤ i ≤ r, 1 ≤ j ≤ n − r ). Proof. The coordinate ring K[U I0 ] is generated by functions pˆ (i 1 , . . . , ir ). The standard products of Pl¨ucker coordinates restrict to the standard double tableaux. If a product of functions pˆ (i 1 , . . . , ir ) is a double tableau which is not standard, we can express as a linear combinations of standard products of Pl¨ucker coordinates, which in turn will express our double tableau as a linear combination of standard double tableaux. This shows that standard double tableaux span K[U I0 ]. To show that the standard double tableaux are linearly independent on U I0 , we notice that if some linear combination of standard double tableaux (say in degree s) is zero, then by multiplying by some power of p(n − r + 1, . . . , n) we can extend each summand to Grass(r, E). Each summand extends to a standard tableau, since adding at the end the row (n − r + 1, . . . , n) does not affect standardness. Thus we get a nontrivial linear combination of standard tableaux on Grass(r, E) which vanishes on U I0 . This is a contradiction. Let us denote by F, G respectively the space of rows and columns of the matrix (∗). The polynomial ring K[U I0 ] is identiﬁed with Sym(F ⊗ G). We want to consider more closely the meaning of Pl¨ucker relations in terms of

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Grassmannians and Flag Varieties

double tableaux. Let us consider the nonstandard tableau T =

i 1 i 2 . . . i a . . . ir . j1 j2 . . . ja . . . jr

More precisely, let us assume that i 1 < . . . < ir , j1 < . . . < jr , i a > ja , and i a ≤ n − r . To express T as a sum of earlier tableaux we use the relation R(i 1 , . . . , i a−1 ; ja+1 , . . . , jr ; j1 , . . . , ja , i a , . . . , ir ) from (3.1.2). Let us suppose that after the restriction to U I0 , p(i 1 , . . . , ir ) becomes an s × s minor of (∗) and p( j1 , . . . , jr ) becomes a t × t minor of (∗). This means that 1 ≤ i 1 < . . . < i s ≤ n − r < i s+1 < . . . < ir ≤ n, 1 ≤ j1 < . . . < jt ≤ n − r < jt+1 < . . . < jr ≤ n. We also assume that s ≤ t. We notice that in the summands of R(i 1 , . . . , i a−1 ; ja+1 , . . . , jr ; j1 , . . . , ja , i a , . . . , ir ) the number of entries that are ≤ n − r in the ﬁrst row is ≥ s. The summands in R(i 1 , . . . , i a−1 ; ja+1 , . . . , jr ; j1 , . . . , ja , i a , . . . , ir ) where the number of entries that are ≤ n − r in the ﬁrst row is equal to s correspond to shufﬂing j1 , . . . , ja with i a , . . . , i s . In all those summands the numbers bigger than n − r stay ﬁxed. This combination of products of minors of size s and t (s ≥ t) is, in terms of double tableaux, the relation of the type θ(a, u, v; F) on the left side of the double tableau with its right side ﬁxed. Our relation says that this combination belongs to the linear span of products of minors of size s + i multiplied by minors of size t − i for various i > 0. This establishes the following proposition. (3.2.3) Proposition. Let us consider the composition given by u F ⊗ s+t−u−v F ⊗ v F ⊗ s G ⊗ t G ↓ 1⊗⊗1⊗1⊗1 u F ⊗ s−u F ⊗ t−v F ⊗ v F ⊗ s G ⊗ t G ↓ m 12 ⊗m 34 ⊗1⊗1 s t F⊗ F⊗ sG⊗ tG ↓ 1⊗t23 ⊗1 s F⊗ sG⊗ tF⊗ tG ↓ ζs ⊗ζt Ss (F ⊗ G) ⊗ St (F ⊗ G) ↓m Ss+t (F ⊗ G), where ζs : s F ⊗ s G −→ Ss (F ⊗ G) sends f i1 ∧ f i2 ∧ . . . ∧ f is ⊗ g j1 ∧ g j2 ∧ . . . ∧ g js to the minor of the matrix Z = (z i, j )i, j ( f i ⊗ g j )i, j corresponding to rows i 1 , . . . , i s and columns j1 , . . . , js . Then each element in

3.2. The Standard Open Coverings of Flag Manifolds

95

Im is a linear combination of products of (s + i) × (s + i) minors and (t − i) × (t − i) minors for i > 0. (3.2.4) Example. 3 2 1 4 + =

5 4 3 2 1

1 2 3 4

4 3 1 2

4 2 1 3

1 2 3 4

1 2 3 4 5

−

5 4 3 2 1

1 2 3 4

−

3 2 1 4

−

−

=

1 2 3 4 5

4 3 2 5 1

1 2 3 4 5

−

4 3 2 1

1 2 3 4

,

5 4 2 3 1

1 2 3 4 5

1 2 3 4

5 3 2 4 1

1 2 3 4 5

5 4 3 2 1

1 2 3 5 4

.

For each partition λ = (λ1 , . . . , λt ) let us consider the maps ζλ :

λ1

F⊗

λ1

G⊗

λ2

F⊗

λ2

G ⊗ ... ⊗

λt

F⊗

λt

G

−→ Sym|λ| (F ⊗ G), where ζλ = ζλ1 ζλ2 . . . ζλt . Now we are ready to restate the characteristic free version of the Cauchy formula. (3.2.5) Theorem. The symmetric power Sm (F ⊗ G) has a natural GL(F) × GL(G)-invariant ﬁltration whose associated graded object equals |λ|=m L λ F ⊗ L λ G. Proof. Let us order all the partitions λ of weight m by the order ≤ (cf. section 1.1). We deﬁne Im ζµ , %λ = Im ζµ %λ = µ≤λ

µ dim F/ X they are zero by the relative version of the Grothendieck theorem ([H1, chapter III, Corollary 11.2]). Let us assume that j > i + 1 and that theorem is proven for smaller j − i. We take β = σi. (α), so β = (α1 , . . . , αi−1 , αi+1 − 1, αi + 1, αi+2 , . . . , αn ). Notice that β still satisﬁes our property, but now the pair (i, j) is changed to (i + 1, j). This means by induction that Ru h ∗ L(β) = 0 for all u ≥ 0. Now we again use the map h i . We notice that L(β) = L(α) ⊗ 'idi (L(α))+1 . and vice d (L(β))+1 versa L(α) = L(β) ⊗ 'i i . Since one of the line bundles L(α), L(β) has degree di which is ≥ −1, we can apply Proposition (4.2.2) to this bundle.

4.2. The Proof of Bott’s Theorem for the General Linear Group

121

We get either Ru h ∗ L(α) = Ru+1 h ∗ L(β) for all u or Ru h ∗ L(α) = Ru−1 h ∗ L(β) for all u. In both cases all higher direct images of Ru h ∗ L(α) are 0. Let us assume now that possibility (2) of (4.1.4) occurs. This means that the weight α + ρ has no repeated entries. Therefore there exists a unique permutation σ such that σ (α + ρ) is a strictly decreasing sequence, so σ . α is a partition β. Let us write a reduced expression σ = σv1 . . . σvl , where l = l(σ ). We deﬁne the weights β s = σvl−s+1 . . . σv.l (α) for s = 1, . . . , l. Obviously β l = β. We also set β 0 = α. We apply Proposition (4.2.2) to pvl−s and to L(β s+1 ). Since σv1 . . . σvl is a reduced expression, we have dls (L(β s+1 )) ≥ 0, so the proposition applies. We get Ru+1 h ∗ L(β s ) = Ru h ∗ L(β s+1 ) for all u and s. Putting these equalities together, we get Ru+l h ∗ L(α) = Ru h ∗ L(β) for all u. Therefore we are reduced to calculating the cohomology of the bundles L(β) where β is a partition. First we show that for such bundles Ru h ∗ L(β) = 0 for u > 0. In order to do this we use the permutation τ (i) = (n + 1 − i). Clearly l(τ ) = n2 . Let us choose the reduced expression τ = σ1 (σ2 σ1 ) . . . (σn−2 . . . σ1 )(σn−1 . . . σ1 ). Using Proposition (4.2.2) repeatedly, as above, we get n

Ru h ∗ L(β) = Ru+(2) h ∗ L(τ . (β))

for each u. Since n2 = dim F/ X , we get Ru h ∗ L(β) = 0 for u > 0. It remains to identify the direct images h ∗ L(β) for every partition β. To do this we ﬁrst notice that the question is local in X . Therefore we can assume that the bundle E is trivial, so E = X × E for some vector space E of dimension n over k. This means that we can identify Flag(E) with X × Flag(E). The map h is just the ﬁrst projection. The direct image h ∗ L(β) becomes O X ⊗ H 0 (Flag(E), L(β)). To conclude the proof we have to show that for every nonincreasing sequence β = (β1 , . . . , βn ) the space of sections

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H 0 (Flag(E), L(β)) is isomorphic to K (β1 ,...,βn ) E ∗ . Notice that we can also assume that βn = 0, because adding 1 to each coordinate corresponds to ten soring our bundle by n E ∗ , and the same is true for the functors K (β1 ,...,βn ) E ∗ . Pick β = (β1 , . . . , βn ). Let {b1 , . . . , bt } = {1 ≤ i ≤ n − 1 | βi > βi+1 }. Consider the graded ring S(β) = K nβ1 ,...,nβn E ∗ . n≥0

This is a homogeneous coordinate ring of the ﬂag variety Flag(b1 , . . . , bt ; E) embedded in a projective space P(L β E ∗ ) by using the line bundle L(β) on Flag(b1 , . . . , bt ; E). (4.2.4) Lemma. The ring S(β) is a domain. Proof. Assume S(β) is not a domain. Then the zero divisors of S(β) are the union of ﬁnitely many prime ideals P1 , . . . , Pm in S(β). The group GL(E) acts on S(β). Since GL(E) is connected, it follows that all ideals Pi are equivariant. This means P1 has to contain a U-invariant. But P1 is prime, so it has to contain a U-invariant in degree 1 – the canonical tableau. Since the representation L β E ∗ is irreducible, P1 contains all elements of degree 1. This is a contradiction proving that S(β) is a domain. By ([H1, chapter II, Exercise 5.14) we see that the normalization of S(β) S(β) = H 0 (Flag(b1 , . . . , bt ; E), L(nβ)). n≥0

Moreover, the same exercise shows that for n >> 0 we have H 0 (Flag(b1 , . . . , bt ; E), L(nβ)) = K nβ1 ,...,nβn E ∗ .

(∗∗)

Now it is easy to ﬁnish the proof. We will show in fact that S(β) is integrally closed. Without loss of generality we can assume that β is not a multiple of another weight. For such β we will show that (∗∗) holds for every n > 0. By (2.2.3) it is enough to show that every U-invariant in S(β) is a power of a canonical tableau cβ . Let x be a U-invariant in S(β). Then a high enough power x m is the power of the canonical tableau cβ , because S(β)/S(β) has to have ﬁnite length since it is supported at the origin. Let x m = cβl . If m divides l, we are done, because S(β) is a domain, so x has to be the power of canonical tableau. If not, then the weight of x is not an integral weight. The proof of Theorem (4.1.4) is complete.•

4.3. Bott’s Theorem for General Reductive Groups

123

4.3. Bott’s Theorem for General Reductive Groups In this section we assume that the reader is familiar with the basic notions concerning reductive groups and root systems. We state here Bott’s theorem for reductive groups. The results of this section will be used only in Chapter 8. We start with recalling some standard notation. Let K be an algebraically closed ﬁeld, and let G be a reductive linear group over K. Let T be a maximal torus, and B a Borel subgroup containing T. We denote by the root system associated to the pair G, T. This is by deﬁnition a ﬁnite set of vectors in Hom K (Lie(T), K), where Lie(T) is the Lie algebra of the torus T. The choice of B determines the subset + of positive roots in . The space HomK (Lie(T), K) is equipped with a nondegenerate scalar product ( , ). We deﬁne, for α, β ∈ HomK (Lie(T), K), (β, α) = 2(β, α)/(α, α). We denote by ( the lattice of integral weights in HomK (Lie(T), K): ( = {γ ∈ HomK (Lie(T), K) | ∀α ∈ , (γ , α) ∈ Z}. The lattice ( contains the cone (+ of dominant integral weights, (+ = {γ ∈ ( | ∀α ∈ + , (γ , α) ∈ Z+ }. Bott’s theorem gives a rule for calculating cohomology groups of the line bundles on the homogeneous space G/B. Such bundles are described by weights. Indeed, for each character γ of T we deﬁne a one dimensional rational B-module V (γ ) by letting the unipotent radical U of B act trivially on V (γ ) and the torus T act by the character γ , and by letting L(γ ) = G ×B V (γ ), where for any rational B-module V we denote by G ×B V the product G × V divided by the equivalence relation (g, v) ∼ (gb, b−1 v) for b ∈ B. We can identify the group of characters of T with the additive subgroup in ( by associating to each character its derivative at identity.

The Weyl group W of G acts naturally on weights. Let ρ = 12 α-0 α be half of the sum of positive roots. We deﬁne the dotted action of W on weights σ . (γ ) = σ (γ + ρ) − ρ. Let us recall that the irreducible representations of G correspond to the dominant integral weights. For a dominant integral weight β we denote by Vβ the irreducible G-module of highest weight β. (4.3.1) Theorem (Bott). Let G, T, B, W , be as above. Let γ be an integral weight, and let L(γ ) be the corresponding line bundle over G/B. Then one

124

Bott’s Theorem

of two mutually exclusive possibilities occurs: (1) There exists σ ∈ W , σ = 1, such that σ . (γ ) = γ . Then the cohomology groups H i (G/B, L(γ )) are zero for i ≥ 0. (2) There exists a unique σ ∈ W such that σ . (γ ) := (α) is a dominant integral weight. In this case all cohomology groups H i (G/B, L(γ )) are zero for i = l(σ ), and H l(σ ) (G/B, L(γ )) = Vα . (4.3.2) Remark. The proof in the general case follows the same scheme as in the case of the general linear group. The role of the ﬂag varieties Flag (1, 2, . . . , i − 1, i + 1, . . . , n − 1; E) is played by the homogeneous space G/Pα , where Pα is a parabolic subgroup corresponding to a simple root α. (4.3.3) Examples. (a) Let us ﬁx a vector space E of dimension n, and let us consider the general linear group G = GL(E). Then we can choose the maximal torus T to be the subgroup of diagonal matrices, and the Borel subgroup B to be the subgroup of upper triangular matrices. The homogeneous space G/B can be identiﬁed with Flag(E), the set ( = Zn , and the Weyl group is isomorphic to n . The statement of Bott’s theorem reduces to Corollary (4.1.2). (b) Let F be a vector space of dimension 2n + 1 with a nondegenerate symmetric bilinear form ( , ). Let us take G = SO(F) to be the special orthogonal group. The lattice ( = Zn . The Weyl group W is a hyperoctahedral group acting on ( by signed permutations. The half sum of the positive roots is ρ = ( 2n−1 , 2n−3 , . . . , 12 ) 2 2 (c) Let F be a vector space of dimension 2n with a nondegenerate skew symmetric bilinear form ( , ) on F. Let us take G = Sp(F) to be the symplectic group associated to F. The lattice ( = Zn . The Weyl group W is a hyperoctahedral group acting on ( by signed permutations. The half sum of the positive roots is ρ = (n, n − 1, . . . , 1) (d) Let F be a vector space of dimension 2n with a nondegenerate symmetric bilinear form ( , ). Let us take G = SO(F) to be the special orthogonal group. The lattice ( = Zn . The Weyl group W is a subgroup of hyperoctahedral group acting on ( by signed permutations with even number of sign changes. The half sum of the positive roots is ρ = (n − 1, n − 2, . . . , 0).

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125

We proceed with some explicit calculations on homogeneous spaces G/P where P is a maximal parabolic subgroup in G. We will need these kinds of calculations in chapter 8. We limit ourselves to some examples related to classical groups. They can be viewed as analogues of (4.1.9). We recall that by the general theory ([Hu2], [Bou]) there are (up to conjugation) ﬁnitely many types of parabolic subgroups in G, and they correspond to subsets of simple roots. We start with the symplectic group. Let F be a vector space of even dimension 2n with a nondegenerate skew symmetric form ( , ). The simple roots of are α j = j − j+1 for j = 1, . . . , n − 1 and αn = 2n . Next we give a concrete description of the homogeneous spaces G/B. Let us recall that a subspace R ⊂ F is isotropic if the restriction of ( , ) to R is zero. We consider the set IFlag(F) = {(R1 , . . . , Rn ) ∈ Flag(1, 2, . . . , n; F) | Rn is isotropic}. The group Sp(F) acts on the set IFlag(F) transitively. To see this, observe that for a ﬂag (R1 . . . , Rn ) ∈ IFlag(F) we can choose a symplectic basis e1 , . . . , en , e¯ n , . . . , e¯ 1 of F so e1 , . . . , ei is a basis of Ri (1 ≤ i ≤ n). Since the symplectic group operates transitively on symplectic bases, we are done. The space IFlag(F) can be identiﬁed with the homogeneous space G/H, where H is a subgroup of elements in Sp(F) stabilizing the ﬂag (R1 , . . . , Rn ). By Borel’s theorem there exists a Borel subgroup B contained in H. Since every parabolic subgroup of a connected reductive group is connected ([Hu2]), we have H = B. This realization of the homogeneous space G/B allows us to develop relative theory in the same spirit as in section 3.3. We just give the deﬁnitions, leaving the details to the reader. Let F be a symplectic vector bundle over a scheme X , i.e. a vector bundle F equipped with a map ( , ) : 2 F → O X for which the restriction to each ﬁber gives a nondegenerate skew symmetric form on it. We can construct a relative isotropic ﬂag variety IFlag(F) with the structure map p : IFlag(F) → X . The statement of Bott’s theorem is true in a relative version with higher direct images replacing cohomology groups. We leave the formulation of this result to the reader. For each j = 1, . . . , n we consider the maximal parabolic subgoup P j in G = Sp(2n) which corresponds to the subset of all simple roots except α j . The space G/P j can be identiﬁed with the isotropic Grassmannian IGrass( j, F) of isotropic subspaces of dimension j in F. This is a closed subset in Grass(F), so we can talk about the tautological subbundle R j on IGrass( j, F). For any

126

Bott’s Theorem

isotropic subspace from IGrass( j, F) we deﬁne the orthogonal complement R ∨ = {x ∈ F | ∀y ∈ R (x, y) = 0}. The space R ∨ contains R and has dimension 2n − j. The correspondence R → R ∨ deﬁnes a tautological bundle R∨j of dimension 2n − j on IGrass ( j, F). We have the inclusions of bundles on IGrass( j, F) R j ⊂ R∨j ⊂ F × IGrass( j, F). The bundle R∨j /R j is a symplectic bundle. Indeed, we have a map of vector bundles 2

R∨j /R j → OIGrass( j,F) ,

which on each ﬁber is induced by the form ( , ), so it is nondegenerate. This means that for each dominant weight µ = (µ1 , . . . , µn− j ) for the root system of type Cn− j we can talk about the bundle Vµ (R∨j /R j ). Its ﬁber over a point corresponding to the isotropic space R is Vµ (R ∨ /R). (4.3.4) Corollary. Let us consider the vector bundle Vβ,µ = K β R j ⊗ Vµ (R∨j /R j ) over IGrass(r, F), where β = (β1 , . . . , β j ) is a dominant integral weight for the root system of type A j−1 and µ = (µ1 , . . . , µn− j ) is the integral dominant weight for the root system of type C n− j . Let us consider the weight γ = (−β j , . . . , −β1 , µ1 , . . . , µn− j ). Then one of the mutually exclusive possibilities occurs: (1) There exists σ ∈ W , σ = 1, such that σ (γ ) = γ . Then all cohomology groups H i (IGrass( j, F), Vβ,µ ) are 0 for i ≥ 0, (2) There exists unique σ ∈ W such that σ . (γ ) := α is a dominant integral weight for the root system of type C n . Then all cohomology groups H i (IGrass( j, F), Vβ,µ ) are 0 for i = l(σ ), and H l(σ ) (IGrass( j, F), Vβ,µ ) = Vα (F). Proof. Let us consider the projection p : G/B → G/P j . Identifying, as above, G/B with the space of isotropic ﬂags and G/P with the isotropic Grassmannian, we see that the ﬁber over a point corresponding to a subspace R is p −1 (R) = Flag(R) × IFlag(R ∨ /R). Therefore we can identify G/B with the relative variety Flag(R j ) × IFlag(R∨j /R j ). Consider the line bundle L(γ ) on G/B. Using Bott’s theorem in relative situation for the types A j−1 and Cn− j

4.3. Bott’s Theorem for General Reductive Groups

127

we see that Ri p∗ (L(γ ) = 0 for i > 0 and R0 L(γ ) = Vβ,µ . Now our statement follows from Bott’s theorem (4.3.1) and the spectral sequence of the composition. (4.3.5) Remark. In the above corollary and in the following calculations we adopt the convention that for a vector space E of dimension m, K (β1 ,...,βm ) E ∗ ∼ = K (−βm ,...,−β1 ) E. Thus the above proposition also allows the calculation of the cohomology groups H i (IGrass( j, F), K β R∗j ⊗ Vα (R∨j /R j )). Let us look more closely at the isotropic Grassmannian IGrass( j, F) as a subset of the Grassmannian Grass ( j, F). (4.3.6) Proposition. The isotropic Grassmannian IGrass( j, F) is locally a complete intersection in Grass( j, F). The structure sheaf of IGrass( j, F) can be resolved by locally free sheaves over Grass( j, F) by means if the Koszul complex 0→

(2j ) 2

Rj

→ ... →

2

ψ

R j → OGrass( j,F) .

Proof. Since F is a symplectic space, we have the following map of locally free sheaves over Grass( j, F): 2

Rj →

2

F × OGrass( j,F) → OGrass( j,F)

with the left map coming from tautological inclusion and the right one induced by the form ( , ). The composition gives us the cosection ψ of 2 R j which deﬁnes our Koszul complex. It is clear by deﬁnition that IGrass( j, F) is equal to the set of zeros of this cosection. Moreover, an easy calculation shows that locally these equations deﬁne a reduced subscheme of Grass( j, F). The dimension count shows that locally the equations give a regular sequence, so the Koszul complex is acyclic. The situation for the orthogonal group is very similar, but there is one difference. The special orthogonal group does not act transitively on the isotropic ﬂags; only the orthogonal group does. This leads to some minor differences, which we highlight below. We just formulate the results we need, as the proofs are the same as in the case of a symplectic group. We ﬁrst consider the group of type Bn . Let F be a vector space of odd dimension 2n + 1 with a nondegenerate symmetric form ( , ). The simple

128

Bott’s Theorem

roots of are α j = j − j+1 for j = 1, . . . n − 1 and αn = n . We take G = SO(F). This group is not simply connected. Since we have in mind some applications to nilpotent orbits in the corresponding Lie algebra, we do not need to discuss spinor groups. We start with a description of the homogeneous spaces G/B in terms of ﬂags. Let us recall that a subspace R ⊂ F is isotropic if the restriction of ( , ) to R is zero. We consider the set IFlag(F) = {(R1 , . . . , Rn ) ∈ Flag(1, 2, . . . , n; F) | Rn is isotropic}. The orthogonal group O(F) acts transitively on the set IFlag(F). Indeed, for each ﬂag (R1 . . . , Rn ) ∈ IFlag(F) we can choose a hyperbolic basis e1 , . . . , en , e, e¯ n , . . . , e¯ 1 of F so e1 , . . . , ei is a basis of Ri (1 ≤ i ≤ n) and e1 , . . . , en , e is a basis of Rn∨ . Now the orthogonal group O(F) operates transitively on hyperbolic bases. We will show that even SO(F) does. Indeed, let g ∈ O(F) be an element from O(F) which in given hyperbolic basis sends for each i ei to ei , e¯ i to e¯ i and e to −e. The element g ﬁxes he ﬂag (R1 , . . . , Rn ) where Ri is spanned by e1 , . . . , ei for i = 1, . . . , n. This means that for every h ∈ O(F) we have h(R1 , . . . , Rn ) = hg(R1 , . . . , Rn ). One of these elements has to lie in SO(F). As for symplectic group we now identify G/B with IFlag(F). Moreover, we can again give the relative version of the whole setup. Let F be an orthogonal vector bundle of dimension 2n + 1 over a scheme X , i.e. a vector bundle F equipped with a map ( , ) : S2 F → O X for which the restriction to each ﬁber gives a nondegenerate symmetric form on it. We can construct a relative isotropic ﬂag variety IFlag(F) with the structure map p : IFlag(F) → X . The relative version of Bott’s Theorem (4.3.1) is true if we replace cohomology groups with higher direct images. We leave the formulation of this result to the reader. For each j = 1, . . . , n we consider the maximal parabolic subgoup P j in G = SO(2n + 1) which corresponds to the subset of all simple roots except α j . The space G/P j can be identiﬁed with the isotropic Grassmannian IGrass( j, F) of isotropic subspaces of dimension j in F. This is a closed subset in Grass(F), so we can talk about the tautological subbundle R j on IGrass( j, F). For any isotropic subspace from IGrass( j, F) we deﬁne the orthogonal complement R ∨ = {x ∈ F | ∀y ∈ R (x, y) = 0}. The space R ∨ contains R and has dimension 2n + 1 − j. The correspondence R → R ∨ deﬁnes a tautological bundle R∨j of dimension 2n + 1 − j

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129

on IGrass( j, F). We have the inclusions of bundles on IGrass( j, F) R j ⊂ R∨j ⊂ F × IGrass( j, F). The bundle R∨j /R j is an orthogonal bundle of dimension 2(n − j) + 1. Indeed, we have a map of vector bundles S2 (R∨j /R j ) → OIGrass( j,F) which on each ﬁber is induced by the form ( , ), so it is nondegenerate. This means that for each dominant weight µ = (µ1 , . . . , µn− j ) for the root system of type Bn− j we can talk about the bundle Vµ (R∨j /R j ). Its ﬁber over a point corresponding to to the isotropic space R is Vµ (R ∨ /R). (4.3.7) Corollary. Let us consider the vector bundle Vβ,µ = K β R j ⊗ Vµ (R∨j /R j ) over IGrass(r, F), where β = (β1 , . . . , β j ) is a dominant integral weight for the root system of type A j−1 and µ = (µ1 , . . . , µn− j ) is the integral dominant weight for the root system of type Bn− j . Let us consider the weight γ = (−β j , . . . , −β1 , µ1 , . . . , µn− j ). Then one of the mutually exclusive possibilities occurs: (1) There exists σ ∈ W , σ = 1 such that σ (γ ) = γ . Then all cohomology groups H i (IGrass( j, F), Vβ,µ ) are 0 for i ≥ 0, (2) There exists unique σ ∈ W such that σ . (γ ) := α is a dominant integral weight for the root system of type Bn . Then all cohomology groups H i (IGrass( j, F), Vβ,µ ) are 0 for i = l(σ ), and H l(σ ) (IGrass( j, F), Vβ,µ ) = Vα (F). The proof of (4.3.7) is identical to that of (4.3.4). Let us look more closely at the isotropic Grassmannian IGrass( j, F) as a subset of the Grassmannian Grass( j, F). (4.3.8) Proposition. The isotropic Grassmannian IGrass( j, F) is locally a complete intersection in Grass( j, F). The structure sheaf of IGrass( j, F) can be resolved by locally free sheaves over Grass( j, F) by means if the Koszul complex 0→

j+1 ( 2 )

ψ

(S2 R j ) → . . . → S2 R j → OGrass( j,F) .

The proof is identical to that of (4.3.6).

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Finally we consider the group of type Dn . Let F be a vector space of even dimension 2n with a nondegenerate symmetric form ( , ). The simple roots of are α j = j − j+1 for j = 1, . . . n − 1 and αn = n−1 + n . We take G = SO(F). Again we are not interested in the spinor group. The description of the homogeneous spaces G/B in terms of ﬂags is now different. We again start with the set IFlag(F) of isotropic ﬂags: IFlag(F) = {(R1 , . . . , Rn ) ∈ Flag(1, 2, . . . , n; F) | Rn is isotropic}. As above, the orthogonal group O(F) acts transitively on the set IFlag(F). However, here IFlag(F) has two connected components and SO(F) operates transitively on each of them. In order to see this, let us ﬁx a hyperbolic basis e1 , . . . , en , e¯ n , . . . , e¯ 1 . We associate to it a ﬂag (R10 , . . . , Rn0 ) where Ri0 is spanned by e1 , . . . , ei . For a given ﬂag (R1 , . . . , Rn ) there exists h ∈ O(F) such that h Ri0 = Ri for i = 1, . . . , n. Both ﬂags are in the same component if h ∈ SO(F). Both components are homogeneous spaces for SO(F), so they are connected. The only thing to show is that they do not coincide. Let (R1 , . . . , Rn ) be in both components. Then there exist elements h 1 ∈ SO(F) and h 2 ∈ O(F) \ SO(F) such that h j Ri0 = Ri for i = 1, . . . , n, j = 1, 2. This 0 means that h −1 2 h 1 ﬁxes Ri for i = 1, . . . , n. Now simple linear algebra shows −1 that det(h 2 h 1 ) = 1, which is a contradiction. We will denote two components of IFlag(F) by IFlag+ (F) and IFlag− (F). We identify G/B with Flag+ (F). Moreover, we can again give the relative version of the whole setup. Let F be an orthogonal vector bundle of dimension 2n over a scheme X , i.e. a vector bundle F equipped with a map ( , ) : S2 F → O X for which the restriction to each ﬁber gives a nondegenerate symmetric form on it. We can construct a relative isotropic ﬂag variety IFlag(F) with the structure map p : IFlag(F) → X . The relative version of Bott’s theorem (4.3.1) is true if we replace cohomology groups with higher direct images. We leave the formulation of this result to the reader. For each j = 1, . . . , n we consider the maximal parabolic subgoup P j in G = SO(2n) which corresponds to the subset of all simple roots except α j . Similar arguments to those for type Bn show that for j = 1, . . . , n − 2 the space G/P j can be identiﬁed with the isotropic Grassmannian IGrass( j, F) of isotropic subspaces of dimension j in F. For j = n the isotropic Grassmannian IGrass(n, F) has two connected components IGrass+ (F) and IGrass− (F). They can be identiﬁed with the homogeneous spaces SO(F)/P j for j = n − 1, n. For the remainder of this section we assume that 1 ≤ j ≤ n − 2. Each SO(F)/P j is a closed subset in Grass( j, F), so we can talk about the

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131

tautological subbundle R j on IGrass( j, F). For any isotropic subspace from IGrass( j, F) we deﬁne the orthogonal complement R ∨ = {x ∈ F | ∀y ∈ R (x, y) = 0}. The space R ∨ contains R and has dimension 2n − j. The correspondence R → R ∨ deﬁnes a tautological bundle R∨j of dimension 2n − j on IGrass( j, F). We have the inclusions of bundles on IGrass( j, F) R j ⊂ R∨j ⊂ F × IGrass( j, F). The bundle R∨j /R j is an orthogonal bundle of dimension 2(n − j) + 1. Indeed, we have a map of vector bundles S2 (R∨j /R j ) → OIGrass( j,F) which on each ﬁber is induced by the form ( , ), so it is nondegenerate. This means that for each dominant weight µ = (µ1 , . . . , µn− j ) for the root system of type Dn− j we can talk about the bundle Vµ (R∨j /R j ). Its ﬁber over a point corresponding to to the isotropic space R is Vµ (R ∨ /R). (4.3.9) Corollary. Let us consider the vector bundle Vβ,µ = K β R j ⊗ Vµ (R∨j /R j ) over IGrass( j, F), where β = (β1 , . . . , β j ) is a dominant integral weight for the root system of type A j−1 and µ = (µ1 , . . . , µn− j ) is the integral dominant weight for the root system of type Dn− j . Let us consider the weight γ = (−β j , . . . , −β1 , µ1 , . . . , µn− j ). Then one of the mutually exclusive possibilities occurs: (1) There exists σ ∈ W , σ = 1 such that σ (γ ) = γ . Then all cohomology groups H i (IGrass( j, F), Vβ,µ ) are 0 for i ≥ 0. (2) There exists unique σ ∈ W such that σ . (γ ) := α is a dominant integral weght for the root system of type Dn . Then all cohomology groups H i (IGrass( j, F), Vβ,µ ) are 0 for i = l(σ ), and H l(σ ) (IGrass( j, F), Vβ,µ ) = Vα (F). The proof of (4.3.8) is identical to that of (4.3.4). Let us look more closely at the isotropic Grassmannian IGrass( j, F) as a subset of the Grassmannian Grass( j, F).

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(4.3.10) Proposition. The isotropic Grassmannian IGrass( j, F) is locally a complete intersection in Grass( j, F). The structure sheaf of IGrass( j, F) can be resolved by locally free sheaves over Grass( j, F) by means if the Koszul complex 0→

j+1 ( 2 )

ψ

(S2 R j ) → . . . → S2 R j → OGrass( j,F) .

The proof is identical to that of (4.3.5).

Exercises for Chapter 4 The General Linear Group 1. (a) Calculate the cohomology groups of bundles L(1, 4, 7, 5), L(3, 2, 1, 5) on G/B for G = GL(4, C). (b) Calculate the cohomology of vector bundles K (3,2,1) Q ⊗ K (7,6,1) R and of K (7,6,6) R on Grass(3, E) with dim E = 6. 2. Calculate the cohomology groups of bundles K λ Q∗ on the Grassmannian with tautological sequence 0 → R → F → Q → 0, where dim R = r , dim Q = q. 3. Let E be an n-dimensional space. Consider the Grassmannian Grass(r ; E) with the tautological sequence 0 → R → E × Grass(r ; E) → Q → 0. Let ξ be a subbundle of S2 E × Grass(r ; E) ﬁtting into the exact sequence 0 → ξ → S2 E × Grass(r ; E) → S2 Q → 0. Notice that ξ also ﬁts into an exact sequence 0 → S2 R → ξ → R ⊗ Q → 0. Calculate the cohomology groups of 2 ξ and 3 ξ using the information from the Schur complexes associated to the map S2 E × Grass(r ; E) → S2 Q. Calculate this cohomology using the ﬁltration induced by the second sequence on 2 ξ , 3 ξ . 4. We recall from Proposition (3.3.5) that the tangent bundle TGrass(r ;E) can be identiﬁed with R∗ ⊗ Q.

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133

(a) Calculate the cohomology groups of the exterior algebra on the tangent bundle TGrass(r ;E) . Prove that the higher cohomology groups vanish. (b) Calculate the cohomology groups of the exterior algebra on the cotan∗ i gent bundle TGrass(r ;E) . Prove that only the group H (Grass(r ; E), j ∗ (TGrass(r ;E) )) is nonzero if and only if when i = j. Prove that ∗ H i (Grass(r ; E), i (TGrass(r ;E) )) consists of P(i, r, n − r ) copies of trivial representation, where P(i, r, n − r ) is the number of partitions of i contained in the r × (n − r ) rectangle. 5. Some characteristic free cases of Bott’s theorem: (a) Let λ = (λ1 , . . . , λn ) be such that for some s < t we have λ1 ≥ . . . ≥ λs > λs+1 = . . . = λt−1 ≥ λt+1 − 1 ≥ . . . ≥ λn − 1 λt = λs+1 + t − s. Then H i (G/B, L(λ)) = 0 for i = t − s − 1 and H t−s (G/B, L(λ)) = L ν E where ν = (λ1 , . . . , λs , λt − (t − s) + 1, λs+1 + 1, . . . , λt−1 + 1, λt+1 , . . . , λn ). (b) Let λ = (λ1 , . . . , λn ) be such that for some s < t we have λ1 ≥ . . . ≥ λs > λs+1 = . . . = λt−1 , λs+1 < λt < λs+1 + t − s. Then H i (G/B, L(λ)) = 0 for all i.

Other Classical Groups 6. Let F be a symplectic space of dimension 2n. Consider the isotropic Grassmannian IGrass( j, F). Let λ = (λ1 , . . . , λ j ) be a partition. Prove that if λ1 ≤ 2n − j + 1, then the cohomology of K λ R j can be zero or can contain only a trivial representation of Sp(F). More precisely, the cohomology is nonzero precisely when λ is one of the partitions occurring in Proposition (6.4.3). 7. Let IGrass(r ; F) be the isotropic Grassmannian of r -dimensional isotropic subspaces in a symplectic space (F, (−, −)) of dimension 2n. Calculate the cohomology groups of the exterior powers of the vector bundle R∨ . 8. Formulate and prove the analogues of exercises 6 and 7 for the even and odd orthogonal groups. 9. Let (F, ( , )) be a symplectic space of dimension 2n. For 1 ≤ r ≤ n, let IGrass(r ; F) be the isotropic Grassmannian of r -dimensional isotropic spaces in F with tautological subbundle R. Consider the ﬁltration 0 ⊂ R ⊂ R∨ ⊂ F × IGrass(r ; F) of the vector bundles on IGrass(r ; F). The factor (F × IGrass(r ; F))/R∨ can be identiﬁed with R∗ . Therefore we

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have an epimorphism π of vector bundles on IGrass(r ; F) which is a composition π : (TGrass(r ;F) )|IGrass(r ;F) = R∗ ⊗ ((F × IGrass(r ; F))/R) 2 → R∗ ⊗ R∗ → R∗ . (a) Prove that the embedding IGrass(r ; F) ⊂ Grass(r ; F) allows one to identify the vector bundle TIGrass(r ;F) with the kernel of π. (b) Prove the exact sequence 0 → R∗ ⊗ (R∨ /R) → TIGrass(r ;F) → D2 R∗ → 0. 10. Let (F, (−, −)) be an orthogonal space of dimension n. For 1 ≤ r ≤ n2 , let IGrass(r ; F) be the isotropic Grassmannian of r -dimensional isotropic spaces in F with tautological subbundle R. Consider the ﬁltration 0 ⊂ R ⊂ R∨ ⊂ F × IGrass(r ; F) of the vector bundles on IGrass(r ; F). The factor (F × IGrass(r ; F))/R∨ can be identiﬁed with R∗ . Therefore we have an epimorphism π of vector bundles on IGrass(r ; F) which is a composition π : (TGrass(r ;F) )|IGrass(r ;F) = R∗ ⊗ ((F × IGrass(r ; F))/R) → R∗ ⊗ R∗ → S2 R∗ . (a) Prove that the embedding IGrass(r ; F) ⊂ Grass(r ; F) allows to identify the vector bundle TIGrass(r ;F) with the kernel of π. (b) Prove the exact sequences 0 → R∗ ⊗ (R∨ /R) → TIGrass(r ;F) →

2

R∗ → 0,

0 → TIGrass(r ;F) → R∗ ⊗ F/R → S2 R∗ → 0. 11. Use the results of exercises 9 and 10 to calculate cohomology groups of exterior powers of tangent and cotangent bundles on isotropic Grassmannians of isotropic subspaces of maximal dimension. Compare to the results of exercise 4.

Tensor Product Multiplicities 12. (Brauer and Klimyk’s formula.) Let λ, µ be two dominant weights for a reductive group G. Prove that the decomposition of the tensor product

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135

Vλ ⊗ Vµ can be calculated as follows. Consider the set of weights (µ) = {ν1 , . . . , ν N } occurring in Vµ (with N = dim Vµ ). For each weight in λ + (µ) = {λ + ν1 , . . . , λ + ν N } calculate the Euler characteristic of the line bundle L(λ + νi ) on G/B. Then make cancellations if the same representation occurs with a positive and a negative sign. The remaining representations occur with positive sign and give the irreducible representations occurring in Vλ ⊗ Vµ .

5 The Geometric Technique

In this chapter we develop the basic technique for calculating syzygies. It applies to the subvarieties Y in an afﬁne space X with a desingularization Z which is a total space of a vector bundle over some projective variety V , which is a subbundle of the trivial bundle X × V over V . In such situation the Koszul complex of sheaves on X × V resolving the structure sheaf of Z has terms that are pullbacks of vector bundles over V . Taking the direct image of this Koszul complex by the projection p : X × V → V , one gets the formula expressing terms on the free resolution of the coordinate ring of Y in terms of cohomology of bundles on V . One also gets interesting complexes by taking direct images of the Koszul complex twisted by a pullback of a vector bundle on V . In this chapter we discuss the general construction and properties of direct images of Koszul complexes. The examples will be given in following chapters. The chapter is organized as follows. In section 5.1 we state the properties of the twisted direct images F(V)• of Koszul complexes. In particular we give the expressions for their terms and homology. We also state the criteria for F(V)• to be acyclic, the duality theorem for such complexes, and the result expressing the codimension and degree of Y in terms of the complex F(V)• . In section 5.2 we give the actual construction of complexes F(V)• . It involves constructing certain double complexes of sheaves on X × V resolving the Koszul complex. In section 5.3 we prove the other statements announced in section 5.1. In some of the proofs in sections 5.2 and 5.3 we rely on the machinery of derived categories. The necessary information is collected in section 1.2.5. Section 5.4 contains the equivariant setup. We prove that if a reductive group G acts on X and the action stabilizes Y , the variety V is a homogeneous space G/P, and the bundle V is a homogeneous bundle, then the terms and homology of the complexes F(V)• also carry an action of G. We also discuss the results of Kempf on rational singularities of the subvarieties Y and on the geometry of the desingularization Z in the case when V is a homogeneous space. 136

5.1. The Formulation of the Basic Theorem

137

In section 5.5 we give more explicit description of the differentials of F(V)• . Section 5.6 describes the technique of degeneration sequences which allows us to compare complexes F(V)• supported in different subvarieties.

5.1. The Formulation of the Basic Theorem Throughout this chapter we work over the algebraically closed ﬁeld K of arbitrary characteristic. Let us consider the projective variety V of dimension m. Let X = AKN be the afﬁne space. The space X × V can be viewed as a total space of trivial vector bundle E of dimension N over V . Let us consider the subvariety Z in X × V which is the total space of a subbundle S in E. We denote by q the projection q : X × V −→ X and by q the restriction of q to Z . Let Y = q(Z ). We get the basic diagram Z ⊂ ↓ q Y ⊂

X×V ↓q X

The projection from X × V onto V is denoted by p, and the quotient bundle E/S by T . Thus we have the exact sequence of vector bundles on V , 0 −→ S −→ E −→ T −→ 0. The dimensions of S and T will be denoted by s, t respectively. The coordinate ring of X will be denoted by A. It is a polynomial ring in N variables over K. We will identify the sheaves on X with A-modules. (5.1.1) Proposition. (a) The locally free resolution of the sheaf O Z as an O X ×V -module is given by the Koszul complex K(ξ )• : 0 →

t

( p∗ ξ ) → . . . →

2

( p ∗ ξ ) → p ∗ (ξ ) → O X ×V

where ξ = T ∗ . The differentials in this complex are homogeneous of degree 1 in the coordinate functions on X . (b) The direct image p∗ (O Z ) can be identiﬁed with the the sheaf of algebras Sym(η) where η = S ∗ . Proof. Let us identify X with the vector space E of dimension N over K. The bundle S is the s-dimensional subbundle of the N -dimensional trivial bundle

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over V . By the universal property (3.3.2) of the Grassmannian, there exists a map f : V −→ Grass(s, E) such that S = f ∗ (R). Let us consider the complex K• (Q∗ ) from (3.3.3). We set K(ξ )• := f ∗ K• (Q∗ ). The proposition follows by the same arguments as in the proof of (3.3.3). The idea of the geometric technique is to use the Koszul complex K(ξ )• to construct for each vector bundle V on V the free complex F(V)• of A-modules with the homology supported in Y . These complexes are the main subject of this book. In many cases the complex F(OV )• gives the free resolution of the deﬁning ideal of Y . In this section we state the theorems establishing the existence and basic properties of complexes F(V)• . The most important is the basic theorem (5.1.2) below, which gives the terms and the precise description of homology of complexes F(V)• . The next two sections will be devoted to the proofs of all the results that follow. Before we state the basic theorem, let us introduce the twisted Koszul complex. For every vector bundle V on V we introduce the complex K(ξ, V)• := K(ξ )• ⊗O X ×V p ∗ V. This complex is a locally free resolution of the O X ×V -module M(V) := O Z ⊗ p ∗ V. Now we are ready to state the basic theorem. (5.1.2) Basic Theorem. For a vector bundle V on V we deﬁne free graded A-modules i+ j H j V, ξ ⊗ V ⊗k A(−i − j). F(V)i = j≥0

(a) There exist minimal differentials di (V) : F(V)i → F(V)i−1 of degree 0 such that F(V)• is a complex of free graded A-modules with H−i (F(V)• ) = Ri q∗ M(V). In particular the complex F(V)• is exact in positive degrees.

5.1. The Formulation of the Basic Theorem

139

(b) The sheaf Ri q∗ M(V) is equal to H i (Z , M(V)) and it can be also identiﬁed with the graded A-module H i (V, Sym(η) ⊗ V). (c) If φ : M(V) → M(V )(n) is a morphism of graded sheaves then there exists a morphism of complexes f • (φ) : F(V)• → F(V )• (n) Its induced map H−i ( f • (φ)) can be identiﬁed with the induced map H i (Z , M(V)) → H i (Z , M(V ))(n). This theorem will be proven in section 5.2. If V is a trivial bundle of rank one on V , then the complex F(V)• is denoted simply by F• . The next theorem gives the criterion for the complex F• to be the free resolution of the coordinate ring of Y . (5.1.3) Theorem. Let us assume that the map q : Z −→ Y is a birational isomorphism. Then the following properties hold: (a) The module q∗ O Z is the normalization of K[Y ]. (b) If Ri q∗ O Z = 0 for i > 0, then F• is a ﬁnite free resolution of the normalization of K[Y ] treated as an A-module. (c) If Ri q∗ O Z = 0 for i > 0 and F0 = H 0 (V, 0 ξ ) ⊗ A = A, then Y is normal and it has rational singularities. The complexes F(V)• satisfy a Grothendieck type duality. Let ωV denote the canonical divisor on V . (5.1.4) Theorem. Let V be a vector bundle on V . Let us introduce the dual bundle t V ∨ = ωV ⊗ ξ ∗ ⊗ V ∗. Then F(V ∨ )• = F(V)∗• [m − t]. This result can be applied to give a criterion for the twisted module to be Cohen–Macaulay. (5.1.5) Corollary. Let us assume that dim Z = dim Y . Assume that for some vector bundle V on V we have Ri q∗ (O Z ⊗ p ∗ V) = 0 for i > 0. Then the

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module R0 q∗ (O Z ⊗ p ∗ V) is a maximal Cohen–Macaulay module supported in Y if and only if Ri q∗ (O Z ⊗ p ∗ V ∨ ) = 0 for i > 0. In that case the module R0 q∗ (O Z ⊗ p ∗ V ∨ ) is also a maximal Cohen–Macaulay module, dual to R0 q∗ (O Z ⊗ p ∗ V) in the sense of (1.2.26). Proof. We apply (5.1.4) to the complexes F(V)• and F(V ∨ )• . Our assumption implies that codim Y = dim X − dim Y = dim X − dim Z = dim X −(dim X + m − t) = t − m. Now (5.1.4) implies that the length of F(V)• equals t − m if and only if F(V ∨ )• has all the terms in nonnegative degrees. This establishes the ﬁrst claim. The duality statement follows because the two complexes are dual to each other. If the complex F(V)• satisﬁes the conditions of Corollary (5.1.5), we say that it has the Cohen–Macaulay property. In particular, when codim Y = 1 the complexes with Cohen–Macaulay property have length one, so they are just matrices. The determinant of such a matrix equals g rank V , where g is an irreducible equation of Y . In that case the complex F(V)• is called a determinantal complex. We will analyze such complexes for the case of discriminants and resultants in chapter 9. We conclude this section by showing that the complex F• contains the information about the codimension and the degree of Y . This fact will be useful in the cases when q is not necessarily a birational map. (5.1.6) Theorem. (a) codim X Y = max { i | Fi = 0 }. (b) Let us assume that dim Z = s + m < dim X = N . Let r = N − m − s. Then we have deg(q ) deg Y =

i+ j (i + j)r j ξ (−1)i+r h V, r! i, j

where by deﬁnition deg(q ) is 0 when dim Y < dim Z . Theorems (5.1.3), (5.1.4), and (5.1.6) as well as Corollary (5.1.5) will be proved in section 5.3.

5.2. The Proof of the Basic Theorem

141

5.2. The Proof of the Basic Theorem Before we prove Theorem (5.1.2) we recall several facts we will need. The ﬁrst one is the result on an equivalence of categories of graded modules and sheaves. Let S be a graded ring with S0 = A a ﬁnitely generated K-algebra and S1 a ﬁnitely generated A-module. For a graded S-module M we denote by M the corresponding sheaf on Proj S. For a sheaf F on X we deﬁne *(Proj S, F(n)). *∗ (F ) = n∈Z

We deﬁne an equivalence relation ≈ on graded S-modules by saying M ≈ M if there exists an integer d such that M≥d / M≥d . Here M≥d = n≥d Mn . We say that a graded S-module M is quasiﬁnitely generated if M is equivalent to a ﬁnitely generated module. In this setting we have (5.2.1) Proposition. The functors and *∗ induce an equivalence of categories between the category of quasiﬁnitely generated graded S-modules modulo the equivalence ≈ and the category of coherent OProj S -modules. Proof. This is exercise 5.9 in section II.5 of [H1]. We proceed with the discussion of some general facts concerning free complexes. Let A be a graded ring with A0 = K. (5.2.2) Proposition. (a) Let G • be a complex of ﬁnitely generated graded free A-modules. The complex G • decomposes into a direct sum G • G • = G • where G • is a minimal complex and G • is exact. The terms of the complex G • are G i = Hi (G • ⊗ A K) ⊗k A. (b) Let M• be a complex of graded A-modules with Mi = 0 for i < i 0 . Then there exists a minimal complex G • of free graded A-modules and a map φ : G • → M• which is a quasiisomorphism. Proof. We will start with the proof of (a). Let G i = 1≤ j≤gi A(−e j,i ), where gi = dimG − i. Let us consider the differential di : G i → G i−1 . We can

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The Geometric Technique

identify di with the matrix over A where the (k, j)th entry is homogeneous of degree e j,i − ek,i−1 . We will prove that we can change the basis in G i in a homogeneous way so the module G i will decompose as follows: G i = Bi ⊕ Ui ⊕ Bi ,

(∗)

so the differential di has a block decomposition 0 0 I di = 0 di 0 . 0 0 0 If this is done, then obviously we can set G i = Ui and G i = Bi ⊕ Bi , and the proposition follows. To get our decomposition we ﬁx an index m, choose a basis in G m in the appropriate way, and then spread this choice to the left and to the right by induction. We start by changing the basis in G m and G m−1 in a homogeneous way to bring dm to the canonical form dm 0 dm = 0 dm where dm is a minimal matrix with homogeneous entries and dm is an identity matrix. This means we can write G m = Wm ⊕ Vm , G m−1 = Wm−1 ⊕ Vm−1 , with dm corresponding to the map Wm → Wm−1 and dm corresponding to the map Vm → Vm−1 . Now we notice that the rows of dm+1 corresponding to Vm are zero because G • is a complex. Therefore it is really a map from G m+1 to Um . Bringing this map to the canonical form as above, we see that we can decompose Wm = Um ⊕ Bm and G m+1 = Wm+1 ⊕ Vm+1 in such way that dm+1 is a direct sum of the minimal map from Wm+1 to Um and the identity map from Vm+1 to Bm . We get the required choice of basis in G m by setting Bm = Vm . In fact we have also chosen the direct summand Bm−1 = Vm−1 in = Vm+1 in G m+1 . G m−1 and the direct summand Bm+1 Next we show how to extend our choice of basis to the right. Assume that we have the block decomposition (∗) for i > j together with the decomposition G j = B j ⊕ W j . We notice that B j ⊂ Ker d j . Therefore we treat d j as a map from W j to G j−1 . We bring it to the canonical form, which means we can write W j = U j ⊕ B j and G j−1 = B j−1 ⊕ W j−1 , so d j is a direct sum of the minimal map from U j to W j−1 and the identity from B j to B j−1 . Similarly, let us assume that the decomposition (∗) is achieved for i < j together with the decomposition G j = W j ⊕ B j . We notice that the image of

5.2. The Proof of the Basic Theorem

143

d j+1 is contained in W j , so we can treat is as a map from G j+1 to W j . Reducing this map to the canonical form, we get the decompositions W j = B j ⊕ U j and G j+1 = W j+1 ⊕ B j+1 such that d j+1 is a direct sum of a minimal map from W j+1 to U j and the identity map from B j+1 to B j . This completes the proof of (a). Let us prove (b). It is enough to construct the quasiisomorphism φ : G • → M• where G • is a complex of free graded A-modules. We can achieve minimality by applying (a) and taking G • as our complex. Let us denote the submodule Ker d in Mi by Z i and the submodule d(Mi+1 ) by Bi . Then we have exact sequences 0 → Bi → Z i → Hi → 0, 0 → Z i → Mi → Bi−1 → 0. Let 0 → Bi → Zi → Hi → 0, 0 → Zi → Mi → Bi−1 → 0 be the short exact sequences of free complexes covering the above maps. For each i we consider the map of complexes ηi given by the composition Mi → Bi−1 → Zi−1 → Mi−1 . It is clear that ηi−1 ηi = 0. We consider the double complex ηi

ηi−1

→ . . . Mi −→Mi−1 −→Mi−2 → . . . . We deﬁne G • to be the total complex of this double complex. By construction it is equipped with the natural map φ to M• . It is clear from the spectral sequence associated to our double complex that φ is a quasiisomorphism. After these preparations we can proceed with the proof of (5.1.2). Let us embed V in a projective space. Let OV (1) be the ample line bundle corresponding to this embedding. We can assume that the higher cohomology of the sheaves OV (n) vanishes for n > 0. This can be achieved by Serre’s theorem ([H1, Proposition III.5.3]). Let us denote by R the homogeneous coordinate ring of V in this embedding. The key step in the construction of complexes F(V)• is the existence of a certain right resolution of the twisted Koszul complex K(ξ, V)• .

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(5.2.3) Lemma. There exists a right resolution 0 → K(ξ, V)• → P(V)•• such that the following properties hold: (a) Each module P(V)i j is a direct sum of sheaves A(i) ⊗ OV (n), where n > 0, and therefore is q∗ -acyclic. (b) Each column 0 → K(ξ, V) j → P(V) r j is a q∗ -acyclic resolution of K(ξ, V) j by coherent O X ×V -modules which is the tensor product of A(− j) with the *-acyclic resolution of j (ξ ) ⊗ V. Proof. Let us start with the dual complex K(ξ, V)∗• . Its j-th term equals K(ξ, V)∗j =

−j ( p∗ ξ ∗ ) ⊗ V ∗ .

This is a complex of sheaves over X × V whose differential is homogeneous of degree 1 with respect to the generators of A. Let us apply to this complex the functor *∗ from (5.2.1). We get a complex of bigraded A ⊗ R-modules. The generators of the j-th term here have A-degree − j. Now we replace this complex by the complex of equivalent modules by cutting out in each module the components in nonpositive R-degrees. We get a complex C(V)• of bigraded A ⊗ R-modules where the j-th term has generators in A-degree − j and in positive R-degree. •• of the complex C(V)• . Next we consider the minimal free resolution C(V) Each module Ci j has generators in positive R-degree. •• , Now we construct P(V)•• by applying the functorto the complex C(V) and dualizing. By (5.2.1) it is the right resolution of the Koszul complex. Properties (a) and (b) are obviously satisﬁed. Consider the double complex q∗ (P(V)•• ). By Lemma (5.2.3) (a) it is a double complex of free graded A-modules. Let us consider the total complex associated to q∗ (P(V)•• ), G(V)• := T ot• (q∗ (P(V)•• )). Since the resolution P(V)•• is q∗ -acyclic, we get H−i G(V)• = Ri q∗ M(V).

5.2. The Proof of the Basic Theorem

145

Now we apply Proposition (5.2.2) to the complex G(V)• . We want to calculate the components of the minimal part of this complex. It is enough to calculate the homology of the complex G(V)• ⊗ K. We consider the double complex of vector spaces q∗ (P•• ) ⊗ K. The horizontal differentials in this double complex are 0 because the horizontal maps in P•• are by Lemma (5.2.3) (a) the matrices with entries of degree 1 in A. By Lemma (5.2.3) (b) the homology of each column q∗ (P• j ) consists of cohomology groups H . (V, j (ξ ) ⊗ V). Therefore H l (G(V)• ⊗ K) =

l+ j ξ ⊗V . H j V,

j≥0

Proposition (5.2.2) applied to G(V)• gives G(V)• = F(V)• L(V)• for some exact complex L(V)• . This proves part (a) of theorem (5.1.2). Let us prove part (b). The ﬁrst part of the statement follows from the fact that the module on X is determined by its global sections. The second part follows from the spectral sequence of the composition of maps X × V → V → ∗, from the fact that p is afﬁne, and from (5.1.1) (b). Before we turn to part (c), let us state some facts about the complexes F(V)• that follow easily from the proof of part (a) of Theorem (5.1.2). (5.2.4) Proposition. (a) The component i+ j (di )( j, j ) : H j V, ξ ⊗ V ⊗k A(−i − j) → Hj

V,

i−1+ j

ξ ⊗ V ⊗k A(−i + 1 − j )

of the differential di (V) : F(V)i → F(V)i−1 is of homogeneous degree j − j + 1. (b) The component (di )( j, j ) is zero if j < j . Proof. Part (a) follows from the fact that di is a homogeneous map of degree 0. Part (b) follows from minimality of the complex F(V)• . We need a result characterizing the complex F(V)• .

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(5.2.5) Proposition. The complex F(V)• is the unique minimal free complex quasiisomorphic to Rq∗ (O Z ⊗ p ∗ V). In particular, the complex F(V)• does not depend on the choice of the sheaf OV (1) and of the resolution P(V)•• . Proof. We constructed the complex F(V)• as a minimal free complex quasiisomorphic to the total complex of the double complex q∗ (P(V)•• ). However, the total complex of P(V)•• is a q∗ -acyclic complex quasiisomorphic to O Z ⊗ p ∗ (V). The statement now follows from (5.2.2) (b) and the fact implicitly contained in (5.2.2) that every quasiisomorphism between minimal free complexes has to be an isomorphism. Now we conclude the proof of part (c) of Theorem (5.1.2). A morphism φ : M(V) → M(V )(n) induces the map Rq∗ (φ) : Rq∗ (O Z ⊗ p ∗ V) → Rq∗ (O Z ⊗ p ∗ V )(n) in the derived category of bounded complexes of A-modules. However, every map between free complexes in this derived category is represented by a genuine map ψ of complexes ([H2], the dual version of Proposition I.4.7, or [GM], the dual version of Theorem 21, chapter III, section 5). The map ψ does not need to be homogeneous. However, since both complexes are graded, the homogeneous component ψ0 of ψ of degree zero will also be a map of complexes. Moreover, since the induced map H (ψ)∗ has degree zero, the map ψ − ψ0 is a map of free complexes inducing the trivial map on homology. Such a map is homotopic to zero, and thus ψ is homotopic to the map ψ0 of degree zero. •

5.3. The Proof of Properties of Complexes F(V)• In this section we prove Theorems (5.1.3), (5.1.4), and (5.1.6). Proof of Theorem (5.1.3). First of all, we notice that parts (b) and (c) follow from Theorem (5.1.2) and part (a) of (5.1.3). Thus it is enough to prove part (a). This statement follows from the following elementary lemma applied to the normalization of Y . (5.3.1) Lemma. Let q : Z → Y be a desingularization of Y . Let us assume that Y is normal. Then q∗ O Z = OY .

5.3. The Proof of Properties of Complexes F(V)•

147

Proof. The question is local on Y , so we can assume that Y = Spec A where A is a normal domain. The sheaf q∗ O Z is the sheaf associated to the ring *(Z , O Z ). Therefore it is enough to show that *(Z , O Z ) = A. Since q is birational, it is clear that *(Z , O Z ) is contained in the ﬁeld of fractions of A. It is also a ﬁnitely generated A-module, because q is proper. This proves the lemma. Proof of Theorem (5.1.4). We use the duality theorem for proper morphisms (Theorem (1.2.22)) for the map f = q : X × V → X , for F • = • ( p ∗ ξ ) ⊗ p ∗ V, and for G • = O X . The complex F(V)• is, by its construction, a free graded minimal representative of the object R f ∗ (F • ). Therefore the right side of the theorem gives R Hom•X (R f ∗ (F • ), G • ) = R Hom•X (F(V)• , O X ). Now R can be dropped because F(V)• is its own projective resolution (we calculate R Hom as R I I R I : compare Lemma 6.3, p. 66 in [H2]), and we are left with the complex F(V)∗• . To identify the right side we notice that by (1.2.21) (c) we have f ! (G • ) = f ∗ (G • ) ⊗ ω X ×V /V [n] = O X ×V ⊗ p ∗ (ωV )[n], because f is smooth. Therefore the inside term on the left side in (1.2.22) can be written as • R Hom•X ×V (F • , f ! (G • )) = R Hom•X ×V ( p ∗ ξ ) ⊗ p ∗ V, p ∗ (ωV )[n] . By proposition 5.16 (p. 113) of [H2], with L = identify the right hand side with R Hom•X ×V (O X ×V , O X ×V ) ⊗

•

( p ∗ ξ ) ⊗ p ∗ V) we can

•

( p ∗ ξ )∗ ⊗ p ∗ V ∗ ⊗ p ∗ (ωV )

which can be written as R Hom•X ×V (O X ×V , O X ×V ) ⊗

•

( p∗ ξ ) ⊗

t

(ξ ∗ ) ⊗ p ∗ V ∗ ⊗ p ∗ (ωV )

where t = rank ξ . We can drop R in the above expression, because in the left place we have a locally free complex (again we calculate R Hom as R I I R I , using Lemma 6.3, p. 66 in [H2]). This means the complex above is quasiisomorphic to • ( p ∗ ξ ) ⊗ t ( p ∗ ξ )∗ ⊗ p ∗ V ∗ ⊗ p ∗ (ωV ). Therefore the left hand side in

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The Geometric Technique

(1.2.22) can be identiﬁed with • t R f∗ ( p ∗ ξ ) ⊗ ( p ∗ ξ )∗ ⊗ p ∗ V ∗ ⊗ p ∗ (ωV ) =F

t

∗

∗

∗

∗

∗

( p ξ ) ⊗ p V ⊗ p (ωV ) •,

as claimed in (5.1.4). Proof of Theorem (5.1.6). We start with part (a). Let us consider the canonical sheaf ω Z . By the adjunction formula ([H1, Proposition II.8.20]), ω Z = ω X ×V | Z ⊗ t ξ ∗ . Since X is just an afﬁne space, ω X ×V = p ∗ K . Therefore ω Z = O Z ⊗ K ⊗ t ξ ∗ . By the Grauert–Riemenschneider theorem (1.2.28) we know that Ri q∗ (ω Z ) = 0 for i > dim Z − dim Y . Therefore the terms F(K ⊗ t ξ ∗ )i are zero for i < dim Y − dim Z . By the duality (5.1.4) this means that Fi = 0 for i > codim X Y . It remains to show that for i = dim Z − dim Y we have Ri q∗ (ω Z ) = 0. After shrinking Y we can assume that q is smooth and projective. Then the last claim follows from the uppersemicontinuity theorem ([H1, III.12.11]) and the adjunction formula ([H1, II.8.20]), since each ﬁber Z y is smooth of dimension i, so H i (Z y , ω Z y ) is one dimensional (hence nonzero) by Serre duality. This proves (a). To prove (b) we consider the graded Hilbert function P(F• , t) =

i i+ j

(−1) t

h

j

i+ j V, ξ (1 − t)−N .

i, j≥0

Writing P(F• , t) = a≥0 P(a)t a , we know that for big a the function P(a) is polynomial in a. We also know that P(F• , t) is the alternating sum of graded Hilbert functions of the homology modules Ri O Z of F• . The homology modules Ri O Z are supported in Y . Moreover, the modules Ri O Z for i > 0 are supported in the locus of points in Y where the ﬁbers of q have dimension at least 1, which is a proper subvariety of Y . The sheaf q∗ O Z is generically of rank deg q . This means that P(a) is a polynomial of degree ≤ N − r and that the highest coefﬁcient of P(a) equals (N − r )! deg q deg Y in the case dim Y = dim Z and is zero otherwise. Statement (b) of (5.1.6) now follows by standard calculation. (5.3.2) Remarks. The geometric method was ﬁrst applied to determinantal varieties ([Ke 0],[L2], [JPW]). The general forms of statements (5.1.2), (5.1.3), (5.1.4), related to derived categories, were ﬁrst used to deal with

5.4. The G-Equivariant Setup

149

examples related to nilpotent orbit closures and discriminants ([W2], [W3], [Br5]). We follow the approach from [Br5] to prove the ﬁrst part of (5.1.2) without derived categories.

5.4. The G-Equivariant Setup In this section we consider the special case of the construction from section 5.1 related to the situation when the variety V is a homogeneous space. This is the most important class of known examples where the geometric method applies. In fact, all examples considered in the following chapters are of this kind. Let G be a linearly reductive group, and let P be a parabolic subgroup in G. We assume that the variety V is the homogeneous space G/P. We also assume that the group G acts linearly on the afﬁne space X , so X can be identiﬁed with a representation of G. Let U be a P-submodule of X . We associate to U the vector bundle Z = G ×P U , which is by deﬁnition the orbit space G × U/P with P acting by p . (g, y) = (gp −1 , py). The projection G × U → G induces the G-equivariant morphism p : Z = G ×P U → V = G/P. Since U is a submodule of a G-module X , we have the embedding Z = G ×P U → G ×P X = G/P × X The identiﬁcation on the right hand side is made by the morphism (g, x) → (gP, gx). We denote by q the projection G/P × X → X , and by q its restriction to Z . As before, we denote Y = q (Z ). This places us in the situation of section 5.1. Let W be another P-module. We associate to W the vector bundle V(W ) := G ×P W . We can apply the construction from section 5.1 to get the complex F(V(W ))• of free graded modules over the ring A = K[X ] = Sym(X ∗ ). Let us recall that F(V(W ))i is given by the formula F(V(W ))i =

H

j

V,

i+ j

ξ ⊗ V(W ) ⊗k A(−i − j).

j≥0

Since the group G acts naturally on the bundles V(W ) and ξ , it acts ratio nally on the cohomology groups H j (V, i+ j ξ ⊗ V(W )) . Therefore G acts rationally on free modules F(V(W ))• via the diagonal action.

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(5.4.1) Theorem. Let G, P, V , X , Z , and V(W ) be as above. Then the complex F(V(W ))• can be constructed in such way that all the differentials di (V(W )) : F(V(W ))i → F(V(W ))i−1 are G-equivariant. Proof. We just have to follow the proof of Theorem (5.1.2) to assure that each step can be made G-equivariant. Before we do that, we need a G-equivariant analogue of (5.2.2). Let A be a graded ring over K with A0 = K. Let us assume that G acts rationally on A, i.e., G acts as a group of automorphisms of the graded ring A, so that each graded component Ai is a representation of G and that the multi

plication maps are G-equivariant. Let M = i≥i0 Mi be a graded A-module. We say that the group G acts rationally on M (compatibly with the action on A ) if each graded component Mi is a G-representation and the structure maps for the A-module M are G-equivariant. We will call a complex M• of ﬁnitely generated modules over A G-equivariant if G acts rationally on each module G i and the differentials are G-equivariant. Let M be a graded A-module on which G acts rationally. Then we can choose a minimal set of generators for M which forms a G submodule in M. Indeed, the projection M → M/A+ M splits as a map of G-modules. This means that every projective graded ﬁnitely generated A-module P on

which G acts rationally is of the form P = j P j ⊗ A(− j) for some ﬁnite dimensional G-representations P j . Therefore a complex G • of ﬁnitely generated graded free A-modules is G-equivariant if each G i is of the form

G i = j G i, j ⊗ A(− j) for some ﬁnite dimensional representations G i, j of G, and the differentials di : G i → G i−1 are G-equivariant. We can now recover all the standard results on minimal free resolutions of graded modules and complexes in G-equivariant form. In particular we have (5.4.2) Proposition. (a) Let G • be a G-equivariant complex of ﬁnitely generated graded free A-modules. The complex G • decomposes into a direct sum G • = G • ⊕ G • , where G • is a minimal complex, G • is exact, and both G • and G • are G-equivariant. The terms of the complex G • are G i = Hi (G • ⊗ A K) ⊗K A.

5.4. The G-Equivariant Setup

151

(b) Let M• be a G-equivariant complex of graded A-modules with Mi = 0 for i < i 0 . Then there exists a minimal G-equivariant complex G • of free graded A-modules and a map φ : G • → M• which is a quasiisomorphism. Proof. To prove both statements we just repeat the proof of (5.2.2). For (a) we notice that the decompositions G i = Bi ⊕ Wi ⊕ Bi

(∗)

can be chosen in G-equivariant way. In the proof of (b) all modules Z i , Bi , etc. are the modules with the rational G actions, so all the exact sequences are G-equivariant. Therefore the covering complexes of free modules can be also choosen in a G-equivariant way. Now we go through all the steps of the proof of (5.1.2). (1) The embedding of V in the projective space can be chosen in a Gequivariant way. Indeed, we can use any positive line bundle L(α) (cf. section 4.3). Then the sheaf OV (1) corresponds to the ample line bundle whose total space admits an action of G. Therefore the homogeneous coordinate ring R of V in this embedding admits a rational G-action. The key step in the construction of complexes F(V)• is the existence of a certain right resolution of the twisted Koszul complex K(ξ, V)• . (2) The twisted Koszul complex K(ξ, V(W ))• of sheaves on X × V consists of vector bundles admitting G-action. Therefore, after applying the functor *∗ to its dual, we get a G-equivariant complex of bigraded A ⊗ R-modules. Therefore its minimal resolution C(V(W ))•• is a Gequivariant complex, and thus the double complex P(V(W ))•• is a G-equivariant complex of sheaves. (3) It follows that the double complex q∗ (P(V)•• ) is a G-equivariant double complex of graded free A-modules. The rest of the proof follows by applying (5.4.2). This concludes the proof of Theorem (5.4.1). (5.4.3) Remark. The construction we applied in this section is also possible when the group G is only assumed to be reductive. In such case we cannot claim that the complex F(V(W ))• is G-equivariant. We get the G-action on the terms of F(V(W ))i , and we can claim the G-equivariance of linear strands of F(V(W ))• . However, the higher degree maps come from the spectral sequence, so some lifting is required. Therefore the higher degree maps need not be G-equivariant.

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5.5. The Differentials in Complexes F(V)• . In this section we discuss the description of differentials in the complexes of type F(V)• . The point is that if one follows through the proof of Proposition (5.2.2), one gets an inductive procedure for calculating the differential in F(V)• which is not convenient to use. The following result, due to Eisenbud and Schreyer, allows us to describe the differential in a closed form. (5.5.1) Theorem ([ES]). Let F be a double complex ↑ ...

→

...

→

↑ dh

F ji+1

→

F ji ↑

→

dv ↑

dh

i+1 F j+1 ↑dv

→ ...

i F j+1 ↑

→ ...

in some abelian category. Assume that F ji = 0 for i & 0. Suppose that the vertical differential of F splits, so that for each i, j there is a decomposition i−1 i i i F ji = G ij ⊕ dv (G i−1 j ) ⊕ H j such that the kernel of dv in F j is H j ⊕ dv (G j ), isomorphically to dv (G ij ). Let us write s : F ji → and such that dv maps G i−1 j H ji for the projection corresponding to this decomposition and p : F ji → i−1 for the composition of the projection with the inverse of dv dv (G i−1 j ) → Gj i−1 restricted to G j . Then the total complex of F is homotopic to the complex ... →

d

H ji →

i+ j=k

with differential d=

H ji → . . .

i+ j=k−1

s(dh p)+ dh .

+≥0

Proof. We write dt = dv ± dh for the differential of the total complex. We i−+ i−+ note ﬁrst that s(dh p) j dh takes H ji to Hi+++1 . Since F j+++1 = 0 for + 1 0, the sum in the deﬁnition of d is ﬁnite. Let F denote F without a differential, i.e. viewed as a bigraded module. We will ﬁrst show that F is the direct sum of three components G ij , dt (G), and H = H ji G= i, j

and that dt is a monomorphism on G.

i, j

5.5. The Differentials in Complexes F(V)• .

153

The same statements with dv replacing dt are true by hypothesis. In particular, any element of F can be written in the form g + dv (G) + h with i g ∈ G ij , g ∈ G i−1 j , h ∈ H j for some i, j. Modulo G + dt (G) + H , such an element can be written as dh (G) ∈ F i−1 + j + 1. Since Fts = 0 for s & 0, we can use induction on i and assume dh (g) ∈ G + dt (G) + H , so we see that F = G + dt (G) + H . Suppose G ij , g∈G= G ij , and h ∈ H = H ji g ∈ G = i, j

i, j

i, j

be such that g + dt (g) + h = 0. We need to show that g = g = h = 0. Write

k−1 with gts ∈ G st . If b − a = −1, then dt = 0 and the desired g = bk=a g+−k result is a special case of the hypothesis. In any case, there is no componnt b−1 . From of g in G b+−b−1 , so the component of dt (g) in G b+−b is equal to dv g+−b b−1 b−1 the hypothesis we see that dv g+−b = 0, so g+−b = 0, and we are done by induction on b − a. This shows that F = G ⊕ dt (G) ⊕ H and that dt is an isomorphism from G to dt (G). The modules G ⊕ dt (G) form a double complex contained in F that we will call G. Since dt : G → dt (G) is an isomorphism, the total complex of G is split exact. It follows that the total complex tot(F) is homotopic to F/tot(G), and the modules in the last complex are isomorphic to i+ j=k H ji . We will complete the proof by showing that the induced differential on tot(F)/tot(G) is the differential d deﬁned in the statement of the theorem. Choose h ∈ H ji . The image of h under the induced differential is the unique element h ∈ H such that dt (h) ≡ h ((mod G) + dt (G)). Now dt h = dh h ≡ sdh h + (dv p)dh h

(mod G).

However, dv p ≡ dh p ≡ s(dh p) + dv p(dh p)

(mod G + dt (G)).

Continuing this way, and using again the fact that F ji = 0 for i & 0, we obtain s(dh p)+ dh h (mod G + dt (G)), dt h ≡ +

as required. Let us specialize to the situation where our abelian category is the category of graded A-modules for some graded ring A = d≥0 Ad , with A0 = K. Assume that the modules F ji above are free A-modules.

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(5.5.2) Corollary. Let F be a double complex of graded free A-modules. Assume that the differential dv is of degree 0 and that dh is minimal. Then the assumptions of Theorem (5.5.1) are satisﬁed, and H is a minimal complex homotopically equivalent to tot(F). Proof. The only claim needing veriﬁcation is that the vertical differential splits. Since A0 = K and the modules F ji are free, each column of F is obtained from some complex of vector spaces over K by tensoring with A. The splitting can also be chosen over K. The most efﬁcient general procedure to calculate the differential on the complexes of type F• (V) consists in applying Corollary (5.5.2) to the complex q∗ (P(V)•• ) constructed in the course of the proof of Theorem (5.1.2) in section 5.2. Still, that procedure cannot be carried to its completion for large complexes. We will see in the following chapters that for equivariant complexes representation theory is the best tool for identifying the differentials. 5.6. Degeneration Sequences So far we discussed the complexes F(V)• and their properties. They often give the terms of the minimal resolution of the module q∗ (O Z ⊗ p ∗ (V)). Sometimes it is useful to consider the exact sequences formed by such modules. This is especially useful in the “equivariant” situations, i.e. when our projective variety V is a homogeneous space. Such analysis allows sometimes to compare the resolutions of two orbit closures Y and Y1 such that Y1 ⊂ Y , i.e., Y1 is a degeneration of Y . Let us consider the basic diagram Z ⊂ ↓ q Y ⊂

X×V ↓q . X

We assume that V = G/P for some reductive algebraic group G and a parabolic subgroup P. The variety Z is a total space of a vector subbundle of X × V which can be identiﬁed with η∗ . Assume that η is a homogeneous bundle, i.e., it is of the form η∗ = G ×P U for some rational P-module U . We denote B := Sym(U ∗ ). This is a polynomial ring with a rational P-action. Let I ⊂ B be a P-equivariant ideal. We have a corresponding G-equivariant sheaf of ideals I ⊂ O Z . The degeneration technique comes from trying to exploit the resolution of B/I as a B-module.

5.6. Degeneration Sequences

155

(5.6.1) Proposition. Assume that we can ﬁnd a P-equivariant resolution 0 → G m → G m−1 → . . . → G 1 → G 0 of B/I with G i = Wi ⊗ B. Then we have an induced exact sequence 0 → Gm → Gm−1 → . . . → G1 → G0

(∗)

of vector bundles G j = (G ×P G j ) ⊗ O Z which is a resolution of O Z /I. Assume that higher cohomology groups H i (G/P, G j ) = 0 for i ≥ 1, 0 ≤ j ≤ m. Then we have a G-equivariant acyclic sequence of A-modules 0 → Mm → Mm−1 → . . . → M1 → M0 where M j = H 0 (G/P, G j ). Proof. Decompose the exact sequence (∗) into short exact sequences, and use long cohomology sequences. The existence of such a P-equivariant resolution is in general a rather subtle question. There are, however, two cases when such a resolution exists. The ﬁrst case occurs when the unipotent radical N of P acts on B trivially. Denote by L a Levi factor of P. (5.6.2) Proposition. Assume that N acts trivially on B and that L is linearly reductive. Then the resolution (∗) of O Z /I exists. Proof. Since N acts on B trivially, B is really an L-module. Since L is linearly reductive, we can construct an L-equivariant resolution of B/I by the arguments used in section 5.4. The other case occurs when the ideal I is a complete intersection deﬁned by some P-semiinvariants. Let’s recall that if an algebraic group acts rationally on a vector space U , then the ring of semiinvariants S I (H, U ) = S I (H, U )χ , χ∈char(H)

where S I (H, U )χ = { f ∈ Sym(U ∗ ) | h ◦ f = χ (h) f ∀h ∈ H }, is the ring of functions that transform according to a certain character of H.

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The Geometric Technique

There is an important special case when one can predict which semiinvariants one should look at. It occurs when the rational H-module U has an open H-orbit. In such case one can classify the H-semiinvariants. This is due to the following result of Sato and Kimura. (5.6.3) Lemma (Sato–Kimura [SK]). Let H be a linear algebraic group acting rationally on a vector space U . Assume that this action has an open orbit. Then the ring S I (H, U ) is a polynomial ring. Moreover, the characters of the generators are linearly independent. The generators of the ring of semiinvariants can be described as follows. Assume O x is the open orbit of H in U . Let U \ O x = D1 ∪ . . . ∪ Dt be a decomposition into irreducible components. Assume that the ﬁrst s components have codimension 1 in U , while the other components have codimension bigger than 1. Then the generators of S I (H, U ) are the irreducible equations v1 , . . . , vs of D1 , . . . , Ds . For the proof of this lemma the reader should consult [Kr1, Theorem 2, section 3.6]. Coming back to our basic situation, i.e. H = P, U = η∗ , and using the Koszul complex, we have (5.6.4) Proposition. Assume that the ideal I is a complete intersection deﬁned by P-semiinvariants v1 , . . . , vs of weights λ1 , . . . , λs . Then the Koszul complex gives a resolution (∗) of length s, with G j = 1≤t1 0. (c) The Hilbert function of the normalization of K[Y ] does not depend on the characteristic of K. Proof. To prove (a) we consider the open subset U = {(φ, R) ∈ Z | rank φ = r }. By the deﬁnition of Z we see that if (φ, R) ∈ U then R = Im φ. Therefore the algebraic map φ → (φ, Im φ) is the inverse of the map q |U . This means q |U is an isomorphism, so q is a birational isomorphism. In order to establish (b) we apply Proposition (5.1.1)(b). We see that q∗ O Z = Sym(F ⊗ R∗ ). Using Theorem (5.1.2)(b) (applied with V = OV ), we see that it is enough to show that H i (V, Sym(F ⊗ R∗ )) = 0

for i > 0.

By the Cauchy formula (3.2.5), every symmetric power St (F ⊗ R∗ ) has a

ﬁltration with associated graded object |λ|=t L λ F ⊗ L λ R∗ . By the version (4.1.12) of Kempf’s theorem for Grassmannians, the higher cohomology of each L λ F ⊗ L λ R∗ vanishes. This proves part (b). Notice that if the characteristic of K is zero, then we can use the version (4.1.9) of the Bott’s theorem for Grassmannians to establish the vanishing of higher cohomology groups

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The Determinantal Varieties

of Sym(F ⊗ R∗ ). Since the dimension of H 0 (V, L λ R∗ ) does not depend on the characteristic of K by Kempf’s theorem, (c) follows from (5.1.2)(b) and part (b). It follows from (6.1.1) and (5.1.3) that the complex F• gives a minimal resolution of the normalization of the coordinate ring K[Y ] as an A-module. We want to calculate the terms of the complex F• . Since the cohomology depends on the characteristic of K, we have to treat the cases of characteristic 0 and p separately. Thus for the remainder of this section we assume that char K = 0. By Theorem (5.1.2) the terms of the complex F• are given by the cohomo logy groups H i (V, i+ j (F ⊗ Q∗ )). Applying the Cauchy formula (2.3.3), we know that t L λ F ⊗ K λ Q∗ . (F ⊗ Q∗ ) = |λ|=t

Therefore we have t L λ F ⊗ H i (V, K λ Q∗ ). H i V, (F ⊗ Q∗ ) = |λ|=t

We calculate the groups H i (V, K λ Q∗ ). Using the version of Bott’s theorem for Grassmannians (Corollary (4.1.9)), we see that we have to apply Bott’s algorithm (4.1.5) to the sequence (0, λ) + ρ = (n − 1, . . . , n − r, λ1 + n − r − 1, . . . , λq ). The ﬁrst r numbers in this sequence are consecutive, and the last q numbers form a decreasing sequence. Let us assume that (0, λ) is regular. Let s be the biggest number such that λs + n − r − s > n − 1. Then λs+1 + n − r − s − 1 < n − r . In terms of λ this means that λs ≥ r + s,

λs+1 < s + 1.

We denote the set of partitions satisfying the above inequalities by P(r, s). Reordering (0, λ) + ρ means that the numbers λ1 + n − r − 1, . . . , λs + n − r − s go in front of n − 1, . . . , n − r . Let us denote the corresponding permutation by w(s). Clearly +(w(s)) = r s. The conditions for λ ∈ P(r, s) mean that if the partition λ has the Durfee square size s (cf. section 1.1.2), then the sequence (0, λ) is regular if and only if λ contains an additional s × r rectangle to the right of the Durfee square. In that case the partition w(s) · (0, λ) = (λ1 − r, . . . , λs − r, s r , λs+1 , . . . , λq ). This means we have proved

6.1. The Lascoux Resolution

163

(6.1.2) Proposition. The i-th term of the complex F• is given by L λ F ⊗ K w(s). (0,λ) G ∗ ⊗K A. Fi = s≥0

λ∈P(r,s), |λ|−r s=i

Let us try to rewrite this result in a more symmetric way. Every partition λ from P(r, s) can be written as λ = (r + s + α1 , . . . , r + s + αs , β1 , . . . , βn−s ) where α and β are two partitions. In this setup we have w(s) · (0, λ) = (s + α1 , . . . , s + αs , s r , β1 , . . . , βq−s ). The dual partition (w(s) · (0, λ)) can in this notation be expressed as ). (w(s) · (0, λ)) = (r + s + β1 , . . . r + s + βs , α1 , . . . , αm−s

The term corresponding to λ appears in Fi with i = s 2 + |α| + |β|. We can rewrite (6.1.2) in terms of partitions α and β. For any s we consider the set Q(s) = {(α, β)|α ⊂ (m − r − s)s , β ⊂ (s)(n−r −s) }. For (α, β) ∈ Q(s) we denote P1 (α, β) = (r + s + α1 , . . . , r + s + αs , β1 , . . . , βn−s ), ). P2 (α, β) = (r + s + β1 , . . . , r + s + βs , α1 , . . . , αm−s

Then (6.1.2) can be rewritten in following way: (6.1.3) Proposition. Fi = s≥0

(α,β)∈Q(s),

L P1 (α,β) F ⊗ L P2 (α,β) G ∗ ⊗K A.

i=s 2 +|α|+|β|

This way of writing the term is symmetric in F and G ∗ . Now we can state the basic result on the syzygies of determinantal varieties. (6.1.4) Theorem (Lascoux [L2]). Assume that char K = 0. The complex F• is a minimal resolution of the coordinate ring K[Yr ]. Therefore the i-th module in this minimal resolution is given by the formula (6.1.3). Proof. The only thing that we have to prove is that K[Yr ] is normal. However, it is clear from (6.1.3) that F0 = A. By (5.1.3)(c) the normality follows. Let us state some other consequences of Theorem (6.1.4).

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The Determinantal Varieties

(6.1.5) Corollary. Assume that char K = 0. (a) The determinantal ideal Ir +1 is a prime ideal. The quotient ring A/Ir +1 is a normal domain. (b) The quotient ring A/Ir +1 is a Cohen–Macaulay ring with rational singularities. (c) Let us assume that m ≥ n. Then the type of the module A/Ir +1 is equal to the dimension of the module L (n−r )(m−n) G ∗ . Therefore A/Ir +1 is a Gorenstein ring if and only if m = n. (d) The t-th homogeneous component of the ring A/Ir +1 decomposes as a GL(F) × GL(G)-module as follows: Lλ F ⊗ LλG∗ (A/Ir +1 )t = |λ|=t, λ1 ≤r

Proof. To prove (a) let us notice that from (6.1.3) it follows at once that F1 = r +1 F ⊗ r +1 G ∗ ⊗K A(−r − 1). The map from F1 to F0 has to be GL(F) × GL(G)-invariant. By the Cauchy formula (3.2.5) it has to be given by (r + 1) × (r + 1) minors of the matrix . Since the variety Y is irreducible, its deﬁning ideal has to be prime. Part (b) follows from (5.1.3)(c). To see that (c) is true, let us assume that m ≥ n. We look at the last module in the resolution. It is clear from (6.1.3) that it is F(m−r )(n−r ) = L (m)(n−r ) F ⊗ L (n)(n−r ) ,(n−r )(m−n) G ∗ ⊗K A(−m(n − r )). The result follows, since the codimension of Yr equals (m − r )(n − r ). This is also a way to see that A/Ir +1 is Cohen–Macaulay without using rational singularities. Finally, (d) follows from (5.1.2)(b) and from Corollary (4.1.9). The remainder of this section is devoted to a more explicit description of the resolution of determinantal ideals in some important special cases. Historically these cases preceeded Lascoux’s paper. We preserve all our notation, assuming again that m ≥ n. (6.1.6) The Eagon–Northcott Complex. Let us consider the case r = n − 1. The complex F• is the resolution of the ideal In of maximal minors of the matrix . We notice that all modules corresponding to elements of Q(s) are zero for s ≥ 2. Obviously the only contribution from the set Q(0) is F0 = L (0) F ⊗ L (0) G ∗ ⊗K A = A.

6.1. The Lascoux Resolution

165

The elements from the set Q(1) give the contribution to the terms Fi for i > 0. In fact Fi = L (n+i−1) F ⊗ L (n,1i−1 ) G ∗ ⊗K A(−n − i + 1) for i ≥ 1, and identifying these Schur functors, we get Fi =

n+i−1

n

F⊗

G ∗ ⊗ Di−1 G ∗ ⊗K A(−n − i + 1).

We used the divided power because in such form the description of our complex will be characteristic free. Since our complex is the minimal resolution of In and the differentials are GL(m) × GL(n)-equivariant (section 5.4), we can use the representation theory to identify completely the differentials in F• . The differential d1 :

n

F⊗

n

G ∗ ⊗K A(−n) −→ A

has to be the composition n

F⊗

n

G ∗ ⊗K A(−n) → Sn (F ⊗ G ∗ ) ⊗K A(−n) → A,

where the left map is the embedding via n × n minors (cf. (3.2.5)) and the right map is the multiplication in A. Similarly, for i ≥ 1 the map di+1 :

n+i

F⊗

n

n+i−1

→

G ∗ ⊗ Di G ∗ ⊗K A(−n − i)

F⊗

n

G ∗ ⊗ Di−1 G ∗ ⊗K A(−n − i + 1)

is determined by its homogeneous component n+i

F⊗

=

n

n+i−1

G ∗ ⊗ Di G ∗ →

F⊗

n

n+i−1

F⊗

n

G ∗ ⊗ Di−1 G ∗ ⊗ A1

G ∗ ⊗ Di−1 G ∗ ⊗ F ⊗ G ∗

and thus has to be (up to a scalar we choose to be equal to 1) the tensor product of diagonalizations n+i

F→

n+i−1

F ⊗ F, Di G ∗ → Di−1 G ∗ ⊗ G ∗

tensored with n G ∗ . One can describe di+1 by an explicit formula. If { f 1 , . . . , f m } is a basis of F, {g1 , . . . , gn } is a basis of G, and i 1 , . . . , i n are nonnegative integers such

166

The Determinantal Varieties

that i 1 + . . . + i n = i, then the image di+1 ( f u 1 ∧ . . . ∧ f u n+i ⊗ g1∗ (i1 ) . . . gn∗ (in ) ) equals (−1)s+1 φu s ,t f u 1 ∧ . . . ∧ fˆu s ∧ . . . ∧ f u n+i ⊗ g1∗ (i1 ) . . . gt∗ (it −1) . . . gn∗ (in ) s,t

One can check easily that for i ≥ 1 one has di di+1 = 0. Our result generalizes to arbitrary characteristic. (6.1.7) Proposition. The complex F• deﬁned above is a minimal free resolution of the A-module A/In over a ﬁeld K of arbitrary characteristic, and thus when K is replaced by an arbitrary commutative ring. Proof. The proof is a repetition of proof of Lascoux’s theorem in a characteristic free setting. We notice that in the case under consideration the bundle Q∗ used in our Koszul complex is a line bundle. Therefore i (F ⊗ Q∗ ) = i F ⊗ Si Q∗ . The cohomology of bundles Si Q∗ has a characteristic free description by Serre’s theorem ([H1, chapter 3, section 5]). This proves that the terms of the complex F• are given by the formulas above. Then the reasoning we have just given in characteristic 0 case allows us to identify the differentials. Of course there exists an alternative proof of the general version based on Buchsbaum–Eisenbud acyclicity criterion (1.2.12) (see for example [BV, section 2C]). (6.1.8) The Gulliksen–Negard Complex ([GN]). This is the case m = n and r = n − 2. The nonzero terms of F• are F0 = L (0) F ⊗ L (0) G ∗ ⊗K A(0) = A(0), F1 = L (n−1) F ⊗ L (n−1) G ∗ ⊗K A(−n + 1), F2 = (L (n) F ⊗ L (n−1,1) G ∗ ⊗K A(−n)) ⊕ (L (n−1,1) F ⊗ L (n) G ∗ ⊗K A(−n)), F3 = L (n,1) F ⊗ L (n,1) G ∗ ⊗K A(−n − 1), F4 = L (n,n) F ⊗ L (n,n) G ∗ ⊗K A(−2n). We give an explicit description of the differentials. The idea is to construct the middle strand of F• (consisting of F3 , F2 , and F1 ) as a complex ﬁrst. We treat the map : F ⊗K A(−1) → G ⊗K A

6.1. The Lascoux Resolution

167

as a complex with nonzero terms appearing in homological degrees 1 and 0. The complex ∗ [−1] : G ∗ ⊗K A → F ∗ ⊗K A(1) has nonzero terms in degrees 0 and −1. Let f 1 , . . . , f m and g1 , . . . gn be the bases of F and G respectively. We have two equivariant maps of complexes Ev : ⊗ ∗ → A, Tr : A → ⊗ ∗ given by formulas Ev( f i ⊗ f j∗ ) = δi, j ,

Ev(gi ⊗ g ∗j ) = δi, j ,

Ev( f i ⊗ g ∗j ) = 0,

Ev(gi ⊗ f j∗ ) = 0,

and Tr(1) =

n i=0

gi ⊗ gi∗ −

m

f j ⊗ f j∗ .

j=0

This implies that the composition Ev Tr = 0. We consider the complex of complexes A[−1] → ⊗ ∗ [−1] → A[−1], and let H• be the complex which is the homology in the middle term. The nonzero terms of the complex H• are F ⊗ G ∗ ⊗K A(−1) → U ⊗K A → G ⊗ F ∗ ⊗K A(1), where U is the homology of the complex of vector spaces K → (F ⊗ F ∗ ) ⊕ (G ⊗ G ∗ ) → K with the maps coming from trace and evaluations according to the formulas given. Now we can see that after tensoring H• by n F ⊗ n G ∗ , shifting the grading by −n, and shifting the homological degree, we get the complex with the terms F3 , F2 , F1 . Reasoning as with the Eagon–Northcott com plex, we see that H• ⊗ n F ⊗ n G ∗ has to be a middle strand of F• . We

168

The Determinantal Varieties

augment our complex from both sides. We set F0 = F0 , F4 = F4 . We deﬁne the maps d1 : F1 → F0 by sending the generator f 1 ∧ f 2 ∧ . . . ∧ fˆi ∧ . . . ∧ f m ⊗ g1∗ ∧ . . . ∧ gˆ∗j ∧ . . . ∧ gn∗ ⊗ 1 to the determinant M(i, j) of the matrix with the i-th row and j-th column deleted. We also deﬁne the map d4 : F4 → F3

n

by sending the generator to i, j=1 (−1)i+ j M(i, j) f i ⊗ g ∗j ⊗ 1. Here we identify L (n,1) F with F, and L (n,1) G ∗ with G ∗ . Reasoning as with the Eagon– Northcott complex, we see that H• ⊗ n F ⊗ n G ∗ has to be a middle strand of F• . (6.1.9) Remarks. (a) The fact that determinantal ideals are perfect was ﬁrst proved by Eagon and Hochster in [HE]. The proof using the straightening law was given by DeConcini, Eisenbud, and Procesi in [DEP1]. (b) The idea of using higher direct images to calculate syzygies is due to Kempf. In his thesis ([Ke1]) he constructed the Eagon–Northcott complex in the way described above, using Serre’s theorem. Lascoux in his groundbreaking paper [L2] constructed the syzygies in the general case (in characteristic 0) using this method. The precise description of the differentials in Lascoux’s resolution was given by P. Roberts in his unpublished preprint [R1.]. (c) Several special cases of the resolutions of determinantal ideals were known before Lascoux’s proof. In addition to the Eagon–Northcott and Gulliksen–Negard complexes mentioned above, Poon ([Pn]) treated the case m = r + 3, n = r + 2. In all these papers the approach was algebraic. Various criteria for the exactness of the complex and localization were used. The resolutions in these cases were proven to be characteristic free.

6.2. The Resolutions of Determinantal Ideals in Positive Characteristic Throughout this section we assume that K is a ﬁeld of characteristic p. We retain the rest of the notation from the previous section. Let us recall that we

6.2. The Resolutions of Determinantal Ideals

169

constructed the free complex F• of A-modules with the i-th term i+ j j ∗ H V, (F ⊗ Q ) ⊗K A, Fi = j≥0

which is a minimal resolution of the normalization of the coordinate ring K[Yr ]. We will use the Schur complexes (section 2.3) to obtain some information about the terms of the complex F• in characteristic p. Our information is far from being complete, because we do not have the analogue of Bott’s theorem. Still, we can prove that K[Yr ] is normal and that the determinantal ideal Ir +1 is radical. This will establish properties (1)–(3) from the beginning of the previous section in characteristic p. We will also show that even though the Hilbert function of K[Yr ] does not depend on the characteristic of K, the resolution F• does. More precisely, for m = n = 5 and r = 1 the complex F• is different in characteristic 0 and 3. This is the example given by Hashimoto. Consider the natural epimorphism π : G ∗ × V → R∗ deﬁned over V = Grass(r, G). Applying (2.4.12)(c), we see that the complex L λ π gives a right resolution of K λ Q∗ . By (2.4.12)(a), the terms of L λ π have a ﬁltration whose associated graded object is µ K λ/µ G ∗ ⊗L µ R∗ . By Corollary (4.1.12) we see that each term of this ﬁltration is *-acyclic. This means that L λ π is a *acyclic resolution of K λ Q∗ , so the complex *(L λ π ) can be used to calculate the cohomology of K λ Q∗ . We want to identify *(L λ π) more precisely. Let us consider the Schur complex L λ (id), where id : G ∗ → G ∗ is the identity map. Applying (2.4.12)(a), we see that there exists a natural subcomplex (L λ (id))(r + 1) whose terms have a ﬁltration with the associated graded object K λ/µ G ∗ ⊗ L µ G ∗ . µ,µ1 >r

(6.2.1) Proposition. The complex *(L λ π ) is naturally isomorphic to L λ (id)/ L λ (id)(r + 1). Proof. We have a commutative diagram id

G ∗ −→ G ∗ ↓ id ↓π π ∗ G −→ R∗ This induces a map of complexes θ¯ : L λ (id) → L λ (π). Applying the functor * to this map, we get a map of complexes *(θ¯ ) : L λ (id) → *(L λ π). We

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The Determinantal Varieties

¯ is an epimorphism with the kernel L λ (id)(r + 1). First want to show that *(θ) of all it is clear that the image of L λ (id)(r + 1) is zero for dimension reasons. This means we get the induced map θ : L λ (id)/L λ (id)(r + 1) → *(L λ π). Let us deﬁne the subcomplexes X 0,

190

The Determinantal Varieties

because Y is an afﬁne variety. Using Theorem (5.1.3)(a), we see that q∗ (O Z 1 ) is the normalization of K[Yra ] and this normalization has rational singularities. a ]. We just saw that q∗ (O Z 1 ) can Finally let us show that q∗ (O Z 1 ) = K[Y2u a 0 be identiﬁed with the A -module H (V, Sym( 2 Q)). Using (2.3.8)(b) and Corollary (4.1.9), we see that this module decomposes into the representations of GL(E) in the following way: 2 0 H V, Sym( Q) = L λ E. λ,λ1 ≤2u, λi even for all i a K[Y2u ]

To prove that the ring is normal it is therefore enough to show that each representation on the right hand side of the above formula occurs in a ]. K[Y2u We use the explicit description of U-invariants in Sym( 2 E) given in the proof of (2.3.8)(b). It is obvious that these functions do not vanish on Y2a u a ]. and therefore the corresponding representations occur in K[Y2u 2 The description of U-invariants in Sym( E) has another application. a consists It is clear from what we just proved that the deﬁning ideal of Y2u of all representations L λ E occurring in (2.3.8)(b) (for arbitrary t) for which λ1 > 2u. Also, by (2.3.8)(b) we see that the U-invariant corresponding to a a such representation is contained in I2u+2 . Since I2u+2 is GL(E)-equivariant, a . This shows that the the whole representation L λ E is contained in I2u+2 a a is equal to I2u+2 . deﬁning ideal of Y2u This completes the proof of ﬁrst two parts of Theorem (6.4.1). 2. The second incidence variety. Let us ﬁx n and r = 2u. We consider V = Grass(n − u, E) with the tautological sequence 0 → R → E × V → Q → 0, where dim R = n − u and dim Q = q = u. For a subspace R in E we denote by i the embedding of R into E. We consider the variety Z 2 = {(φ, R) ∈ X × V | i ∗ φ i = 0 }. The variety Z 2 is the total space of the bundle S2 deﬁned by the following sequence: 0 → S2 →

2

E∗ × V →

2

R∗ → 0.

This means that we are in the setting of the section 5.1 with E = 2 E ∗ × V . 2 ∗ R . It follows that T2 = Note that the variety Y , which is by deﬁnition q(Z 2 ), is not a priori equal a to Y2u .

6.4. The Determinantal Ideals for Skew Symmetric Matrices

191

We consider the basic diagram Z2 ⊂ ↓ q Y ⊂

X×V ↓q X

Applying Proposition (5.1.1), we see that the resolution of O Z as an O X ×V module is given by the Koszul complex 2 K R •:0→ 1 2 (n−u)(n−u−1)

p

∗

2

R

→...→ p

∗

2

R → O X ×V .

We calculate the terms of the complex F• . By the formula (2.3.9)(b) we see that t 2 K λ R. R = λ∈Q −1 (2t)

Therefore we have to calculate the cohomology groups H i (V, K λ R) for λ ∈ Q −1 . Using Corollary (4.1.9), we see that we have to consider a sequence (0, λ) + ρ = (n − 1, . . . , n − u, λ1 + n − u − 1, . . . , λn−u ). The ﬁrst u numbers in this sequence are consecutive, and the last n − u numbers form a decreasing sequence. Let us assume that (0, λ) is regular. Let s be the biggest number such that λs + n − u − s > n − 1. Then λs+1 + n − u − s − 1 < n − u. In terms of λ this means that λs ≥ u + s,

λs+1 < s + 1.

The only nonzero cohomology group of K λ R will be H us (V, K λ R) = K µ E, where µ = (λ1 − u, . . . , λs − u, s u , λs+1 , . . . , λn−u ). Now we use the fact that λ ∈ Q −1 . This means that there exists a partition α (with α1 ≤ n − 2u − s + 1) such that ). λ = λ(α, s) = (α1 + s + u, . . . , αs + s + u, s u+1 , α1 , . . . , αn−2u−s+1

In terms of α the partition µ equals ). µ = µ(α, s) = (α1 + s, . . . , αs + s, s 2u+1 , α1 , . . . , αn−2u−s+1

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The Determinantal Varieties

The term corresponding to µ(α, s) occurs in F• in the place |α| + 12 s(s + 1). We have proved the following proposition. (6.4.3) Proposition. The terms of the complex F• are given by the formula Fi = K µ(α,s) E ⊗K Aa . (s,α), α1 ≤n−2u−s−1, i=|α|+ 12 s(s+1)

We proceed with the analysis of the complex F• . First of all, it is clear that Fi = 0 for i < 0. Therefore F• is acyclic. Let us analyze the terms F0 and F1 . Clearly F0 = K 0 E and F1 = K (12u+2 ) E. This means that the complex F• a is the resolution of K[Y2u ], and the last part of Theorem (6.4.1) follows.• We ﬁnish this section by giving two examples in low codimension. (6.4.4) The Buchsbaum–Eisenbud complex ([BE3]). This is the case n = 2t + 1, r = 2t − 2. The complex G • gives the resolution of the module Aa /I2ta . The calculation of the terms of the complex G • from Theorem (6.4.1) gives G 3 = L (2t+1,2t+1) E ⊗ Aa (−2t − 1), G 2 = L (2t+1,1) E ⊗ Aa (−t − 1), G 1 = L (2t) E ⊗ Aa (−t), G 0 = Aa (0). The middle strand G 2 −→ G 1 is just ⊗

n

E. where ∗

: E ⊗K A (−1) → E ⊗K Aa a

is the generic map. We augment the complex G 2 −→ G 1 by two maps d1 : G 1 −→ G 0 and d3 : G 3 −→ G 2 given by the formulas ˆ d1 (e1 ∧ . . . ∧ eˆ j ∧ . . . ∧ e2t+1 ) = Pf( j) d3 (1) =

2t+1

ˆ j (−1) j Pf( j)e

j=1

ˆ is the Pfafﬁan of the 2t × 2t skew symmetric matrix obtained by where Pf( j) deleting in the j-th row and the j-th column. In the second formula we identify L (2t+1,1) E with E. One can check directly that these formulas deﬁne the GL(E)-equivariant complex with the same terms as G • . Then the argument we used in (6.1.6) for the Eagon–Northcott complex applies, and we see that G • is isomorphic to G • and thus it is a resolution of Aa /I2ta . One can use the Buchsbaum–Eisenbud acyclicity criterion (1.2.12) to

6.4. The Determinantal Ideals for Skew Symmetric Matrices

193

show that the complex deﬁned in this way gives a resolution of Aa /I2ta when K is replaced by an arbitrary commutative ring. (6.4.5) Remarks. Buchsbaum and Eisenbud proved in [BE3] that the complex G • deﬁned above is the universal resolution of Gorenstein ideals of codimension 3. This means that if S is a commutative ring and J is a Gorenstein ideal of codimension 3 in S, then there exists a homomorphism of rings ψ : Aa → S such that J = ψ(I2ta ), i.e., J is the ideal of 2t × 2t Pfafﬁans of the (2t + 1) × (2t + 1) matrix ψ(). The resolution of S/J as an S-module is given by G • ⊗ Aa S. (6.4.6) The J´ozeﬁak–Pragacz complex ([JP2]). This is the complex of length 6 which arises when n = 2t + 2, r = 2t − 2. The complex G • gives the resolution of Aa /I2ta . The nonzero terms are G 0 = Aa (0), G 1 = L (2t) E ⊗K Aa (−t), G 2 = L (2t+1,1) E ⊗K Aa (−t − 1), G 3 = (L (2t+2,1,1) E ⊗K Aa (−t − 2) ⊕ (L (2t+1,2t+1) E ⊗K Aa (−2t − 1), G 4 = L (2t+2,2t+1,1) E ⊗K Aa (−2t − 2), G 5 = L (2t+2,2t+2,2) E ⊗K Aa (−2t − 3), G 6 = L (2t+2,2t+2,2t+2) E ⊗K Aa (−3t − 3). As before, we identify the linear strands of G • using Schur complexes and trace and evaluation maps. The complex G • has two linear strands (apart from the trivial ones occurring at both ends). The ﬁrst one is the complex H•1 with the terms L (2t+2,1,1) E ⊗K Aa (−t − 2), L (2t+1,1) E ⊗K Aa (−t − 1), L (2t) E ⊗K Aa (−t). Since is a skew symmetric matrix, we can identify and ∗ [−1]. We will denote the identifying isomorphism by τ . We have a map EV of complexes 2

1⊗τ

EV

() −→ ⊗ −→ ⊗ ∗ [−1]−→Aa [−1].

Deﬁne B(1,1) () := Ker EV. We deﬁne H•1 = B(1,1) () ⊗ This is the ﬁrst linear strand of G • .

2t+2

E(−t)[−1].

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The Determinantal Varieties

The second linear strand, H•2 , is the complex with the terms L (2t+2,2t+2,2) E ⊗K Aa (−2t − 3), L (2t+2,2t+1,1) E ⊗K Aa (−2t − 2), L (2t+1,2t+1) E ⊗K Aa (−2t − 1). We have a map TR of complexes TR

1⊗τ

Aa [−1]−→ ⊗ ∗ [−1]−→ ⊗ −→ S2 (). Deﬁne A2 () = Coker TR. Finally, H•2 = A2 () ⊗

2t+2

E ⊗2 (−2t − 1)[−3].

Of course we can deﬁne the 0th linear strand to be H•0 Aa (0), and the strand H•3 = L (2t+2,2t+2,2t+2) E ⊗ Aa (−2t − 3). Notice that up to grading and homological shifts, H•i is dual to H•3−i . It remains to deﬁne the maps di from H•i to H•i−1 The map d1 is nonzero on the term L (2t) E ⊗ Aa (−t), and it sends the generator e1 ∧ . . . ∧ eˆi ∧ . . . ∧ ˆ j), ˆ where the last symbol denotes the Pfafﬁan of the eˆ j ∧ . . . ∧ en to Pf(i, 2t × 2t skew symmetric matrix we get from by deleting the i-th and j-th rows and the i-th and j-th columns. The map d3 can be deﬁned as the dual of d1 (up to shifts in grading and homological degree). The most difﬁcult part is the deﬁnition of d2 . We just sketch its construction. We start with the map of complexes: ⊗TR

1⊗1⊗

S2 ()[−1] −→ ⊗ ⊗ S2 () −→ ⊗ ⊗ ⊗ 2 2 m 1,3 ⊗m 2,4 −→ () ⊗ (). Here m 1,3 ⊗ m 2,4 denotes the map which multiplies the ﬁrst factor by the third one, and the second factor by the fourth one. We can check easily that this map induces the map of complexes A2 ()[−1] −→ B(1,1) () ⊗ B(1,1) () However, up to grading and homological shifts, B(1,1) () is the same as H•1 . Therefore we have a map d1 : B(1,1) () −→ Aa . We deﬁne d2 as a composition 1⊗d1

A2 ()[−1] −→ B(1,1) () ⊗ B(1,1) ()−→B(1,1) ().

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195

Now it is easy to check that the maps di deﬁne a GL(E)-equivariant double complex H•3 −→ H•2 −→ H•1 −→ H•0 . In order to check that this complex is isomorphic to G • it is enough to show that the components of the differentials are nonzero when restricted to every summand L λ E ⊗ Aa . Indeed, inducting on the homological and the homogeneous degree, we see by exactness of G • that in given homological and homogeneous degrees the module of cycles in G • has only one irreducible representation of the required kind, so it has to he the one covered by the differential we just constructed. Checking that the differentials deﬁned above have the required properties can be carried out using the U-invariant elements in each representation. (6.4.7) Remarks. The minimal resolutions of ideals of Pfafﬁans are not, in general, characteristic free. In fact, Kurano showed in [K3], [K4] that the relations between 4 × 4 Pfafﬁans of an 8 × 8 skew symmetric matrix are not spanned by linear relations when char K = 2. The more general family of similar examples is provided in [Ha5]. In the case of complex (6.4.6) Pragacz showed in [Pi ] that the ranks of syzygies do not depend on characteristic of K.

6.5. Modules Supported in Determinantal Varieties We preserve the notation from section 6.1. We will study the GL(F) × GL(G)equivariant modules supported in determinantal varieties. We are interested in the structure of such modules. An interesting family of modules supported in determinantal varieties are the direct images of equivariant sheaves on the desingularization we studied in section 6.1. Our constructions allow us to calculate the minimal resolutions of such modules, given by the appropriate complexes F(V)• . It turns out that in addition to the desingularization used in section 6.1, we have two other resolutions of singularities of the determinantal variety Yr , leading to three such families. The choice of desingularization does not make a difference when investigating the coordinate rings Ar := K[Yr ], but different desingularizations lead to different families of equivariant modules. It turns out that each of these families generates the Grothendieck group of graded GL(F) × GL(G)-equivariant modules supported in Yr . This means that, at least in principle, the resolution of an equivariant GL(F) × GL(G)module supported in Yr can be obtained from complexes F(V)• .

196

The Determinantal Varieties

We show that some of the terms of the complexes F(V)• depend on the characteristic of the ﬁeld K. When studying determinantal varieties Yr in section 6.1, we used a desingularization ¯ ∈ X × Grass(r, G) | Im φ ⊂ R}. Z r(2) = {(φ, R) In fact we could have made another choice: Z r(1) = {(φ, R) ∈ X × Grass(m − r, F) | φ |

R¯

= 0}.

The choice of desingularization was irrelevant when studying the coordinate rings of determinantal varieties, but it makes a difference when looking at twisted complexes F(V)• . In order to make the situation symmetric it is also necessary to study the ﬁbered product ¯ ∈ X × Grass(r, G) Z r = Z r(1) × X r Z r(2) = {(φ, R, R) × Grass(m − r, F) | Im φ ⊂ R, φ | R¯ = 0}. Throughout this section we denote by 0 → R → F × Grass(m − r, F) → ¯ → Q → 0 the tautological sequence on Grass(m − r, F), and by 0 → R ¯ G × Grass(r, G) → Q → 0 the tautological sequence on Grass(r, G). Let Cr (F, G) be the category of graded Ar -modules with rational GL(F) × GL(G)-action compatible with the module structure, and equivariant degree 0 maps. We denote by K 0 (Ar ) the Grothendieck group of the category Cr (F, G). For an equivariant graded module M ∈ Ob(Cr (F, G)) and for q ∈ Z, we denote by M(q) the module M with gradation shifted by q, i.e. M(q)n = Mq+n . For M ∈ Ob(Cr (F, G)) we deﬁne the graded character of M, char(Mn ) q n ∈ Rep(GL(F) × GL(G ∗ ))[[q]][q −1 ]. char(M) = n∈Z

where Rep(GL(F) × GL(G ∗ )) denotes the representation ring of GL(F) × GL(G ∗ ) and char is the product of character maps described in (2.2.10). We recall from section 2.2, that an integral weight for GL(m) is just an mtuple α = (α1 , . . . , αm ) of integers. The weight α is dominant if α1 ≥ . . . ≥ αm . Let α = (α1 , . . . , αm ) be an integral weight for GL(F). We set α (1) = (α1 , . . . , αr ), α (2) = (αr +1 , . . . , αm ). Let β = (β1 , . . . , βn ) be an integral weight for GL(G ∗ ). We deﬁne β (1) = (β1 , . . . , βr ), β (2) = (βr +1 , . . . , βn ). Let α = (α (1) , α (2) ). Assume that both α (1) and α (2) are dominant. Let β be a dominant weight for GL(G ∗ ). For each such pair (α, β) we deﬁne a sheaf M(α, β) = p (1)∗ (L α(1) Q ⊗ L α(2) R) ⊗ L β G ∗ ⊗ O Z (1)

6.5. Modules Supported in Determinantal Varieties

197

of graded modules on Z (1) . Symmetrically, let α be a dominant weight for GL(F), and let β be a weight for GL(G ∗ ) such that β (1) and β (2) are both dominant. We deﬁne a sheaf ¯ ⊗ O Z (2) N (α, β) = L α F ⊗ p (2)∗ (L β (1) Q¯ ⊗ L β (2) R) of graded modules on Z (2) . Finally, let α, β be weights such that α (1) , α (2) , β (1) , β (2) are dominant. We deﬁne a sheaf ¯ ⊗ OZ P(α, β) = p (1)∗ (L α(1) Q ⊗ L α(2) R) ⊗ p (2)∗ (L β (1) Q¯ ⊗ L β (2) R) of graded modules on Z . We deﬁne the equivariant graded modules M(α, β) = H 0 (Z (1) , M(α, β)), N (α, β) = H 0 (Z (2) , N (α, β)), P(α, β) = H 0 (Z , P(α, β)). Let us start with providing some examples of modules from three families. We look ﬁrst at the family M(α, β). Since in this case M(α, β) = M(α, (0)) ⊗K L β G ∗ , we can assume that β = (0). (6.5.1) Example. Let α (2) = (0). The sheaf M(α, β) equals L α(1) Q ⊗ O Z (1) . Therefore the direct image p∗(1) M(α, β) equals L α(1) Q ⊗ Sym(Q ⊗ G ∗ ). Using the straightening formula (2.3.2) and the characteristic free version [Bo2] of the Littlewood–Richardson rule, we see that the higher cohomology of M(α, β) vanishes. The j-th homogeneous component M j (α, (0)) has a ﬁltration with the associated graded object (L α(1) F ⊗ L µ F)≤r ⊗ L µ G ∗ , |µ|= j

where (L α(1) F ⊗ L µ F)≤r denotes a factor of (L α(1) F ⊗ L µ F) consisting of all L ν F in the tensor product with ν1 ≤ r . In characteristic zero we can write the direct sum instead of ﬁltration. (6.5.2) Example. The simplest examples of modules M(α, (0)) are provided by the ones corresponding to line bundles on Grass(m − r, F). The line bundles on Grass(m − r, F) correspond to tensor powers OGrass(m−r,F) (t) = ( r Q)⊗t for t ∈ Z. For t ≥ 0 this is a special case of the example (6.5.1). Let us describe the minimal free resolution of the module M((n r ), (0)). Assume that the characteristic of K is zero. The terms come from the cohomology of the vector bundles ( r Q)⊗t ⊗ • (R ⊗ G ∗ ). Decomposing by Cauchy’s

198

The Determinantal Varieties

formula we see that one needs to apply Bott’s theorem (4.1.9) to the weights (t r , λ1 , . . . , λm−r ). The description is similar to the description of syzygies of determinantal varieties. The surviving terms decompose to families depending on a parameter s saying how many parts λ1 , . . . , λs have to move in front of r parts t. Notice that if λ1 ≤ t, there are no exchanges and we have s = 0. The condition for s terms being exchanged is λs ≥ s + t + r , λt+1 ≤ s + t. In such case we write λ := λ(t, s, α, β) = (α1 s + t + r, . . . , αt + s + t + r, β1 , . . . , βm−r −t ). The corresponding term is K µ F ⊗ L λ G ∗ where µ := µ(t, s, α, β) = ((α1 + s + t, . . . , αt + s + t, (s + t)r β1 , . . . , βm−r −t ). The term corresponding to the set of data t, s, α, β occurs in homological degree |α| + |β| + s 2 + st. In particular, take r = 3, t = 1, s = 2. Take α = (3, 1), β = (3, 2). The partitions λ, µ are X X λ= ◦ ◦

• • • •

X X X X ◦ ◦ ◦

,

X X X • • • X X X • .

µ= ◦ ◦ ◦ ◦ ◦

Here the boxes corresponding to α are ﬁlled by •, the boxes corresponding to β are ﬁlled by ◦, and the boxes corresponding to the (s + t) × s rectangle both partitions have to contain are ﬁlled by X . (6.5.3) Theorem. The group K 0 (Ar ) is generated by the classes of the modules of each of the families M(α, β)(q), N (α, β)(q), P(α, β)(q) where α and β are both dominant weights and q ∈ Z. (6.5.4) Theorem. (a) The group K 0 (Ar ) is isomorphic to the additive subgroup of the ring Rep(GL(F) × GL(G ∗ ))[[q]][q −1 ] generated by shifted graded characters of modules M(α, β) (or N (α, β), or P(α, β)(q)) for α, β dominant, (b) The group K 0 (Ar ) is isomorphic to the additive group of the ring Rep(GL(F) × GL(G ∗ ))[q][q −1 ].

6.5. Modules Supported in Determinantal Varieties

199

We provide the proofs only for the family M(α, β). The proofs for the family N (α, β) are symmetric. The proofs for the family P(α, β) and the transition formulas are given in [W6]. We start with basic observations about the cohomology groups of the sheaves M(α, β). Since p (1) is an afﬁne map, R i p∗(1) O Z (1) = 0 for i > 0. We also have (1) p∗ O Z (1) = Sym(Q ⊗ G ∗ ). Using the Leray spectral sequence and a projection formula (assuming α arbitrary, β dominant), we have H i (Z (1) , M(α, β)) = H i (Grass(m − r, F), K α(1) Q ⊗ K α(2) R ⊗ Sym(Q ⊗ G ∗ )). (6.5.5) Proposition. Let α, β be dominant weights for GL(F), GL(G ∗ ) respectively. Then H i (Z (1) , M(α, β)) = 0 for i > 0. Proof. By deﬁnition M(α, β) = p (1)∗ (L α(1) Q ⊗ L α(2) R) ⊗ L β G ∗ ⊗ O Z (1) . Using this, we see that p∗(1) M(α, β) = L α(1) Q ⊗ L α(2) R ⊗ L β G ∗ ⊗ Sym(Q ⊗ G ∗ ). Using the straightening law (3.2.5) and the fact that the tensor product of two Schur functors has a ﬁltration whose associated graded object is a direct sum of Schur functors ([Bo2]), we are reduced to proving that if α is dominant then H i (Grass(m − r, F), L α(1) Q ⊗ L α(2) R) = 0 for i > 0. In characteristic 0 this follows at once from (4.1.9). The characteristic free version follows from (4.1.12). We can get additional information about the cohomology of M(α, β) when K is a ﬁeld of characteristic zero. Let α be a weight with α (1) , α (2) dominant. We deﬁne the number l(α) as follows. Consider the weight α + ρ where ρ = (m − 1, m − 2, . . . , 1, 0) = (u 1 , . . . , u m ). Deﬁne, by reverse induction on s (from s = r to s = 1) the numbers δs = min{t | t ≥ δs+1 , t +m −s ∈ / {δs+1 + m − s − 1, . . . , δr + m − r, αr +1 + m − r −1, . . . , αm }}.

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The Determinantal Varieties

By construction the weight (δ1 , . . . , δr , αr +1 , . . . , αm ) + ρ is not orthogonal to any root. By (4.1.9) there exists a unique l such that H l (Grass(m − r, F), K δ Q ⊗ K α(2) R) = 0. We deﬁne l(α) := l. (6.5.6) Proposition. Let K be a ﬁeld of characteristic zero. Let α = (α (1) , α (2) ) be a weight for GL(F) with α (1) , α (2) dominant. Let l(α) be deﬁned as above. Assume that β is a dominant weight for GL(G ∗ ). (a) H i (Z (1) , M(α, β)) = 0

for i > l(α).

(b) H l(α) (Z (1) , M(α, β)) = 0. Proof. We can assume that β = (0) because, by the projection formula, tensoring with K β G ∗ commutes with taking cohomology. This means we are reduced to calculating the cohomology H ∗ (Grass(m − r, F), L α(1) Q ⊗ L α(2) R ⊗ Sym(Q ⊗ G ∗ )). This can be rewritten as H ∗ (Grass(m − r, F), L δ Q ⊗ L α(2) R ⊗ L γ G ∗ ). δ∈α (1) ⊗γ

By the Littlewood–Richardson rule (2.3.4), every weight occurring in the tensor product α (1) ⊗ γ is bigger than or equal to α (1) termwise. Also, since dim Q = r ≤ dim G ∗ , all such weights δ will occur in α ⊗ γ for some γ . Consider α = (α (1) , α (2) ) satisfying the assumptions of the proposition. Let δ0 be the weight constructed in deﬁning l(α). This, by deﬁnition, is the termwise minimal weight for which L δ0 Q ⊗ L α(2) R has nonzero cohomology. This cohomology occurs in degree l(α). Also it is clear from (4.1.9) that for δ that is bigger termwise than δ0 the cohomology of L δ Q ⊗ L α(2) R, if nonzero, occurs in degree ≤ l(α). This proves both parts of the proposition. We also get information on the support of cohomology modules H i (Z (1) , M(α, β)). In oder to state the result we need to introduce one more notion. The permutation σ ∈ m is an r -Grassmannian permutation if σ (1) > . . . > σ (r ), σ (r + 1) > . . . > σ (m). For each r -Grassmannian permutation σ we denote by Cσ the Weyl chamber of all weights (γ1 , . . . , γm ) such that the entries in

6.5. Modules Supported in Determinantal Varieties

201

γ + ρ are pairwise different and ordered in the same way as the sequence σ (1), . . . , σ (m). Let σ be an r -Grassmannian permutation of length i. For α arbitrary and β dominant we deﬁne the Ar -module H i (Grass(m − r, F), M(α, β))σ to be the part of H i (Grass(m − r, F), M(α, β)) consisting of all cohomology modules of sheaves K δ Q ⊗ K α(2) R ⊗ K γ G ∗ for which the weight (δ, α (2) ) ∈ Cσ . It is clear that this is a direct summand of the Ar -module H i (Z (1) , M(α, β)). (6.5.7) Proposition. Let K be a ﬁeld of characteristic 0. Asume that α is arbitrary and β is dominant. (a) The module H i (Z (1) , M(α, β))σ is nonzero if and only if there exists δ = (δ1 , . . . , δr ) such that δ ≥ α (1) (termwise) and (δ, α (2) ) ∈ Cσ . (b) The support of H i (Z (1) , M(α, β))σ is the determinantal variety Ys−1 for s = σ (r + 1). Proof. As in the proof of (6.5.6), we can assume that β = (0). The ﬁrst part of the proposition follows as in the proof of (6.5.6). Let us choose an r Grassmannian permutation σ of length i. We are interested in the support of the cohomology modules of M(α, β)σ = K δ Q ⊗ K α(2) R ⊗ K γ G ∗ . γ

δ∈α (1) ⊗γ , δ∈Cσ

Let σ (r + 1) = s. Then we can increase δ1 , . . . , δs−1 as we please to still get the weights (δ, α (2) ) from Cσ . On the other hand, the indices δs , . . . , δr can increase only by a limited number if we are to get a weight from Cσ . Now the use of Littlewood–Richardson rule and (6.1.5)(d) shows that the support of the module H i (Z (1) , M(α, β))σ equals Ys−1 . We proceed to prove Theorems (6.5.3) and (6.5.4) for the family M(α, β). Each equivariant sheaf M on Z (1) has its Euler characteristic class χ(M) = (−1)i [H i (Z (1) , M)] ∈ K 0 (Ar ). i≥0

(6.5.8) Proposition. The group K 0 (Ar ) is generated by the Euler characteristic classes χ (M(α, β)) for (α (1) , α (2) , β dominant). Proof. Let M be a graded Ar -module with a rational GL(F) × GL(G ∗ ) action. Then the natural morphism M → (q∗(1) )(q (1)∗ M)

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The Determinantal Varieties

has a kernel and cokernel supported in Yr −1 . It is therefore enough to show. (1) If M is a module supported in Yr −1 , then its class in K 0 (Ar ) is in the span of classes χ(M(α, β)) for (α (1) , α (2) ), β dominant. (2) The class of q (1)∗ M is contained in the span of sheaves M(α, β) in the Grothendieck goup of equivariant sheaves on Z (1) . We start with the proof of (1). Since in this argument all constructions will commute with tensoring by L β G ∗ , we will drop it from our notation, dealing with sheaves M(α (1) , α (2) ) := M(α (1) , α (2) , (0)). It is enough to show that the Euler characteristic of each sheaf of type ˆ is in the subgroup of K 0 (Ar ) generated by M(α), for Yr −1 denoted by M(α), Euler characteristics of sheaves M(α) for Yr . Consider the Grassmannian Grass(m − r + 1, F) with the tautological sequence ˆ → F × Grass(m − r + 1, F) → Qˆ → 0. 0→R Consider the ﬂag variety Flag(m − r, m − r + 1; F) with universal ﬂag ˆ ⊂ F. Let 0⊂R⊂R v1 : Flag(m − r, m − r + 1; F) → Grass(m − r, F), v2 : Flag(m − r, m − r + 1; F) → Grass(m − r + 1, F) denote the natural projections. We have by deﬁnition ˆ (1) , . . . , α (1) , α (2) , . . . , α (2) M(α 1 r −1 1 m−r +1 ) ˆ ⊗ L (2) = v2∗ (L α(1) Qˆ ⊗ L (2) (R/R)

(2) R (α2 ,...,αm−r +1 )

α1

⊗ Sym(Qˆ ⊗ G ∗ )).

The higher direct images of the tensor product sheaf on the right hand side vanish. This sheaf has a Koszul type resolution of locally free modules over Sym(Q ⊗ G ∗ ) on Flag(m − r, m − r + 1; F) with terms ˆ ⊗ L (α(2) ,...,α(2) L α(1) Qˆ ⊗ L α(2) (R/R) ⊗

•

1

2

m−r +1 )

R

ˆ ⊗ G ∗ ) ⊗ Sym(Qˆ ⊗ G ∗ ), (Ker(Q → Q)

which can be rewritten as ˆ ⊗ L (α(2) ,...,α(2) L α(1) Qˆ ⊗ L α(2) +• (R/R) 1

2

m−r +1

)R ⊗

•

(G ∗ ) ⊗ Sym(Qˆ ⊗ G ∗ ),

6.5. Modules Supported in Determinantal Varieties

203

ˆ is isomorphic to R/R, ˆ because Ker(Q → Q) as can be seen from the commutative diagram 0 0

→ R → ↓ ˆ → → R

F × Flag(m − r, m − r + 1; F) → Q → 0 ↓ ↓ F × Flag(m − r, m − r + 1; F) → Qˆ → 0

of vector bundles over Flag(m − r, m − r + 1; F). We push down the terms of the resolution by v1∗ . The bundle Q is induced from Grass(m − r, F). Thus each term ˆ L α(1) Qˆ ⊗ L α(2) +t (R/R) ⊗ L (α(2) ,...,α(2) 1

2

m−r +1

)R ⊗

t (G ∗ ) ⊗ Sym(Qˆ ⊗ G ∗ )

gives us a sheaf M(γ (1) , γ (2) ), possibly with sign. Taking Euler characteristics ˆ (1) , α (2) ) as a combination of terms of the type gives an expression of χ (M(α (1) (2) χ (M(γ , γ ). To prove statement (2) we notice that q1∗ is a sheaf of graded Sym(Q ⊗ G ∗ )-modules. We can take its free GL(Q) × GL(R) × GL(G ∗ )equivariant resolution. Its terms are, up to ﬁltration, direct sums of sheaves of type M(α (1) , α (2) , β). This completes the proof of Proposition (6.5.4). (6.5.9) Remarks. Notice that the proof of Proposition (6.5.8) shows that if ˆ (1) , α (2) , β) is (α (1) , α (2) ) is dominant, then the Euler characteristic of M(α in the subgroup of K 0 (Ar ) spanned by the Euler characteristics of sheaves M(α, β) with α, β dominant. Theorem (6.5.3) for modules of type M(α, β) follows from the following statement: (6.5.10) Proposition. The classes χ (M(α (1) , α (2) , β)) such that (α (1) , α (2) ) is dominant generate the group K 0 (Ar ). Proof. We ﬁrst reduce the proof to the case when the characteristic of K equals zero. First of all we notice that the representation ring of the general linear group GL(F) is spanned by the classes of Schur modules L α F. This follows from the description of irreducible representations of GL(F) given in Theorem (2.2.9). Moreover, the Littlewood–Richardson rule holds in the representation ring of GL(F) in a characteristic free way, by the remark following Theorem (2.3.4). Finally, even though Bott’s theorem is true only over a ﬁeld of characteristic zero, the Euler characteristic of any line bundle over a ﬂag variety is characteristic free. Our statement involves only Euler

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The Determinantal Varieties

characteristics and the characters of modules M(α, β) for α, β dominant. Both these notions are independent of the characteristic and can be described using only Schur modules and the Littlewood–Richardson rule. We see that the proof in characteristic zero implies that the same statement holds over a ﬁeld of arbitrary characteristic. Let H be the subgroup of K 0 (Ar ) spanned by the Euler characteristic classes of sheaves M(α, β) with α and β dominant. Consider an arbitrary sheaf M(α (1) , α (2) , β). We use induction on s := s(α (1) , α (2) ) = α1(2) − αr(1) . If s ≤ 0, then (α (1) , α (2) ) is dominant and there is nothing to prove. Suppose that for (γ (1) , γ (2) ) with smaller s the corresponding sheaves are in H . We identify M(α (1) , α (2) , β) with its direct image p1∗ M(α (1) , α (2) , β) = L α(1) Qˆ ⊗ L α(2) (R) ⊗ Sym(Q ⊗ G ∗ ). Consider the subsheaf M<s (α (1) , α (2) , β) of M(α (1) , α (2) , β) consisting of all summands L γ (1) Q ⊗ L γ (2) R ⊗ Sδ G ∗ such that γ1(2) − γr(1) < s. It follows from the Littlewood–Richardson rule (2.3.4) that it is a Sym(Q ⊗ G ∗ ) submodule. Denote by Ms (α (1) , α (2) , β) the factor M(α (1) , α (2) , β)/ M<s (α (1) , α (2) , β). We have an exact sequence 0 → M<s (α (1) , α (2) , β) → M(α (1) , α (2) , β) → Ms (α (1) , α (2) , β) → 0. The support of all cohomology groups of the sheaf Ms (α (1) , α (2) , β) is contained in Yr −1 . Indeed, if we multiply the summand L γ (1) Q ⊗ L γ (2) R ⊗ Sδ G ∗ by r Q ⊗ r G ∗ corresponding to r × r minors, we add one to each entry of γ (1) , so s has to decrease. This means that the ideal of r × r minors annihilates all cohomology groups of Ms (α (1) , α (2) , β). Now Remark (6.5.9) and an induction on r imply that the class χ (Ms (α (1) , α (2) , β)) is in H . Consider the relative version of the GL(Q) × GL(R) × GL(G ∗ )equivariant free resolution of the sheaf M<s (α (1) , α (2) , β). Its i-th term Fi is a direct sum of sheaves M(γ (1) , γ (2) , β) and each term occurring in Fi has smaller s than (α (1) , α (2) ). Indeed, by induction on i and by the Littlewood– Richardson rule (2.3.4) it follows that on multiplying M(γ (1) , γ (2) , β) occurring in Fi by summands of Sym(Q ⊗ G ∗ ), the invariant s in all resulting summands can only decrease. We conclude that, by induction on s, the Euler characteristic χ(M<s (α (1) , (2) α , β)) is in H and Proposition (6.5.10) is proven. Proof of Theorem (6.5.4). Part (a) is a consequence of the fact that the characters of M(α, β)(−i) are linearly independent in Rep(GL(F) × GL(G ∗ ))

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[[q]][q −1 ]. To prove (b) we deﬁne the homomorphism of abelian groups : Rep(GL(F) × GL(G ∗ ))[q][q −1 ] → K 0 (Ar ) by sending the class [(L α F ⊗ L β G ∗ )q i ] to [M(α, β)(−i)]. By Theorem (6.5.3) the homomorphism is an epimorphism. It is also a monomorphism, because the characters of modules M(α, β)(−i) are linearly independent in Rep(GL(F) × GL(G ∗ ))[[q]][q −1 ] and therefore in K 0 (Ar ). Theorem (6.5.3) implies that for any graded GL(F) × GL(G ∗ )-equivariant Ar -module M the class [M] can be expressed as a linear combination of classes [M(α, β)(−i)]. In particular we can describe the class of Ar −1 . (6.5.11) Proposition. The class of Ar −1 in K 0 (Ar ) is given by the formula [Ar −1 ] = [Ar ] −

n−r

[M((i + 1, 1r −1 ), (1r +i , 0n−r −i ))].

i=0

Proof. Let A denote the sheaf of algebras Sym(Q ⊗ G ∗ ) over Grass(m − r, F). Consider the relative Eagon–Northcott complex over A, 0 → En−r → En−r −1 → . . . → E1 → E0 → A, where Ei = Di Q ⊗

r

Q⊗

n

G ∗ ⊗ A(−i − r ).

This is a sheaf resolution of a sheaf B of algebras. Using the relative version of the straightening law (3.2.5), we deduce that the sheaf B has a ﬁltration whose associated graded object is L λ Q ⊗ L λ G ∗ . [B] = λ=(λ1 ,...,λr −1 )

Kempf’s theorem (4.1.10) implies that the higher cohomology of B vanishes and that the sections of B are isomorphic to Ar −1 . We also have the identiﬁcation Ei = M((i + 1), 1r −1 ), (1r +i , 0n−r −i )). This concludes the proof. (6.5.12) Remarks. The relative Eagon–Northcott complex used in the proof of (6.5.11) and its sections is the simplest example of a degeneration sequence discussed in section 5.6.

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For the remainder of this section we assume that K is a ﬁeld of characteristic 0. We calculate the depth of modules M(α, β). Of course the depth of M(α, β) is independent of β, so we assume that β is zero and consider M(α) := M(α, (0)). We start with arbitrary (α (1) , α (2) ), not necessarily dominant. We assume only that the sheaf M(α (1) , α (2) ) has no higher cohomology. (6.5.13) Proposition. The sheaf M(α (1) , α (2) ) has no higher cohomology if (2) and only if αr(1) ≥ α1(2) − t where t is such that α1(2) = . . . = αt(2) > αt+1 . Proof. Let δ1 , . . . , δr be the numbers deﬁned before the statement of Proposition (6.5.6). The condition of Proposition (6.5.13) means that δr ≥ α1(2) , and that is equivalent to l(α) = 0. We want to ﬁnd the projective dimension of M(α (1) , α (2) ), because by the Auslander–Buchsbaum formula (1.2.7) we have depth( A, M) + pd A (M) = mn. The resolution of M(α (1) , α (2) ) is given by the complex F(L α(1) Q ⊗ L α(2) R)• , whose i-th term is given by i+ j j ∗ (1) (2) (R ⊗ G ) ⊗ A(−i − j). H Grass(m − r, F), L α Q ⊗ L α R ⊗ j≥0

In order to study the top of the resolution, we look ﬁrst at the last term of the Koszul complex. The corresponding GL(F)-weight is (2) + n). t(α) = (α1(1) , . . . , αr(1) , α1(2) + n, . . . , αm−r

Let’s look at the weight t(α) + ρ = (a1 , . . . , ar , b1 , . . . , bm−r ). For j = 1, . . . , m − r we deﬁne the sequences t j (α) inductively by setting t j (α) = (a1 , . . . , ar , d1 , . . . , d j , b j+1 , . . . , bm−r ), where d j+1 is deﬁned as d j+1 = max{t | t ≤ b j+1 , b < d j , t ∈ / {a1 , . . . , ar }}. Notice that in this deﬁnition the condition t < d j could be skipped, because each b j is essentially lowered to the ﬁrst possible number that is not one of

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the ai ’s or previous dK ’s. Let us also deﬁne numbers q j = n − (b j − dn ), p j = q j − #{i | ai < d j } for j = 1, . . . , m − r . (6.5.14) Lemma. For each j = 1, . . . , m − r we have p j ≥ n − r . Proof. Let us imagine that we construct the sequences t j (α) by the following process. We look at b j and start lowering it by 1 until we reach a number that is not equal to any ai or dK for k < j. Then every step of lowering by one accounts for some ai satisfying d j + 1 ≤ ai ≤ b j for some dK (which comes from a previous aK > b j ). This means that we have a set of b j − d j aK ’s which is disjoint from the set {i | ai < d j }. Therefore p j = n − (b j − d j ) − #{i | ai < d j } ≥ n − r. This concludes the proof of the lemma. (6.5.15) Theorem. Let K be a ﬁeld of characteristic zero. Let (α (1) , α (2) ) satisfy the condition of Proposition (6.5.13). Then the projective dimension

of M(α (1) , α (2) ) over A equals m−r j=1 p j . Proof. Let us decompose the terms of the complex F(L α(1) Q ⊗ L α(2) R)• using the Cauchy formula (2.3.3) and the Littlewood–Richardson rule (2.3.4). The GL(F) weights of all terms have the form (α (1) , δ (2) ) with δ (2) containing α (2) such that the difference between corresponding terms of δ (2) and α (2) does not exceed n. Also, all such weights occur. For all weights of this form we need to ﬁnd the supremum of the numbers |δ (2) | − |α (2) | − l(w) where w is a permutation ordering the weight (α (1) , δ (2) ) + ρ. It is clear tht the top number is obtained from the sequence tm−r (α). We need therefore to ﬁnd in which term F(L α(1) Q ⊗ L α(2) R)i the corresponding term occurs. The homo

geneous degree is m−r j=1 (n − (b j − d j )), and the length of the permutation w

m−r is j=1 #{i | ai < d j }. The statement of the theorem follows. (6.5.16) Corollary. Let K be a ﬁeld of characteristic zero. Assume that the weight (α (1) , α (2) ) satisﬁes the condition of Proposition (6.5.13). The depth of

(1) (2) M(α (1) , α (2) ) equals mn − m−r j=1 p j . The module M(α , α ) is a maximal Cohen–Macaulay module over Ar if and only if every number p j equals n − r.

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Let us look more closely at the weights giving maximal Cohen–Macaulay modules over Ar . Let us look at the process of getting sequences t j (α) by lowering the numbers b j to d j . At each stage we modify the sequence (a1 , . . . , ar ) ( j) ( j) as follows. We deﬁne inductively the sequences (a1 , . . . , ar ) by setting (a1(0) , . . . , ar(0) ) = (a1 , . . . , ar ), ( j−1) ( j−1) ( j) ( j−1) (a1 , . . . , aˆ i , . . . ar , d j ) if b j = ai , ( j) (a1 , . . . , ar( j) ) = ( j−1) ( j−1) ) otherwise. (a1 , . . . ar Now p j = n − r for j = 1, . . . , m − r if and only if we have b j ≥ ( j−1) ( j−1) } for each j = 1, . . . , m − r . Indeed, each ai either inmax{a1 , . . . ar duces a number between b j and d j + 1 or is smaller than d j , so the overall number by which the projective dimension decreases at the j-th stage is n − r . Stating the last result, we have (6.5.17) Corollary. Let (α (1) , α (2) ) satisfy the condition of Proposition ( j) ( j) (6.5.13). Deﬁne the sets (a1 , . . . ar ) as above. Then M(α (1) , α (2) ) is a maximal Cohen–Macaulay Ar -module if and only if for every j = 1, . . . , m − r we have ( j−1)

b j ≥ max{a1

, . . . ar( j−1) }.

We conclude the discussion of modules M(α) by showing an example of a module of type M(α) whose Cohen–Macaulay property depends on the characteristic of the ﬁeld K. (6.5.18) Example. Let us set r = 2, m = 3, n = 4. Consider the module M((2, 0, 0)). It consists of sections of the sheaf M = S2 Q ⊗ Sym(Q ⊗ G ∗ ) over Grass(1, F). The module M((2, 0, 0)) is maximal Cohen–Macaulay over ﬁelds of characteristic = 2, but in characteristic 2 it is not maximal Cohen– Macaulay. Proof. The higher cohomology groups of M(2, 0, 0) vanish. This follows from the Cauchy formula (3.2.5), the Kempf vanishing theorem (4.1.10), and the fact that the tensor product of Schur functors has a ﬁltration with factors isomorphic to Schur modules. (In characteristic 0 one could just use Bott’s theorem (4.1.9).)

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209

A minimal free resolution of M(2, 0, 0) is given by the complex F((2, 0, 0))• = S2 Q ⊗

•

(R ⊗ G ∗ ).

In this case dim R = 1, so for each i, 0 ≤ i ≤ 4, we deal with the cohomology of S2 Q ⊗ Si R ⊗ i G ∗ . In characteristic 0 we get the resolution 0 → L 3,2 F ⊗

3

G ∗ ⊗ A(−3) → L 3,1 F ⊗

2

G ∗ ⊗ A(−2) → S2 F ⊗ A.

In fact one can see that this complex is acyclic over all ﬁelds K with char K = 2. If char K = 2, then the bundle S2 Q ⊗ S4 R has nonzero cohomology. In fact H 1 (Grass(1, F), S2 Q ⊗ S4 R) = H 2 (Grass(1, F), S2 Q ⊗ S4 R) = L 3,3 F. Therefore the resolution of M((2, 0, 0)) over a ﬁeld of characteristic 2 is 0 → L 3,3 F ⊗ ⊗

3

4

G ∗ ⊗ A(−4) → L 3,3 F ⊗

G ∗ ⊗ A(−3) → L 3,1 F ⊗

2

4

G ∗ ⊗ A(−4) ⊕ L 3,2 F

G ∗ ⊗ A(−2) → S2 F ⊗ A.

We see that M((2, 0, 0)) is a maximal Cohen–Macaulay module over Ar over ﬁelds of characteristic = 2, but in characteristic 2 its depth drops by one. 6.6. Modules Supported in Symmetric Determinantal Varieties We preserve the notation from section 6.3. To construct a family of modules supported in Yrs we use the ﬁrst incidence variety from section 6.3. Let us ﬁx n and r . We take Y = Yrs and V = Grass(n − r, E) with the tautological sequence 0 → R → E × V → Q → 0. Here dim R = n − r and dim Q = q = r . For a subspace R in E, we denote by i the embedding of R into E. We consider the variety Z 1 = {(φ, R) ∈ X × V | φ i = 0}. As usual, we have the diagram Z1 ⊂ ↓ q Y ⊂

X×V ↓q X

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The variety Z 1 is the total space of the bundle S1 = S2 Q∗ . The bundle T1 is deﬁned by the exact sequence 0 → S2 Q∗ → S2 E ∗ × V → T1 → 0. Let Crs (E) be the category of graded Ars : = K[Yrs ]-modules with rational GL(E)-action compatible with the module structure, and equivariant degree 0 maps. We denote K 0 (Ars ) the Grothendieck group of the category Crs (E). For an equivariant graded module M ∈ Ob(Crs (E)) and for q ∈ Z, we denote by M(q) the module M with gradation shifted by q, i.e. M(q)n = Mq+n . For M ∈ Ob(Crs (E)) we deﬁne the graded character of M, char(M) = char(Mn ) q n ∈ Rep(GL(E))[[q]][q −1 ], n∈Z

where Rep(GL(E)) denotes the representation ring of GL(E), and char denotes the character map described in (2.2.10). Let α = (α1 , . . . , αn ) be an integral weight for GL(E). We set α (1) = (α1 , . . . , αr ), α (2) = (αr +1 , . . . , αn ). Let α = (α (1) , α (2) ). Assume that both α (1) and α (2) are dominant. We deﬁne a sheaf Ms (α) = p ∗ (L α(1) Q ⊗ L α(2) R) ⊗ O Z 1 of graded modules on Z 1 . We deﬁne the equivariant graded Ars -modules M s (α) = H 0 (Z 1 , Ms (α)). (6.6.1) Example. The simplest examples of modules M s (α) are provided by the pushdowns of line bundles on Grass(n − r, E). The line bundles on Grass(n − r, E) correspond to tensor powers OGrass(n−r,E) (m) = ( r Q)⊗m for m ∈ Z. Notice that the formula (5.1.4) for the dual bundle implies that the canonical module K Ars = M((r + 1)r , 0n−r ) ⊗ n F ⊗−r −1 . For m ≥ 0 the corresponding modules occur as a subset of a family from the next example. (6.6.2) Example. Let α (2) = (0). The sheaf Ms (α) equals L α(1) Q ⊗ O Z 1 . Therefore the direct image p∗ Ms (α) equals L α(1) Q ⊗ Sym(S2 Q). The higher cohomology of Ms (α) vanishes by Bott’s theorem (4.1.9). The j-th

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211

homogeneous component M sj (α) decomposes as follows: (L α(1) E ⊗ L µ E)≤r , |µ|= j, µi even for all i,µ1 ≤r

where (L α(1) E ⊗ L µ E)≤r denotes a factor of (L α(1) E ⊗ L µ E) consisting of all L ν E in the tensor product with ν1 ≤ r . The main results of this section are the analogues of the results from the previous section. (6.6.3) Theorem. The Grothendieck group K 0 (Ars ) is generated by the classes of the modules M s (α)(q), where α is a dominant weight and q ∈ Z. (6.6.4) Theorem. (a) The Grothendieck group K 0 (Ars ) is isomorphic to the additive subgroup of the ring Rep(GL(E))[[q]][q −1 ] generated by shifted graded characters of modules M s (α) for α dominant. (b) The group K 0 (Ars ) is isomorphic to the additive group of the ring Rep(GL(E))[q][q −1 ]. The proof of Theorems (6.6.3) and (6.6.4) follows the same steps as the proof of Theorems (6.5.3) and (6.5.4). We only give the statements, leaving the detais as an exercise to the reader. (6.6.5) Proposition. Let α be a dominant weight for GL(E). Then H i (Z 1 , Ms (α)) = 0 for i > 0. Proof. This is an analogue of (6.5.5). Let α be a weight with α (1) , α (2) dominant. We deﬁne the number l(α) as follows. Consider the weight α + ρ, where ρ = (n − 1, n − 2, . . . , 1, 0) = (u 1 , . . . , u n ). Deﬁne, by reverse induction on s (from s = r to s = 1), the numbers δs = min{t | t ≥ δs+1 , t +n−s ∈ / {δs+1 + n − s − 1, . . . , δr + n − r, αr +1 + n − r − 1, . . . , αn }}. By construction the weight (δ1 , . . . , δr , αr +1 , . . . , αn ) + ρ is not orthogonal to any root. By (4.1.9) there exists a unique l such that H l (Grass (n − r, E), K δ Q ⊗ K α(2) R) = 0. We deﬁne l(α) := l.

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(6.6.6) Proposition. Let K be a ﬁeld of characteristic zero. Let α = (α (1) , α (2) ) be a weight for GL(F) with α (1) , α (2) dominant. Let l(α) be deﬁned as above. Then H i (Z 1 , Ms (α)) = 0 for i > l(α). Proof. This is an analogue of the ﬁrst statement of (6.5.6). Notice that the analogue of the second statement need not be true. Each equivariant sheaf Ms on Z 1 has its Euler characteristic class (−1)i [H i (Z 1 , Ms )] ∈ K 0 (Ars ). χ (Ms ) = i≥0

(6.6.7) Proposition. The group K 0 (Ars ) is generated by the Euler characteristic classes χ(Ms (α)) for (α (1) , α (2) ) dominant. Proof. This is an analogue of (6.5.8). For the proof one has to use the same type of induction, using the ﬂag variety Flag(n − r, n − r + 1; E). (6.6.8) Remarks. Notice that, as in the previous section, we can also show ˆ s (α (1) , α (2) ) that if (α (1) , α (2) ) is dominant, then the Euler characteristic of M s is in the subgroup of K 0 (Ar ) spanned by the Euler characteristics of sheaves Ms (α, β) with α dominant. Theorem (6.6.3) follows from the following statement: (6.6.9) Proposition. The classes χ (Ms (α (1) , α (2) ) such that (α (1) , α (2) ) is dominant generate the group K 0 (Ars ). Proof. The proof is analogous to (6.5.10). We proceed by induction on s := s(α (1) , α (2) ) = α1(2) − αr(1) . Proof of Theorem (6.6.4). Part (a) is a consequence of the fact that the characters of M s (α)(−i) are linearly independent in Rep(GL(E))[[q]][q −1 ]. To prove (b) we deﬁne the homomorphism of abelian groups : Rep(GL(E))[q][q −1 ] → K 0 (Ars ) by sending the class [(L α E)q i ] to [M s (α)(−i)]. By Theorem (6.6.3) the homomorphism is an epimorphism. It is also a monomorphism, because the characters of modules M s (α)(−i) are linearly independent in Rep(GL(E)) [[q]][q −1 ] and therefore in K 0 (Ars ).

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We ﬁnish this section with a criterion for a module M s (α) to be maximal Cohen–Macaulay. We use Corollary (5.1.5). Before we start, let us work out the duality statement for the variety Z 1 . To this end we need to calculate the bundle m ∗ ξ ⊗ K , where m = rank ξ . Using (3.3.5), we get K V = ( r Q)−n+r n−r Vr m ∗ r (r +1)/2 n(n+1)/2 ⊗( R) . Similarly, ξ = S2 Q ⊗ S2 F ∗ . Putting these facts together, we conclude that the dualizing bundle is −n+r +1 r ∗ n . ω Z 1 = K (−n−1+r ) F ⊗ Q Therefore, for the weight (2) ), α = (α1(1) , . . . , αr(1) , α1(2) , αn−r

the dual weight is given by the formula (2) , . . . , −α1(2) ). α ∨ = (−n + r + 1 − αr(1) , . . . − n + r + 1 − α1(1) , −αn−r

(6.6.10) Proposition. The module M s (α) is maximal Cohen–Macaulay provided that l(α) = l(α ∨ ) = 0. Proof. Let’s denote V(α) = K α(1) Q ⊗ K α(2) R. One applies Corrolary (5.1.5) to the complexes F(V(α))• and F(V(α ∨ ))• . The vanishing of the higher cohomology is assured by Proposition (6.6.6). (6.6.11) Example. Let us assume that α (2) = 0. Then M s (α) has nonvanishing higher cohomology by Proposition (6.6.6). The condition l(α ∨ ) = 0 is true when the inequality −n + r + 1 − α1(1) ≥ −n + r is satisﬁed, i.e. when α1(1) ≤ 1. This proves that the modules u s u 0 M (1 ) = H Grass(n − r, F), Q ⊗ Sym(S2 Q) are maximal Cohen–Macaulay for 0 ≤ u ≤ r .

6.7. Modules Supported in Skew Symmetric Determinantal Varieties We preserve the notation of section 6.4. To construct a family of modules supported in Yra we use the ﬁrst incidence variety from section 6.4.

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Let us ﬁx n and r , with r even. We take Y = Yra and V = Grass(n − r, E) with the tautological sequence 0 → R → E × V → Q → 0. Here dim R = n − r and dim Q = q = r . For a subspace R in E we denote by i the embedding of R into E. We consider the variety Z 1 = { (φ, R) ∈ X × V | φ i = 0 }. Let Cra (E) be the category of graded Ara : = K[Yra ]-modules with rational GL(E)-action compatible with the module structure, and equivariant degree 0 maps. We denote by K 0 (Ara ) the Grothendieck group of the category Cra (E). For an equivariant graded module M ∈ Ob(Cra (E)) and for q ∈ Z we denote by M(q) the module M with gradation shifted by q, i.e. M(q)n = Mq+n . For M ∈ Ob(Cra (E)) we deﬁne the graded character of M, char(M) = char(Mn ) q n ∈ Rep(GL(E))[[q]][q −1 ], n∈Z

where Rep(GL(E)) denotes the representation ring of GL(E) and char denotes the character map deﬁned in (2.2.10). Let α = (α1 , . . . , αn ) be an integral weight for GL(E). We set α (1) = (α1 , . . . , αr ), α (2) = (αr +1 , . . . , αn ). Let α = (α (1) , α (2) ). Assume that both α (1) and α (2) are dominant. We deﬁne a sheaf Ma (α) = p ∗ (L α(1) Q ⊗ L α(2) R) ⊗ O Z 1 of graded modules on Z 1 . We deﬁne the equivariant graded Ara -modules M a (α) = H 0 (Z 1 , Ma (α)). (6.7.1) Example. The simplest examples of modules M a (α) are provided by the pushdowns of line bundles on Grass(n − r, E). The line bundles on Grass(n − r, E) correspond to tensor powers OGrass(n−r,E) (m) = ( r Q)⊗m for m ∈ Z. For m ≥ 0 the corresponding modules occur as a subset of a family from the next example. (6.7.2) Example. Let α (2) = (0). The sheaf Ma (α) equals L α(1) Q ⊗ O Z 1 . Therefore the direct image p∗ Ma (α) equals L α(1) Q ⊗ Sym(S2 Q). The higher cohomology of Ma (α) vanishes by Bott’s theorem (4.1.9). The j-th

6.7. Modules Supported in Skew Symmetric Determinantal Varieties 215

homogeneous component M aj (α) decomposes as follows: (L α(1) E ⊗ L µ E)≤r , |µ|= j, µi even for all i, µ1 ≤r

where (L α(1) E ⊗ L µ E)≤r denotes a factor of (L α(1) E ⊗ L µ E) consisting of all L ν E in the tensor product with ν1 ≤ r . (6.7.3) Theorem. The Grothendieck group K 0 (Ara ) is generated by the classes of the modules M a (α)(q), where α is a dominant weight and q ∈ Z. (6.7.4) Theorem. (a) The Grothendieck group K 0 (Ara ) is isomorphic to the additive subgroup of the ring Rep(GL(E))[[q]][q −1 ] generated by shifted graded characters of modules M a (α) for α dominant. (b) The group K 0 (Ara ) is isomorphic to the additive group of the ring Rep(GL(E))[q][q −1 ]. Again the proofs of (6.7.3) and (6.7.4) follow the same steps as the proofs of (6.5.3) and (6.5.4). We just give the statements, leaving the proofs to the reader. (6.7.5) Proposition. Let α be a dominant weight for GL(E). Then H i (Z 1 , Ma (α)) = 0 for i > 0. Proof. This is the analogue of (6.5.5). Let α be a weight with α (1) , α (2) dominant. We deﬁne the number l(α) as follows. Consider the weight α + ρ, where ρ = (n − 1, n − 2, . . . , 1, 0) = (u 1 , . . . , u n ). Deﬁne, by reverse induction on s (from s = r to s = 1) the numbers δs = min{t | t ≥ δs+1 , t +n−s ∈ / {δs+1 + n − s − 1, . . . , δr + n − r, αr +1 + n − r − 1, . . . , αn }}. By construction the weight (δ1 , . . . , δr , αr +1 , . . . , αn ) + ρ is not orthogonal to any root. By (4.1.9) there exists a unique l such that H l (Grass(n − r, E), K δ Q ⊗ K α(2) R) = 0. We deﬁne l(α) := l. (6.7.6) Proposition. Let K be a ﬁeld of characteristic zero. Let α = (α (1) , α (2) ) be a weight for GL(F) with α (1) , α (2) dominant. Let l(α) be deﬁned as above.

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Then H i (Z 1 , Ma (α)) = 0

for i > l(α).

Proof. This is the analogue of the ﬁrst statement of (6.5.6). Notice that the analogue of the second statement need not be true. Each equivariant sheaf Ma on Z 1 has its Euler characteristic class (−1)i [H i (Z 1 , Ma )] ∈ K 0 (Ara ). χ (Ma ) = i≥0

(6.7.7) Proposition. The group K 0 (Ara ) is generated by the Euler characteristic classes χ(Ma (α)) for (α (1) , α (2) dominant). Proof. This is the analogue of (6.5.8). We use a similar induction, using the ﬂag variety Flag(n − r, n − r + 2; E). (6.7.8) Remarks. Notice that, as in section 6.5, we can also show that if ˆ a (α (1) , α (2) ) is (α (1) , α (2) ) is dominant, then the Euler characteristic of M a in the subgroup of K 0 (Ar ) spanned by the Euler characteristics of sheaves Ma (α, β) with α dominant. Theorem (6.7.3) follows from the following statement (6.7.9) Proposition. The classes χ (Ma (α (1) , α (2) ) such that (α (1) , α (2) ) is dominant generate the group K 0 (Ara ). Proof. This is the analogue of (6.5.10). Again we induct on s := s(α (1) , α (2) ) = α1(2) − αr(1) . Proof of Theorem (6.7.4). Part (a) is a consequence of the fact that the characters of M a (α)(−i) are linearly independent in Rep(GL(E))[[q]][q −1 ]. To prove (b) we deﬁne the homomorphism of abelian groups : Rep(GL(E))[q][q −1 ] → K 0 (Ara ) by sending the class [(L α E)q i ] to [M a (α)(−i)]. By Theorem (6.7.3) the homomorphism is an epimorphism. It is also a monomorphism, because

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the characters of modules M a (α)(−i) are linearly independent in Rep(GL(E)) [[q]][q −1 ] and therefore in K 0 (Ara ). Next we give a criterion for a module M a (α) to be maximal Cohen– Macaulay. We use Corollary (5.1.5). Before we start, let us work out the duality statement for the variety Z 1 . To this end we need to calculate the bundle m ∗ ξ ⊗ K where m = rank ξ . Using (3.3.5), we get K V = ( r Q)−n+r n−r Vr m ∗ r (r −1)/2 2 n(n−1)/2 2 ∗ ⊗( R) . Similarly, ξ = S Q⊗ F . Putting these facts together, we conclude that the dualizing bundle is ωZ1

−n+r −1 r = K (−n+1+r )n F ⊗ . Q ∗

Therefore for the weight (2) α = (α1(1) , . . . , αr(1) , α1(2) , αn−r ),

the dual weight is given by the formula (2) , . . . , −α1(2) ). α ∨ = (−n + r − 1 − αr(1) , . . . − n + r − 1 − α1(1) , −αn−r

(6.7.10) Proposition. The module M a (α) is maximal Cohen–Macaulay provided that l(α) = l(α ∨ ) = 0. Proof. Let’s denote V(α) = K α(1) Q ⊗ K α(2) R. One applies Corrolary (5.1.5) to the complexes F• (V(α)) and F• (V(α ∨ )). The vanishing of the higher cohomology is assured by Proposition (6.7.6). (6.7.11) Example. Let us assume that α (2) = 0. Then M a (α) has nonvanishing higher cohomology by Proposition (6.6.6). Let us also assume α1(1) ≤ 1. This means we consider the modules v 2 Q ⊗ Sym Q . M a (1v ) = H 0 Grass(n − r, F), Then condition l(α ∨ ) = 0 is equivalent to v ≥ 3 (which implies r = 2u ≥ 4). This proves that the modules M a (1v ) are maximal Cohen–Macaulay for 3 ≤ v ≤ r.

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Exercises for Chapter 6 Analogues of Determinantal Varieties for Other Classical Groups The Symplectic Group Let F be a symplectic space of dimension 2n, let G be a vector space, dim G = m. Consider X = HomK (F, G). This space can be identiﬁed with the set of m-tuples of vectors from F. 1.

For 1 ≤ r ≤ n deﬁne Yr = {φ ∈ X | ∃R ∈ IGrass(r, F), φ(R) = 0 }. Prove that Yr has a resolution of singularities which is a total space of a vector bundle over IGrass(r, F)

2.

Prove, by using Theorem (5.1.2)(b), that in that case Yn is normal and has rational singularities.

3.

Let m ≤ n. Calculate the complex F• , and use it to show m that the variety Yn is a complete intersection given by the vanishing of 2 Sp(F)-invariants of degree 2, given by the representation 2 G ∗ .

4.

Use exercises 2 and 3 to show that there exists an acyclic complex K (n, λ; F)• resolving the irreducible representation Vλ F of the group Sp(F) in terms of Schur functors, with the i-th term K λ/µ F. K (n, λ; F)i := µ∈Q −1 (2i)

5.

Let r = n, dim G = m > n. Calculate the terms of the complex F• . Prove in a determinantal variety that the variety Yn is a complete intersection m of matrices of rank ≤ n, given by 2 Sp(F)-invariants.

6.

Let r < n. Prove that Yr is normal and has rational singularities. Assume that m ≤ 2n − r + 1. Then the terms of F• contain only trivial Sp(F)representations. More precisely, F• is a specialization of the resolution of 2(n − r + 1) Pfafﬁans of the generic skew symmetric m × m matrix (described in section 6.3), where our m × m skew symmetric matrix is a matrix of Sp(F)-invariants in A = Sym(F ⊗ G ∗ ), given by the repre sentation 2 G ∗ ⊂ A2 .

7. For the example 2n = 6, r = 2, m = 6 calculate the terms of the complex F• and prove that Yr is not a complete intersection in a determinantal variety.

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8. Prove that the deﬁning ideal of Yr is generated by 2(n − r + 1) Pfafﬁans of the skew symmetric m × m matrix of Sp(F)-invariants in A = Sym(F ⊗ G ∗ ), given by the representation 2 G ∗ ⊂ A2 . 9. Let 1 ≤ r < n. Deﬁne the variety Y2n−r = {φ ∈ X | ∃R ∈ IGrass(r, F), φ(R ∨ ) = 0}. Prove that Y2n−r has a resolution of singularities which is a total space of a vector bundle over IGrass(r, F). Prove that Y2n−r is normal , with rational singularities. 10. Prove that the deﬁning ideal of Y2n−r is generated by Sp(F)-invariants in A2 (given by the representation 2 G ∗ ) and the (r + 1) × (r + 1) minors of the matrix φ. The Orthogonal Group We will formulate the exercises for the even orthogonal group. The formulations for the odd orthogonal group are left to the reader. Let F be an orthogonal space of dimension 2n; let G be a vector space, dim G = m. Consider X = HomK (F, G). This space can be identiﬁed with the set of m-tuples of vectors from F. 11. For 1 ≤ r ≤ n let Yr = {φ ∈ X | ∃R ∈ IGrass(r, F), φ(R) = 0 }. Prove that Yr has a resolution of singularities which is a total space of a vector bundle over IGrass(r, F). (Hint: ξ = R ⊗ G ∗ .) 12. Prove, by using Theorem (5.1.2)(b), that in that case Yn is normal and has rational singularities. 13. Let m ≤ n. Calculate the complex F• to show that the variety Yn is a complete intersection given by the vanishing of m+1 O(F)-invariants 2 ∗ of degree 2, given by the representation S2 G . 14. Let λ be a dominant weight for SO(F) with integer coordinates. Use exercises 12 and 13 to show that there exists an acyclic complex K (n, λ; F)• resolving the irreducible representation Vλ F of the group SO(F) in terms of Schur functors, with the i-th term K λ/µ F. K (n, λ; F)i := µ∈Q 1 (2i)

15. Let r = n, dim G = m > n. Calculate the terms of the complex F• . Prove that the variety Yn is a complete intersection in a determinantal variety SO(F)-invariants. of matrices of rank ≤ n, given by n+1 2

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16. Let r < n. Prove that Yr is not normal, but that its normalization has rational singularities. Assume that m ≤ 2n − r . Then the terms of F• contain only trivial SO(F)-representations. 17. For the example 2n = 8, r = 2, m = 6 calculate the terms of the complex F• . 18. Prove that the deﬁning ideal of Yr is generated by (n − r + 1) × (n − r + 1) minors of the symmetric m × m matrix of SO(F)-invariants in A = Sym(F ⊗ G ∗ ), given by the representation S2 G ∗ ⊂ A2 . 19. Let 1 ≤ r ≤ n. Deﬁne the variety Y2n−r = { φ ∈ X | ∃R ∈ IGrass(r, F), φ(R ∨ ) = 0 }. Prove that Y2n−r has a resolution of singularities which is a total space of a vector bundle over IGrass(r, F). (Hint: ξ = R∨ ⊗ G ∗ .) Prove that Y2n−r is normal, with rational singularities. 20. Prove that the deﬁning ideal of Y2n−r is generated by O(F)-invariants in A2 (given by the representation S2 G ∗ ) and the (r + 1) × (r + 1) minors of the matrix φ. The First Fundamental Theorem for the General Linear Group Let E be a vector space of dimension n. Let X = E ⊗m ⊕ E ∗⊗ p . We identify X with HomK (G, E) ⊕ HomK (E, H ) where G, H are two vector spaces, dim G = m, dim H = p. We have A = Sym(G ⊗ E ∗ ⊕ E ⊗ H ∗ ). We can identify X with the set of m-tuples of vectors from E and p-tuples of covectors from E ∗ . 21. For each pair (r, s), r + s = n, r ≤ m, s ≤ p, consider the variety Yr,s = {(φ, ψ) ∈ X | ψφ = 0, rank φ ≤ r, rank ψ ≤ s }. Prove that Yr,s has a desingularization Z r,s with V = Grass(r, E), ξ = G ⊗ Q∗ ⊕ R ⊗ H ∗ . Prove that Yr,s is normal and has rational singularities. 22. Choose m + p = n, r = m, s = p. Prove that in this case the complex F• is a Koszul complex on the GL(E)-invariants in A of bidegree (1, 1) which correspond to the representation G ⊗ H ∗ . 23. Let λ = (λ1 , . . . , λr ), µ = (µ1 , . . . , µs ) be two partitions. Take the isotypic component of type K λ G ⊗ K µ H ∗ to obtain the complex K (r, s, λ, µ; E)• with the following properties: (a) K (r, s, λ, µ; E)• is acyclic.

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(b) The i-th term of K (r, s, λ, µ; E)• is K (r, s, λ, µ; E)i = K λ/ν E ∗ ⊗ K µ/ν E. |ν|=i

(c) The complex K (r, s, λ, µ; E)• resolves the representation K (µ,−λ) E, where (µ, −λ) = (µ1 , . . . , µs , −λr , . . . , −λ1 ). Isotropic Grassmannians Revisited 24. Prove that the equations of isotropic Grassmannians deﬁned in exercises 1, 2, 3 in chapter 4 generate (together with the Pl¨ucker relations) the deﬁning ideals of the cones over isotropic Grassmannians. Differentials in the Resolutions of Ideals of Minors of a Generic Matrix 25. We work with the notation of section 6.1. Recall that by (6.1.3) the i-th term in the Lascoux complex equals Fi = L P1 (α,β) F ⊗ L P2 (α,β) G ∗ ⊗K A. s≥0

(α,β)∈Q(s), i=s 2 +|α|+|β|

Denote Fi(s) =

(α,β)∈Q(s),

L P1 (α,β) F ⊗ L P2 (α,β) G ∗ ⊗K A,

i=s 2 +|α|+|β|

so Fi = s≥0 Fi(s) . Prove that the differential di : Fi → Fi−1 has only (s) and components of degree components of degree 1 taking Fi(s) to Fi−1 (s) (s−1) r + 1 taking Fi to Fi−1 . 26. Prove that the only possible nonzero components of the degree 1 part of (s) restricted to L P1 (α,β) F ⊗ L P2 (α,β) G ∗ the differential di1,s : Fi(s) → Fi−1 ⊗K A go to the terms L P1 (γ ,δ) F ⊗ L P2 (γ ,δ) G ∗ ⊗K A with γ ⊂ α, δ ⊂ β, |α/γ | = 1, |β/δ| = 1. 27. Prove that the only possible nonzero components of the degree r + 1 part (s−1) of the differential dir +1,s : Fi(s) → Fi−1 are as follows. The map dir +1,s restricted to the term L P1 (α,β) F ⊗ L P2 (α,β) G ∗ ⊗K A is zero unless αs = 0, α1 ≤ s − 1, βs = 0, β1 ≤ s − 1. If these conditions are satisﬁed, the only nonzero component of d r +1,s restricted to L P1 (α,β) F ⊗ L P2 (α,β) G ∗ ⊗K A (s−1) , where γ = (α1 + 1, . . . , goes to L P1 (γ ,δ) F ⊗ L P2 (γ ,δ) G ∗ ⊗K A from Fi−1 αs−1 + 1), δ = (β1 + 1, . . . , βs−1 + 1). The coefﬁcients of that component are the linear combinations of (r + 1) × (r + 1) minors of .

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Differentials in the Resolutions of Ideals of Minors of a Generic Symmetric Matrix 28. We work with the notation of section 6.3. By Theorem (6.3.1) the i-th term G i of the resolution of the ideal of (r + 1) × (r + 1) minors of a generic symmetric matrix is given by Gi = L λ(α,u) E ⊗K As . u≥0

λ(α,u); i=|α|+2u 2 −u

Denote

G i(u) = λ(α,u);

L λ(α,u) E ⊗K As ,

i=|α|+2u 2 −u

so G i = u≥0 G i(u) . Prove that the differential di : G i → G i−1 has only (u) components of degree 1 taking G i(u) to G i−1 and components of degree (u) (u−1) r + 1 taking G i to G i−1 . 29. Prove that the only possible nonzero components of the degree 1 part of (u) the differential di1,u : G i(u) → G i−1 restricted to L λ(α,u) ⊗K As go to the s terms L λ(β,u) ⊗K A with β ⊂ α, |α/β| = 1. 30. Prove that the only possible nonzero components of the degree r + 1 part (u−1) of the differential dir +1,u : G i(u) → G i−1 are as follows. The map dir +1,u s restricted to the term L λ(α,u) ⊗K A is zero unless α2u−1 = α2u = 0, α1 ≤ 2u − 1. If these conditions are satisﬁed, the only nonzero component of d r +1,u restricted to L λ(α,u) ⊗K As goes to L λ(β,u−1) ⊗K As , where β = (α1 + 2, . . . , α2u−2 + 2). The coefﬁcients of that component are the linear combinations of (r + 1) × (r + 1) minors of .

Differentials in the Resolutions of Ideals of Pfafﬁans of a Generic Skew Symmetric Matrix 31. We work with the notation of section 6.4. By Theoerm (6.4.1) the i-th term G i of the resolution of the ideal of (2r + 2) × (2r + 2) Pfafﬁans of a generic skew symmetric matrix is given by Gi = L λ(α,v) E ⊗K Aa . v≥0

Denote G i(v) =

λ(α,v); i=|α|+ 12 v(v+1)

λ(α,v); i=|α|+ 12 v(v+1)

L λ(α,v) E ⊗K Aa ,

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223

so G i = v≥0 G i(v) . Prove that the differential di : G i → G i−1 has only (v) components of degree 1 taking G i(v) to G i−1 and components of degree (v) (v−1) r + 1 taking G i to G i−1 . 32. Prove that the only possible nonzero components of the degree 1 part of (v) the differential di1,v : G i(v) → G i−1 restricted to L λ(α,v) ⊗K Aa go to the a terms L λ(β,v) ⊗K A with β ⊂ α, |α/β| = 1. 33. Prove that the only possible nonzero components of the degree r + 1 (v−1) part of the differential dir +1,v : G i(v) → G i−1 are as follows. The map r +1,v a restricted to the term L λ(α,v) ⊗K A is zero unless αv = 0, α1 ≤ di v − 1. If these conditions are satisﬁed, the only nonzero component of d r +1,v restricted to L λ(α,v) ⊗K Aa goes to L λ(β,v−1) ⊗K Aa , where β = (α1 + 1, . . . , αv−1 + 1). The coefﬁcients of that component are the linear combinations of (2r + 2) × (2r + 2) Pfafﬁans of . Maximal Cohen–Macaulay Modules with Linear Resolutions 34. Consider the twisted sheaf M((n − r )r , (0)) = K (n−r )r Q ⊗ O Z r(1) deﬁned in section 6.5. (a) Prove that the sheaf M((n − r )r , (0)) has no higher cohomology, so the twisted complex F(K (n−r )r Q)• provides a minimal resolution of M((n − r )r , (0)). (b) Show that the complex F(K (n−r )r Q)• has length (m − r )(n − r ) and that it has a linear differential. More precisely, the only nonvanishing terms of the complex F(K (n−r )r Q)• are i 0 F(K (n−r )r Q)i = H Grass(m − r, F), K (n−r )r Q ⊗ ξ for 0 ≤ i ≤ (m − r )(n − r ). (c) Use the duality from exercise 18, chapter 2, changing the Schur functors on F to the Schur functors on F ∗ , to identify the complex F(K (n−r )r Q)• with the Schur complex L (m−r )(n−r ) (∗ ). (d) Use Buchsbaum–Eisenbud acyclicity criterion (1.2.12) to prove that over a ﬁeld K of arbitrary characteristic the Schur complex L (m−r )(n−r ) (∗ ) is acyclic. Note that the Euler characteristic of this complex in the relative situation (i.e. when F, G are replaced by vector bundles over some scheme) gives the class occurring in the Porteous formula for the cohomology class of the degeneracy locus X r . The module M((n − r )r , (0)) is a maximal Cohen–Macaulay module supported in X r with a linear resolution.

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35. Consider the twisted sheaf M((n − r )r , n − r − 1, n − r − 2, . . . , 1, 0) deﬁned in section 6.6. It is supported in the symmetric determinantal variety X rs of symmetric n × n matrices of rank ≤ r . (a) Prove that the sheaf M((n − r )r , n − r − 1, n − r − 2, . . . , 1, 0) has no higher cohomology, so the twisted complex F(K (n−r )r Q ⊗ K (n−r −1,n−r −2,...,1,0) R)• provides a minimal resolution of M((n − r )r , n − r − 1, n − r − 2, . . . , 1, 0). (b) Show that the complex F(K (n−r )r Q ⊗ K (n−r −1,n−r −2,...,1,0) R)• has length n−r2+1 and that it has a linear differential. More precisely, the only nonvanishing terms of the complex F(K (n−r )r Q ⊗ K (n−r −1,n−r −2,...,1,0) R)• are i 0 H Grass(n − r, F), K (n−r )r Q ⊗ K (n−r −1,n−r −2,...,1,0) R ⊗ ξ . for 0 ≤ i ≤ n−r2+1 . Conclude that M((n − r )r , n − r − 1, n − r − 2, . . . , 1, 0) is a maximal Cohen–Macaulay module supported in X rs with a linear resolution. 36. Consider the twisted sheaf M((n − r − 1)r , n − r − 1, n − r − 2, . . . , 1, 0) deﬁned in section 6.7. It is supported in the skew symmetric determinantal variety X ra of skew symmetric n × n matrices of rank ≤ r . Here we assume that r = 2u is even. (a) Prove that the sheaf M((n − r − 1)r , n − r − 1, n − r − 2, . . . , 1, 0) has no higher cohomology, so the twisted complex F(K (n−r −1)r Q ⊗ K (n−r −1,n−r −2,...,1,0) R)• provides a minimal resolution of M((n − r − 1)r , n − r − 1, n − r − 2, . . . , 1, 0). (b) Show that the complex F(K (n−r −1)r Q ⊗ K (n−r −1,n−r −2,...,1,0) R)• has length n−r and that it has a linear differential. More precisely, 2 the only nonvanishing terms of the complex F(K (n−r −1)r Q ⊗ K (n−r −1,n−r −2,...,1,0) R)• are i H 0 Grass(n − r, F), K (n−r −1)r Q ⊗ K (n−r −1,n−r −2,...,1,0) R) ⊗ ξ for 0 ≤ i ≤ n−r . Conclude that M((n − r − 1)r , n − r − 1, n − r − 2 2, . . . , 1, 0) is a maximal Cohen–Macaulay module supported in X ra with a linear resolution. Resolutions of K λ (Φ) 37. Consider the Grassmannian Grass(m − n, F). Consider the incidence variety Z m = {(φ, R) ∈ X × Grass(m − r, F) | φ | R = 0}.

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225

Denote by ˆ → F × Grass(m − n, F) → Qˆ → 0 0→R the tautological sequence on Grass(m − n, F). Deﬁne X (λ) := K λ Qˆ ⊗ O Z m The corresponding modules of sections are X (λ) := H 0 (Z m , K λ Qˆ ⊗ O Z m ). (a) Prove that H i (Z m , X (λ)) = 0 for i > 0, (b) Prove that p∗ X (λ) := K λ Qˆ ⊗ Sym(Qˆ ⊗ G ∗ ) with Ri p∗ X (λ) = 0 for i > 0. ˆ • gives a minimal free resolution of the (c) Conclude that F(K λ (Q)) module X (λ). (d) Assume that the partition λ has i boxes on the diagonal, i.e. λi+1 ≤ i ≤ λi . Let us also assume that λ has exactly s nonzero parts. Show that the module X (λ) has a minimal presentation K (λ,1n+1−s ) F ⊗

n+1−s

G ∗ ⊗K A(−n − 1 + s)

→ K λ F ⊗K A → X (λ) → 0. (e) Show that the projective dimension of X (λ) is equal to (m − n)i. ˆ • can be aug(f) Assume that i = 1. Show that the complex F(K λ Q) ∗ mented by one more map K λ F ⊗K A → K λ G ⊗K A(|λ|) to get a longer minimal free resolution. In other words, the module X (λ) turns out to be the ﬁrst syzygy of the module C(λ) := Coker(K λ ()). The modules C(λ) are therefore perfect modules supported in In (φ). 38. Let i = 1. Write λ = (q, 1 p−1 ). (a) Use the formula (2.32)(b) and exercise 5 of chapter 4 to prove that the argument from exercise 37 can be made characteristic free. Writing λ = (q, 1 p−1 ), the terms at the right end of the characteristic free version of the resolution of C(λ) (deﬁned as the cokernel of L ( p,1q−1 ) (φ)) are as follows: F0 = L ( p,1q−1 ) G ⊗K A( p + q − 1), F1 = L ( p,1q−1 ) F ⊗K A, Fi = L (i+n−1,1q−1 ) F ⊗ (L (n− p+1,1i−2 ) G)∗ ⊗K A(−n + p + i − 1) for 2 ≤ i ≤ m − n + 1,

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(b) Use exercise 17 of chapter 2 to describe the differential in the resolution of C( p, 1q−1 ) explicitly. Prove that C( p, 1q−1 ) is a perfect module.This resolution was ﬁrst described in [BE2]. The free resolutions of the modules C(λ) for partitions with more boxes on the diagonal were analyzed in [Ar1] and [Ar2]. None of these modules are perfect. 39. Let λ = (k n ). (a) Prove that the module X (λ) ⊗K ( n G ∗ )⊗k is isomorphic (as an equivariant GL(F) × GL(G ∗ )-module) to the k-th power of the ideal In (φ) of maximal minors of φ. The projective dimension of In (φ)k equals (m − n) min(k, n). ˆ • are characteristic free and linear (b) Prove that the complexes F(L λ Q) when there are no terms coming from higher cohomology, i.e. when λi+1 = . . . = λn = i. Resolutions of Powers of the Ideal of 2t × 2t Pfafﬁans of a (2t + 1) × (2t + 1) Skew Symmetric Matrix 40. Take dim E = 2t + 1, V = Grass(1, E) with the tautological sequence ˆ → E × V → Qˆ → 0. 0→R Deﬁne Z 2n = {(φ, R) ∈ X × V | φ |R = 0}, ˆ into E × V . Denote by p, q, q the where i denotes the embedding of R usual projections. Then Xλa = K λ Qˆ ⊗ O Z 2n on Z 2n , where λ = (λ1 , . . . , λ2t ) is a partition into at most 2t parts. Denote X a (λ) = H 0 (Z 2t , X a (λ)). (a) H i (Z 2t , X a (λ)) = 0 for i > 0. ˆ with Ri p∗ X a (λ) = 0 for i > 0. (b) p∗ X a (λ) := K λ Qˆ ⊗ Sym( 2 Q) ˆ • gives a minimal free resolution of the module (c) The complex F(K λ (Q) a X (λ). (d) Let λ = (k 2t ). Show that the module X a (λ) = I2ta (φ)k has a linear free resolution for k even and has one dimensional representation outside the linear strand for k odd, k < 2t + 1. (e) Show k for k even, 2t pd A (X (k )) = k + 1 for k odd.

Exercises for Chapter 6

227

More precisely, the terms in the resolution are Fi = L (k 2t ,i) F ⊗K A(−i) for 0 ≤ i ≤ min(k, 2t), with the additional term k + 1 + 2t Fk+1 = L ((k+1)2t+1 ) F ⊗K A − 2 occurring for k odd, k < 2t. (f) Show, using Kempf’s vanishing theorem and exercise 5 of chapter 4, that the Betti numbers of powers of I2ta (φ)k do not depend on the characteristic of K. The differentials in these resolutions were described explicitly in [BS] and [KU].

7 Higher Rank Varieties

In this chapter we investigate the higher rank varieties. They are the analogues of determinantal varieties for more complicated representations L λ E. They were ﬁrst considered in the paper [Po] of Porras. In section 7.1 we look at the general case. We prove that higher determinantal varieties have rational singularities, and we ﬁnd equations deﬁning them set-theoretically. We also classify the rank varieties whose deﬁning ideals are Gorenstein. In section 7.2 we investigate the rank varieties for symmetric tensors of degree bigger than two. We prove that in this case the deﬁning equations described in section 7.1 generate the radical ideal. We also analyze the cases of tensors of rank one, which correspond to the cones over multiple embeddings of projective spaces. In section 7.3 we look at rank varieties for skew symmetric tensors of degree bigger than two. An interesting feature is that the normality of these rank varieties depends on the characteristic of the base ﬁeld. We pay particular attention to the special case of syzygies of Pl¨ucker ideals deﬁning the cones over Grassmannians embedded into projective space by Pl¨ucker embeddings.

7.1. Basic Properties Let λ be a partition. Let E be a vector space of dimension n over K. Consider the representation X = K λ E ∗ as an afﬁne space over K. Its coordinate ring can be identiﬁed with Aλ = K[X ] = SymK (L λ E). For λ1 ≤ r < n we deﬁne the rank variety Yrλ ⊂ X of tensors of rank ≤ r , Yrλ = {φ ∈ K λ E | ∃S ⊂ E, dim S = r, φ ∈ K λ S ⊂ K λ E }. This means the tensor φ has a rank ≤ r if there exists a basis {e1 , . . . , en } of E such that φ can be written using the tensors involving e1 , . . . , er . The condition r ≥ λ1 assures that X r = ∅. 228

7.1. Basic Properties

229

(7.1.1) Examples. (a) Let λ = (2). Then Aλ = Sym(S2 E), and Yrλ is the rank variety Yrs for symmetric matrices analyzed in section 6.3. (b) Let λ = (12 ). Then Aλ = Sym( 2 E). Assume that r is even. Then Yrλ is the rank variety Yra for skew symmetric matrices considered in section 6.4. If r is odd, we get Yrλ to be Yra−1 . In order to analyze the variety Yrλ we use the obvious incidence variety Z rλ = {(φ, S) ∈ K λ E × Grass(r, E ∗ ) | φ ∈ K λ S ⊂ K λ E ∗ }. We can identify the Grassmannian Grass(r, E ∗ ) with Grass(n − r, E). We write the tautological sequence 0 → R → E × Grass(n − r, E) → Q → 0 with dim R = n − r , dim Q = r . The subspace S becomes a ﬁber of Q∗ . The variety Z rλ is an analogue of varieties Z 1 from sections 6.3, 6.4. We can use our incidence variety in the same way as in chapter 6. (7.1.2) Proposition. Let K be a ﬁeld of characteristic 0. (a) The coordinate ring K[Yrλ ] is normal and has rational singularities. In particular, K[Yrλ ] is Cohen–Macaulay. (b) The ideal Irλ of functions vanishing on Yrλ is a span of all representations L µ E with µ1 > r inside of Sym(L λ E). Proof. Let us use the notation from section 5.1, denoting by p : Z rλ → Grass (n − r, E), q : X × Grass(n − r, E) → X , and q : Z rλ → Yrλ the projections. We will write ξ λ for ξ and ηλ for η. The bundle ηλ can be identiﬁed with L λ Q, and ξ λ ﬁts into an exact sequence 0 → ξ λ → L λ E → L λ Q → 0. Using Theorem (5.1.2) (b) we see that it is enough to show that H i (Grass (n − r, E), Sym(ηλ )) = 0 for i > 0. But since the ﬁeld K has characteristic zero, it is clear that for each j, Sym j (ηλ ) = Sym j (L λ Q) decomposes to a direct sum (with multiplicities) of Schur functors L µ Q. By Corollary (4.1.9) we get the vanishing. It is also clear that the ring H 0 (Grass(n − r, E), Sym(ηλ )) is a factor of Sym(L λ E) obtained by factoring out all representations L µ E with µ1 > r . This proves the normality of H 0 (Grass(n − r, E), Sym(ηλ )) and part (b).

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Higher Rank Varieties

It is an interesting but difﬁcult problem to determine the deﬁning equations of varieties Yrλ . It turns out that in general case we have an easy set of equations deﬁning Yrλ set theoretically. Let us look at the map 1⊗Tr

m⊗1

L λ/(1) E → L λ/(1) E ⊗ E ⊗ E ∗ → L λ E ⊗ E ∗ ,

n ei ⊗ ei∗ , and m is where 1 ⊗ Tr is the multiplication by the trace element i=1 an epimorphism m : L λ/(1) E ⊗ E → L λ E sending the element T ⊗ u where T is a tableau of shape λ/(1) to a tableau T with u inserted in the upper left corner. The presentation of L λ/(1) E by generators and relations from section 2.1 implies the existence of such an epimorphism. The map above induces the map of free Aλ -modules λ : L λ/(1) E ⊗K Aλ (−1) → E ∗ ⊗K Aλ . (7.1.3) Proposition. The variety Yrλ is deﬁned set-theoretically by (r + 1) × (r + 1) minors of λ . Proof. Let φ ∈ K λ E ∗ . Denote by λ (φ) the linear map L λ/(1) E → E ∗ obtained from λ by substituting for each linear function on K λ E ∗ (identiﬁed with an element of degree one in Aλ ) its value on a tensor φ. If a tensor φ ∈ K λ S ⊂ K λ E for some subspace S of dimension r in E ∗ , then the image Im λ (φ) is clearly contained in S. Indeed, choose a basis {e1 , . . . , en−r } of Ker(E → S ∗ ), and complement it to the basis {e1 , . . . , en } ∗ ∗ of E, so en−r +1 . . . . , en are a basis of S. Consider the basis of L λ E consisting of standard tableaux with respect to this basis. If a tableau contains a number ≤ n − r , it is zero when evaluated on φ. Therefore the image of λ (φ) is ∗ ∗ contained in the span of en−r +1 , . . . , en which is S. The other implication follows similarly. If rank λ (φ) ≤ r , then there exists a subspace S of dimension r in E ∗ containing the image of λ (φ). Choosing a basis {e1 , . . . , en } as above, we can conclude that if a standard tableau T contains a number ≤ n − r , then the number in the upper left corner is ≤ n − r , so T was gotten from inserting that number in an upper left corner of a tableau T of shape λ/(i). If T evaluated on φ were not zero, then the image λ (φ)(T ) would not be contained in S. Next we give a criterion for the projection q : Z rλ → Yrλ to be a birational isomorphism. (7.1.4) Proposition. Let λ be a partition, r a number such that λ1 ≤ r < n. Assume that λ is not one of the partitions (2), (r − 2), (1), (r − 1). Then the projection q : Z rλ → Yrλ is a birational isomorphism.

7.1. Basic Properties

231

Proof. Let λ be a partition with λ1 ≤ r < n. First we notice that if there is a tensor φ ∈ K λ E such that rank λ (φ) = r , then the projection q : Z rλ → Yrλ is a birational isomorphism. Indeed, the map sending φ to (φ, Im λ (φ)) is the inverse map to q on an open subset of X r consisting of tensors φ for which rank λ (φ) = r . This subset is nonempty by our assumption. The proposition follows now from the next statement. (7.1.5) Lemma. Let λ be a partition with λ1 ≤ n. Assume λ is not one of the partitions (1), (2), (n − 2), (n − 1). Then there exists in K λ E a tensor φ of rank n. Proof. Let X n−1 be the subset of K λ E of tensors of rank ≤ n − 1. It is enough to show that if λ is not one of the partitions listed in the proposition, then X n−1 = K λ E. If λ = (d), then the tensor e1d + . . . + end is easily seen to be of rank n. If λ1 = n, it is clear that every tableau of shape λ in order to be nonzero has to have all n numbers from [1, n] in the ﬁrst row, so every nonzero tensor φ ∈ K λ E has rank n. This means that when writing λ = (λ1 , . . . , λs ) we can assume 2 ≤ λ1 ≤ n − 1. λ λ Consider the modiﬁcation Z n−1 . It is enough to show that dim Z n−1 < λ λ dim K λ E. The dimension of Z n−1 equals dim Z n−1 = n − 1 + dim K λ E , where E is a vector space of dimension n − 1. It is therefore enogh to show that if λ is not one of the partitions listed in the proposition then dim K λ E − dim K λ E > n − 1. First consider the case λ = (t). Since t t n n−1 n−1 E − dim E = − = , dim t t t −1 we see that we are done in the case 2 < t < n − 1. Notice that for t = 2 and t = n − 2 we have the quality above. Let λ = (λ1 , . . . , λs ) be an arbitrary partition with s ≥ 2, 2 ≤ t = λ1 ≤ n − 1. It is enough to show that dim K λ E − dim K λ E > dim

t

E − dim

t

E .

(∗)

In order to show this fact, we recall that the dimension of K λ E is the number of standard tableaux of shape λ with entries from [1, n]. For every tableau S of shape (t) with number n occurring in S we construct the standard tableau T (S) by setting T (S)(i, j) = S(1, j) for all (i, j) ∈ D(λ). This proves

232

Higher Rank Varieties

that the weak inequality (∗) holds. In order to prove that the inequality is sharp, we produce one more standard tableau of shape λ with entries [1, n], containing n. This will be the tableau U given by setting U (i, λi + 1 − u) = n + 1 − u in the case when the partition λ is not rectangular. If λ is rectangular, then it is easy to produce a standard tableau U containing n which is not constant in columns by taking U (i, λi + 1 − u) = n + 1 − u for i ≥ 2 and U (1, j) = j. Lemma 7.1.5 is proved. (7.1.6) Remark. Propositions (7.1.2) and (7.1.3) generalize to several tensors. If X = K λ(1) E ∗ ⊕ . . . ⊕ K λ(t) E ∗ , we could deﬁne the subvariety Yr to be the set of t-tuples of tensors (φ1 , . . . , φt ) ∈ X which can be simultaneously expressed using tableaux involving r basis vectors. The role of the map λ is played by the map φλ(1) ,...,λ(t) : L λ(1) E ⊗K A ⊕ . . . ⊕ L λ(t) E ⊗K A → E ∗ ⊗K A deﬁned on the j-th component using the tensor φ j . Finally we address the question when the deﬁning ideal of Yrλ is Gorenstein. (7.1.7) Theorem. The variety Yrλ is deﬁned by Gorenstein ideals in the following cases: (a) n =

|λ| dim L λ Q . r

In the remaining cases n > (|λ| dim L λ Q)/r : (b1) λ = (r k , 12 ), r > 1, and n − (|λ| dim L λ Q)/r is positive, divisible by |λ|/2, (b2) λ = (r k , 2), r > 2 is even, and n − (|λ| dim L λ Q)/r is positive, divisible by |λ|, (b3) λ = (r k , (r − 1)2 ), r ≥ 1, and n − (|λ| dim L λ Q)/r is divisible by |λ|/2, (b4) λ = (r k , r − 2), r > 2 is even, and n − (|λ| dim L λ Q)/r is divisible by |λ|, (b5) λ = (r k ), n > k, and n is divisible by k. Proof. We use the duality statement (5.1.4). Denote t = dim ξ . We cal culate the bundle t ξ ∗ ⊗ ωV . By (3.3.5) ωV = OV (−n). Also we have t ∗ ∼ ξ = OV ((|λ| dim L λ Q)/r ). Therefore the dualizing bundle is given by OV ((|λ| dim L λ Q)/r − n). This means that in the case n = (|λ| dim L λ Q)/r the deﬁning ideal is Gorenstein.

7.1. Basic Properties

233

Also in the case n < (|λ| dim L λ Q)/r it cannot happen that the deﬁning ideal is Gorenstein, because the module of sections of the sheaf OV ((|λ| dim L λ Q)/r − n) ⊗ Sym(L λ Q) has a representation of dimension > 1 in degree 0, so it cannot be isomorphic to K [Yrλ ]. However, for n > (|λ| dim L λ Q)/r it can still happen that the deﬁning ideal is Gorenstein. It happens when the module of sections of OV ((|λ| dim L λ Q)/r − n) ⊗ Sym(L λ Q) is isomorphic to K [Yrλ ]. The sections of OV ((|λ| dim L λ Q)/r − n) ⊗ Sym(L λ Q) are given by all representations L µ Q from Sym(L λ Q) such that for the conjugate partition µ = (µ1 , . . . , µr ) we have |λ| dim L λ Q . r Such situation can occur only when the ideal consisting of representations L µ Q satisfying the above condition is generated by the one dimensional bundle L µ Q with µ = (r x ), where x = n − (|λ| dim L λ Q)/r . This can happen only when for L λ F, with dim F = r , the variety of tensors of rank < r has codimension 1. Let us classify such cases. µr ≥ n −

(7.1.8) Lemma. Let dim F = r . Let λ be a partition with λ1 < r . Then the subvariety Yrλ−1 has codimension 1 in the following cases: (b1) (b2) (b3) (b4)

λ = (12 ), λ = (2), r even, λ = ((r − 1)2 ), λ = (r − 2), r even.

Proof. Let us assume that codim Yrλ−1 = 1. Then we have the inequality dim L λ F ≤ 1 + r − 1 + dim L λ F or, equivalently, dim L λ F − dim L λ F ≤ r, where F is a vector space of dimension r − 1. But setting F = F ⊕ K we get that dim L µ F = r. µ,λ/µ∈VS, λ/µ=∅

The case r = 1 is trivial, as λ = ∅. In the case r = 2 we have λ = (1d ) and the set of tensors of rank ≤ 1 is a cone over a rational normal curve of

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Higher Rank Varieties

degree d by exercise 5 of chapter 5. Its codimension is equal to d − 1. The case d = 2 is covered under (b1). Let us assume r ≥ 3. If λ has at least two rows of different lengths, then there are at least three partitions µ on the left hand side of the last formula. Also, observe that if a partition µ has a row of length < r − 1, then dim L µ F ≥ r − 1. Thus our equality cannot happen. Thus all rows of λ have the same length. However if there are at least three of them, we still have at least three possible partitions on the left hand side, so the equality cannot happen. Thus λ has two rows or one row. Now we can analyze the situation directly to see that if there are two rows, they have to have length r − 1 or 1, giving cases (b1) and (b3). If there is one row, we can see that the only possibilities are that the length is 1, 2, r − 2, r − 1. But the cases r − 1 and 1 are eliminated because there Yrλ−1 has codimension zero. This completes the proof of the lemma. Now we can conclude the proof of Theorem (7.1.7). Indeed, we have four possible cases, but we have to take into account that the partition λ can have some rows of length r . Moreover, we have an additional case λ = (r k ) where also the codimension of tensors of rank < r is one. This leads to the cases (b1)–(b5) of Theorem (7.1.7). The divisibility condition comes from the fact that we need x to be divisible by the degree of the generating invariant of the subring of SL(F)-invariants in Sym(L λ F). This completes the proof of Theorem (7.1.7).• 7.2. Rank Varieties for Symmetric Tensors In this section we consider rank varieties for symmetric tensors. Some of the results of the previous section can be strengthened in this case. Again E denotes a vector space over K of dimension n. We take λ = (1d ) and so d X = Dd E ∗ , A(1 ) = Sym(Sd E). We assume that d ≥ 3, because in the case d = 2 we get the determinantal varieties for symmetric matrices which were discussed in section 6.3. First we consider the case r = n − 1. Following the paper [Po] of Porras, (1d ) . The incidence we describe the whole minimal free resolution of the ideal In−1 variety deﬁned in section 7.1 becomes d

(1 ) = {(φ, S) ∈ X × Grass(n − 1, E ∗ ) | φ ∈ Dd S ⊂ Dd E ∗ }. Z n−1

The tautological sequence we use is 0 → R → E × Grass(1, E) → Q → 0. We recall that η = Sd Q.

7.2. Rank Varieties for Symmetric Tensors

235 d

(7.2.1) Proposition. In the case of symmetric tensors we have ξ (1 ) = R ⊗ Sd−1 E. d

Proof. The bundle ξ (1 ) ﬁts into the exact sequence d

0 → ξ (1 ) → Sd E → Sd Q → 0. This means we need to show the exactness of the sequence 0 → R ⊗ Sd−1 E → Sd E → Sd Q → 0. The maps are easily deﬁned. The composition of both maps is zero. To check the exactness we need to do it locally. There the sequence is exact because of the direct sum decomposition (2.3.1) for the symmetric power. This means the calculation of the cohomology for the exterior powers of d ξ (1 ) reduces to the calculation of cohomology of line bundles on the projective space. This is provided by Serre’s theorem [H1, chapter III, Theorem 5.1]. Let us summarize. (7.2.2) Proposition. The complex F• has terms given by d

F0 = A(1 ) , Fi = K i,1n−1 E ⊗

n−1+i

d

(Sd−1 E) ⊗K A(1 ) (−i − n + 1) −n+ for i ≥ 1. The length of the complex is dim Sd−1 E − n + 1 = n+d−2 d−1 1. In fact, F• is just an Eagon–Northcott complex associated to the maximal minors of the map (1d ) : Sd−1 E ⊗K A(1 ) (−1) → E ∗ ⊗K A(1 ) . d

d

d

(7.2.3) Corollary. The ideal Ir(1 ) is generated by (r + 1) × (r + 1) minors of the matrix (1d ) . Proof. For r = n − 1 it follows from Proposition (7.2.2). To prove the general d case we use descending induction on r . Let Ir (1 ) be the ideal generated by (r + d d 1) × (r + 1) minors of the matrix (1d ) . Assume that Ir(1+1) = Ir+1 (1 ) . Let us d also assume that A/Ir (1 ) contains nilpotents. The set of nilpotents in this ring is a GL(E)-stable subspace. Therefore it has to contain a U+ -invariant. But the weight of this U+ -invariant contains ≤ r + 1 basis vectors, otherwise it d d would be in Ir(1+1) = Ir+1 (1 ) . This means such U+ -invariant (which is nilpotent d modulo Ir (1 ) ) exists already for dim E = r + 1. This is a contradiction with Proposition (7.2.2).

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Higher Rank Varieties

(7.2.4) Remark. Proposition (7.2.2) and Corollary (7.2.3) are true for several symmetric tensors of degrees d1 , . . . , dt . The role of the map (1d ) is played by the map φ1d1 ,...,1dt : Sd1 −1 E ⊗K A(1 ) ⊕ . . . ⊕ Sdt −1 E ⊗K A(1 ) → E ∗ ⊗K A(1 ) , d

d

d

where the j-th component is deﬁned using the j-th tensor. We also state the criterion for rank varieties of symmetric tensors to be Gorenstein. d

(7.2.5) Theorem. The variety Yr(1 ) is Gorenstein in the following cases: (a) n = r +d−1 , d−1 r +d−1 (b) n > d−1 , r = 1, n divisible by d. Proof. This follows from Theorem (7.1.7). Analyzing all cases, we see that λ = (1d ) can occur only in cases (a) and (b3) (with r = 1). This last case gives case (b) of our statement. The most interesting varieties are those of tensors of rank ≤ 1. The variety Y1(d) is the cone over the d-tuple embedding of Pn−1 = P(E) into the projective space P(Sd E). There is a lot of interest in the resolutions of these varieties, especially trying to determine for which p they satisfy the property Np of Green and Lazarsfeld introduced in [GL]. This is equivalent to asking about the smallest i for which H 2 ( i ξ ) = 0. The reader may consult [B1], for some recent results. In general the description of the resolution seems to be rather difﬁcult, as it is connected to the problem of inner plethysm. The composition d series of the bundle ξ (1 ) induced by the tautological exact sequence will always involve the term Sd R, so the exterior powers are related to higher plethysm. One might hope, however, to determine all pairs (i, j) for which the d cohomology group H i (Grass(n − r, E ∗ ), i+ j ξ (1 ) ) = 0. One might hope that for every such pair (i, j) there will be a representation that will occur only few times in the spectral sequence and thus will allow one to determine that d H i (Grass(n − r, E ∗ ), i+ j ξ (1 ) ) = 0. Below we do it for the embeddings of P2 . We use the approach of Ottaviani and Paoletti [OP]. Let us look at the resolutions of d-tuple embeddings of projective spaces in more detail. In section 6.3 we exhibited the resolutions of such embeddings for d = 2. Proposition (7.2.2) provides the answer for the embeddings of P1 . d d It is not difﬁcult to locate the top part of the resolution of A(1 ) /Ir (1 ) .

7.2. Rank Varieties for Symmetric Tensors

237

(7.2.6) Proposition ([OP]). Let us ﬁx n, d. We choose the number j to be the minimal number such that ( j + 1)d ≥ n. Then the top part of the complex F• is ⊗(n+d−1 (d+n−1 )−1− j n n−1 n ) d n− j (1 ) ∗ ξ E . Grass(1, E), = S( j+1)d−n E ⊗ H

Proof. Using the duality statement (5.1.4) and taking into account that the canonical bundle on Pn−1 is O(−n), we see that it is enough to locate the rightmost part of the complex F(Sd−n−1 Q)• . But this is equivalent by (5.1.2) (b) to ﬁnding the cohomology of j≥0 Sd−n+ jd Q. It is now clear by (4.1.9) that the higher cohomology has to vanish and that the module H 0 (Grass(1, E), j≥0 Sd−n+ jd Q) is generated by the representation S( j+1)d−n V in degree j described in our statement. For the remainder of this section we assume that char K = 0. (7.2.7) Example. Let us take d = n = 3. Then j = 0, and Proposition (7.2.6) says that the top of the resolution is one dimensional. This means that the 3 resolution in question is self-dual and the ideal I1(1 ) is Gorenstein. But this means that H 0 and H 2 strands of the resolution consist of one copy of one dimensional representation each. This allows to describe the terms of the d resolution by calculating Euler characteristics of exterior powers of ξ (1 ) . The terms of the complex F• are F0 = (0, 0, 0), F1 = (4, 2, 0), F2 = (4, 3, 2) ⊕ (5, 3, 1) ⊕ (5, 4, 0) ⊕ (6, 2, 1), F3 = (5, 4, 3) ⊕ (5, 5, 2) ⊕ (6, 3, 3) ⊕ (6, 4, 2) ⊕ (6, 5, 1) ⊕ (7, 3, 2) ⊕(7, 4, 1), F4 = (6, 5, 4) ⊕ (6, 6, 3) ⊕ (7, 4, 4) ⊕ (7, 5, 3) ⊕ (7, 6, 2) ⊕ (8, 4, 3) ⊕(8, 5, 2), F5 = (7, 6, 5) ⊕ (8, 6, 4) ⊕ (8, 7, 3) ⊕ (9, 5, 4), F6 = (9, 7, 5), F7 = (9, 9, 9), d

where we write (a, b, c) instead of K a,b,c V ⊗ A(1 ) . The homogeneous degree is easily seen from the size of each partition.

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Higher Rank Varieties

Let us ﬁx n = 3. We want to determine the minimal j for which the d cohomology modules H 2 (Grass(1, E) j+2 ξ (1 ) ) = 0. (7.2.8) Proposition (Ottaviani–Paoletti, [OP, Theorem 2.1]). Let n = 3. d We have H 2 (Grass(1, E), j+2 ξ (1 ) ) = 0 for j ≥ 3d − 2. d Proof. It is enough to show that H 2 (Grass(1, E), 3d ξ (1 ) ) = 0. Indeed, applying the duality (5.1.4) takes the H 2 strand to the dual of the H 0 strand, and since the H 0 strand has to be linearly exact we know that if the j-th term in this strand is nonzero, then all terms in degrees ≤ j also have to be nonzero. By Serre duality it is enough to show that H 0 (Grass(1, E), Sd−3 Q ⊗ d(d−3)/2 (1d ) ξ ) = 0. Now everything follows from the following j (1d ) ⊗ St Q has nonzero sections for 1 ≤ j ≤ (7.2.9) n+d−1Lemma. The sheaf n+t−1 ξ − 1, j + 1 ≤ and t ≥ 1. d n−1 But we have an exact sequence 0→

j

d

ξ (1 ) →

j j−1 d (Sd V ) × Grass(1, E) → Sd Q ⊗ ξ (1 ) → 0.

d This means that the sections of j ξ (1 ) ⊗ St Q can be identiﬁed with the kernel j j−1 at (Sd V ⊗ St V → (Sd V ) ⊗ St+d V ). Ker We use a Koszul complex ... →

j+1 j j−1 (Sd V ) ⊗ St−d Q → (Sd V ) ⊗ St Q→ (Sd V ) ⊗ St+d Q → . . .

with the differential being a composition j+1

→

j

(Sd V ) ⊗ St−d Q →

(Sd V ) ⊗ Sd Q ⊗ St−d Q →

j

(Sd V ) ⊗ Sd V ⊗ St−d Q

j

(Sd V ) ⊗ St Q.

Notice that if t = pd + q, this is just a twisted symmetric power Sq Q ⊗ S j+ p (Sd V → Sd Q). The existence of this complex means that for t ≥ d the j+1 j (1d ) sections of (Sd V ) ⊗ St−d Q give the sections of ξ ⊗ St Q. In particular, for d = t we get for each family of polynomials s0 , . . . , s j an

7.3. Rank Varieties for Skew Symmetric Tensors

239

element j

(−1)i s0 ⊗ . . . ⊗ sˆi ⊗ . . . ⊗ s j ⊗ si

i=0

in Ker at . Let 1 ≤ t < d. If we can factor si = uwi with deg u = d − t, we see that the element j

(−1)i s0 ⊗ . . . ⊗ sˆi ⊗ . . . ⊗ s j ⊗ wi

i=0

d gives a nonzero section of j ξ (1 ) ⊗ St Q. This construction is possible as soon as we can ﬁnd j + 1 linearly independent polynomials of degree ≤ t, .• i.e. if j + 1 ≤ n+t−1 n−1

7.3. Rank Varieties for Skew Symmetric Tensors In this section we consider rank varieties for skew symmetric tensors. The results on the equations are less precise than for symmetric tensors because the ideal of minors of the map λ is not radical in this case. We follow the paper [Po] of Porras in extracting information about the resolutions and equations of the deﬁning ideals of tensors of rank ≤ n − 1. One can describe the generators of the deﬁning ideals quite precisely in the case of skew symmetric tensors of degree 3. We also pay special attention to the skew symmetric tensors of degree d of minimal possible rank d. The variety Yd(d) in this case is the cone over a Grassmannian Grass(r, E) embedded via Pl¨ucker embedding. The problem of ﬁnding higher syzygies of these ideals was posed by Study over 100 years ago. As before E denotes a vector space over K of dimension n. We take λ = (d) and so X = d E ∗ , A(d) = Sym( d E). We assume that n − 3 ≥ d ≥ 3, because in the remaining cases the only possible varieties we can get are the determinantal varieties for skew symmetric matrices which were considered in section 6.4. For d ≤ r < n we deﬁned in 7.1 the rank varieties Yr(d) . First we consider the case r = n − 1. The incidence variety deﬁned in section 7.1 becomes (d) = {(φ, S) ∈ X × Grass(n − 1, E ∗ ) | φ ∈ Z n−1

d

S⊂

d

E ∗ }.

240

Higher Rank Varieties

The tautological sequence we use is 0 → R → E × Grass(1, E) → Q → 0. We recall that η(d) = d Q. (7.3.1) Proposition. In the case of skew symmetric tensors of rank ≤ n − 1 we have ξ (d) = R ⊗ d−1 Q. Proof. The bundle ξ (d) ﬁts into the exact sequence 0 → ξ (d) →

d

E→

d

Q → 0.

This means we need to show the exactness of the sequence 0→R⊗

d−1

Q→

d

E→

d

Q → 0.

The maps are easily deﬁned. The composition of both maps is zero. To check the exactness we need to do it locally. There the sequence is exact because of the direct sum decomposition (2.3.1) for the exterior power. We can also easily determine the top term of the resolution. (7.3.2) Proposition([Po]). = n − 1. De Let us assume 3 ≤ d ≤ n − 3. Let rn−1 n−1 note m = dim ξ = d−1 . The top term of the complex F• is H (Grass(n − n−2 1, F), m ξ ) = K (m−n+1,q+1,...,q+1) E ⊗ A(d) (−m), where q = d−2 . In particular the only case in which In−1 is a Gorenstein ideal is when d = 3, n = 6. Proof. The term in question comes from the cohomology of the top exterior power of ξ . It occurs in Fm−n+1 . It is clear by (5.1.6) (a) and by (7.1.3) that Fi = 0 for i > m − n + 1. It remains to show that the term listed above is the only term in Fm−n+1 . The only possible cohomology groups are m− j n−1− j ξ Grass(n − 1, F), H for j > 0. But by (7.3.1), using the isomorphism we see that we look at the weights

m− j

ξ=

m

ξ⊗

j

ξ ∗,

(q − µn−1 , . . . , q − µ1 , m − j), where µ is a partition such that K µ Q∗ occurs in j ( d−1 Q∗ ). We also assume that the weight in question does not have a nonzero (n − 1)st cohomology

7.3. Rank Varieties for Skew Symmetric Tensors

241

group. This means q − µn−1 ≥ m − j − n + 2. This implies

n−2 j ≥m−q −n+2= − n + 2. d −1

If we can show that the inequality above implies j ≥ n − 1, we are done, as we have eliminated all possibilities for j. However the inequality n−2 −n+2≥n−1 d −1 fails only for n = 6, d = 3 and for n = 7, d = 3, 4. These three cases can be handled explicitly. Proposition (7.3.1) means that the calculation of the cohomology for the exterior powers of ξ (d) is not as easy as in the symmetric case. In order to perform the explicit calculation we would need to know the decomposition of i ( d−1 Q) into Schur functors. This, as explained in section 2.3, is a very difﬁcult problem. Still, we have an explicit formula for plethysm when d − 1 = 2. It also allows us to describe in some cases, as for symmetric tensors, the pairs (i, j) for which H i (Grass(1, E), i+ j ξ (d) ) = 0. Let us assume d = 3. Then the plethysm formula (2.3.9) (b) makes the problem of calculating a complex F• a combinatorial exercise. Still, the analysis of the whole complex F• is quite complicated. Here we just analyze the (d) . Interested readers should consult [Po]. deﬁning equations of the ideal of Yn−1 (d) of the variety (7.3.3) Proposition ([Po]). Let us assume d = 3. The ideal In−1 (d) has generators in all degrees i satisfying [n/2] ≤ i ≤ (1 + 2n − (1+ Yn−1 8n)1/2 )/2.

Proof. We need to show that the terms occurring in the term F1 of the complex F• appear in homogeneous degrees i satisfying the above inequalities and that for each such i we get a nonzero contribution. We know that i i−1 Grass(n − 1, E), F1 = H ξ ⊗ A(d) (−i). i>0

We also have by (2.3.9) (b) i

ξ=

µ∈Q −1 (2i)

K µ Q ⊗ Si R.

242

Higher Rank Varieties

This means we need to apply (4.1.9) to the weights (µ1 , . . . , µn−1 , i) with µ ∈ Q −1 (2i). Notice that such term gives a contribution to H i−1 if and only if µn−i = 0 and µn−i+1 = 0. Therefore we need to estimate for which i we have a partition in Q −1 (2i) with exactly n − i parts. Let us look for a lower bound for i. For n = 2t even the smallest partition of this kind is clearly µ = (t − 1, 22 , 1t−3 ) ∈ Q −1 (2t). For n = 2t + 1 the smallest such partition is clearly µ = (t, 1t ) ∈ Q −1 (2t). This proves the lower bound of the proposition. Let us seek the biggest possible partition in Q −1 (2i) with n − i parts. Any such partition has to be contained in the rectangle ((n − i − 1)n−i ), so we must have the inequality 2i ≤ (n − i)(n − i + 1), which gives the upper bound in the proposition. Of course, for any i satisfying the inequalities of the proposition we can ﬁnd the appropriate µ ∈ Q −1 (2i) by choosing any partition from Q −1 (2i) containing the partition giving the lower bound and contained in the rectangle ((n − i + 1)n−i ). (7.3.4) Corollary ([Po]). Let d, n be as above, with 3 ≤ d ≤ n − 3. For any r satisfying 3 ≤ r ≤ n − 1 the ideal of (r + 1) × (r + 1) minors of (d) is not radical. Proof. It is enough to show that Ir(d) has some nonzero elements in degrees ≤ r . This is clear, since the representations generating Ir(d) for r + 1 dimensional space will give the representations from Ir(d) for n dimensional space. By (7.3.3) they occur in degrees 1 + 2(r + 1) − (1 + 8(r + 1))1/2 1 + 2(r + 1) − 3 ≤ = r. 2 2 The corollary follows. Let us also state when the ideal Ir(d) is Gorenstein. (7.3.5) Theorem. The ideal Ir(d) is Gorenstein in the following cases: r −1 (a) n = d−1 , (b) d = 2, and r is even, r −1 r −1 , , d = r − 2, and n − d−1 is divisible by r −2 (c) n > d−1 2 (d) d = r . Proof. This is a special case of Theorem (7.1.7). Cases (b2), (b4), and (b5) of (7.1.7) lead to cases (b), (c), and (d) of our statement.

7.3. Rank Varieties for Skew Symmetric Tensors

243

For the remainder of this section we investigate the case r = d. In this case η = d Q is one dimensional and therefore K (n d ) E. A(d) /Id(d) = n≥0

Let us describe the top term of the resolution. (7.3.6) Proposition. The top term in the resolution F• is H

d(n−d)−n+1

(Grass(n − d, E),

n ( d )−n

ξ ) = K (((n−1)−1)n ) E. d−1

Proof. Let us use the duality statement (5.1.4). The canonical n sheaf K Grass(n−d,E) is equal to OGrass(n−d,E) (−n). The top exterior power (d )−1 ξ is isomorphic (up to the twist by a power of determinant of E) to OGrass(n−d,E) (1). Therefore the dualizing bundle will be OGrass(n−d,E) (−n + 1) or, in terms of tautological bundles d Q⊗(−n+1) . Calculating the cohomology of the sheaf d ⊗(−n+1) Q ⊗ Sym(η), we see that the generator occurs in degree n − 1 and is a trivial representation. This means that by (5.1.4) the top term of the resolution occurs in homogeneous degree dim ξ − n + 1, and it is a one dimensional representation. Since the variety Yd(d) is normal with rational singularities by (5.1.2) (b) and (5.1.3), we see n that this term has to occur in (d) the term Fi with i = dim X − dim Yd = d − d(n − d) − 1. Calculating the powers of the determinants involved, one gets the proposition. (7.3.7) Remark. Note that for d = 2 we constructed the resolution of the Pl¨ucker ideal in section 6.4. It is the resolution of 4 × 4 Pfafﬁans of a generic skew symmetric n × n matrix. Indeed, the cone over the Grassmannian Grass (2, E) can be identiﬁed with the set of skew-symmetric matrices of rank ≤ 1. We ﬁnish this section with the analysis of the case d = 3, n = 6. We will analyze the rank variety Y5(3) . The main goal is to show that this variety is not normal over a ﬁeld K of characteristic 2, so the conclusion of Proposition (7.1.2) fails in positive characteristic. Thus E is a vector space of dimension 6, and we work over the Grassmannian Grass(1, E) with tautological sequence 0 → R → E × Grass(1, E) → Q → 0. We have η(3) =

3

Q, ξ (3) = R ⊗

2

Q.

244

Higher Rank Varieties

(7.3.8) Proposition. Let char K = 0. The complex F• has the following nonzero terms: F0 = A(d) ,

F1 = L (6,3) E ⊗ A(d) (−3),

F3 = L (62 ,5,1) E ⊗ A(d) (−6),

F2 = L (6,5,1) E ⊗ A(d) (−4),

F4 = L (63 ,3) E ⊗ A(d) (−7),

F5 = L (65 ) E ⊗ A(d) (−10). (7.3.9) Proposition. Let K be a ﬁeld of characteristic 2. Then the only nonzero cohomology groups of S2 R ⊗ 2 Q are 2 2 1 H Grass(1, E), S2 R ⊗ Q 2 2 6 = H Grass(1, E), S2 R ⊗ Q = E. 2

Proof. We notice that in characteristic 2, L (3,1) Q is not irreducible. In fact we have the natural exact sequence 0 → M(2,1,1) Q → L (3,1) Q →

4

Q → 0.

Also, we have a natural map ψ : L 3,1 Q →

2 2 Q

induced by the composition map 3

⊗1

Q⊗Q →

2

1⊗m

Q⊗Q⊗Q →

2

Q⊗

2

Q.

This map, however, is not an isomorphism in characteristic zero. Its image is isomorphic to 4 Q; its kernel, to M(2,1,1) Q. Next we notice that H s (Grass(1, E), S2 R ⊗ L 3,1 Q) = 0 for all s ≥ 0, by exercise 5 of chapter 4. Also, the only nonzero cohomology group of S2 R ⊗ 4 Q is H 1 (Grass(1, E), S2 R ⊗ 4 Q) = 6 E. This we can deduce from identifying S2 R with D2 R and using the *-acyclic resolution 2 (E → Q) of D2 R, tensored with 4 Q. The sections of this resolution

Exercises for Chapter 7

give a complex 0 → D2 E ⊗

4

E→E⊗E⊗

4

E→

2

245

E⊗

4

# E

6

E,

which has only one homology – 6 E in degree 1. Now, using the short exact sequences 0 → M(2,1,1) Q → L (3,1) Q → and 0→

4

4

Q→0

2 2 Q→ Q → M(2,1,1) Q → 0

tensored with S2 R and the long exact sequences of cohomology they induce, we deduce the proposition. (7.3.10) Proposition. Let char K = 2. Then the ring A(3) /J5 is not normal. Proof. Recall that H 0 (Grass(1, E), Sym( 3 Q)) can be identiﬁed with the normalization of A/J5 . By Proposition (7.3.9) the complex F• has a nontrivial term in homological degree 0 and in homogeneous degree 2. It follows that the natural map 3 3 0 S2 ( E) → H Grass(1, E), Sym2 Q is not onto, and therefore the normalization of A(3) /J5 is not generated as an A(3) -module by a unit in degree 0.

Exercises for Chapter 7 Minimal Resolutions of the Ideal I3(3) for n = 6, 7 1. Let X = d E ∗ be the set of skew symmetric tensors. Denote by Y ⊂ X the set of 1-decomposable tensors, i.e. the set of tensors φ in X such that φ = ψ ∧ l where l ∈ E ∗ is a linear form and ψ ∈ d−1 E ∗ . Prove that Y has a desingularization which is a total space of a vector bundle over the Grassmannian Grass(n − 1, F). Identify ξ = d R and η = d−1 Q⊗ R. Prove that Y is normal and has rational singularities.

246

Higher Rank Varieties

2. In the situation of exercise 1, consider the twisted complex F(Sd Q∗ )• . Prove that its homology modules are H−i (F(Sd Q∗ )• ) = ⊕ j≥0 H i Grass(n − 1, E), d−1 Sd Q∗ ⊗Sym j Q ⊗ R . Prove that the nonzero homology occurs for i = −d + 1, . . . , 0. Prove that H−d+1 (F(Sd Q∗ )• ) = A/Id(d) (−1). 3. In the situation of exercise 2, specialize to d = 3. Prove that the only nonzero homology modules of F(S3 Q∗ )• are H−2 and H0 . Writing N = H0 (F(S3 Q∗ )• ), prove that the j-th graded component of N is Nj = K ( j−3,a,a,b,b,...) E. a+b+...= j, j−3≥a

4. In the situation of exercise 3, specialize to n = 6. We have K ( j−3,a,a,b,b,0) E. Nj = a+b+...= j, j−3≥a≥b

Prove that the complex F(S3 Q∗ )• has the following nonzero terms (06 ) 4

(2, 14 , 0) ↑ (3, 2, 14 ) ⊕ (24 , 1, 0) ↑ 4 (32 , 22 , 12 ) 4

(5, 25 ) ↑ (5, 33 , 22 ) ⊕ (43 , 23 ) ↑ (5, 42 , 32 , 2) ↑ (52 , 43 , 2) ↑ (55 , 2)

where we just write the partitions of the occurring Weyl functors. The vertical arrows denote the maps of degree 1 and skew arrows denote the maps of degree 2.

Exercises for Chapter 7

247

5. We still work with X = 3 E ∗ , dim E = 6. Consider the variety Y5(3) of tensors of rank ≤ 5 in X . Let Z 5(3) be its desingularization considered in section 7.3: $ 3 3 (3) ∗ ∗ S⊂ E Z 5 = (φ, S) ∈ X × Grass(5, E ) | φ ∈ . We write 0 → R →

3

E × Grass(5, E) → Q → 0

for the tautological sequence on Grass(5, E). We have η = 3 Q , ξ = R ⊗ 2 Q . Consider the twisted complex F ( 5 Q⊗3 )• . Prove that it is acyclic, and show that the A-module H0 (F ( 5 Q⊗3 )• ) is isomorphic to N twisted by 6 E ⊗−3 . Prove that the complex F ( 5 Q⊗3 )• has the 6 ⊗3 E ): following nonzero terms (after twisting back by (35 , 0) ↑ 2 3 (4 , 3 , 1) ↑ (5, 42 , 32 , 2) ↑ 3 2 (6, 4 , 3 ) ⊕ (53 , 33 ) ↑ 4 (7, 45 ) 4

(62 , 52 , 42 ) ↑ 4 (6 , 5, 4) ⊕ (7, 6, 54 ) ↑ (7, 64 , 5)

4 (76 )

The vertical arrows denote the maps of degree 1, and the skew arrows denote the maps of degree 2. 6. Prove that there exists a map of the complex constructed in exercise 5 to the complex constructed in exercise 4 induced by the isomorphism H0 (F(S3 Q∗ )• ) = N . Prove that the cone of gives a minimal resolution of the A(3) -module A(3) /I3(3) .

248

Higher Rank Varieties

7. Let us specialize the situation from exercise 3 to n = 7. We have Nj = K ( j−3,a,a,b,b,c,c) E. a+b+c...= j, j−3≥a≥b≥c

The components N j are nonzero for j ≥ 5. Let N be the span of all summands with c ≥ 1. Prove that N is an A(3) -submodule of N . Denote N = N /N . Prove that the minimal resolution of N equals F(Q∗ )• ⊗ 7 F, in the notation of exercise 1. 8. Prove that the minimal resolution of the A(3) -module N from exercise 7 can be obtained as follows. Take V = Grass(, E) with tautological sequence 0 → R → E × Grass(2, E) → Q → 0. Here dim R = 2, dim Q = 5. Take the bundle ξ to be the kernel 0 → ξ →

3

3

E × Grass(2, E) →

Q → 0.

The bundle ξ also ﬁts into the exact seuence 0→

2

R ⊗ Q → ξ → R ⊗

2

Prove that the minimal resolution of N is F ((

Q → 0.

5

Q )⊗3 )• .

9. Use the information obtained in exercise 8 to calculate the terms in the linear strand of the minimal resolution of A(3) /I3(3) for n = 7. 10. Consider the twisted sheaf M = K ((r −1)(n−r −1))r Q ⊗ K ((r −1)(n−r −1),(r −1)(n−r −2),...,(r −1),0) R ⊗ O Z r(r ) supported in the rank variety Yr(r ) . Prove that M has no higher cohomology. Show that M := H 0 (Z r(r ) , M) is a maximal Cohen–Macaulay module with a linear resolution supported in Yr(r ) . 11. Finish the proof of Proposition (7.3.2). Calculate explicitly the resolution in the case n = 6, d = 3 and in the cases n = 7, d = 3, 4. 12. Let X = Sd E ∗ be the set of symmetric tensors. Denote by Y ⊂ X the set of 1-decomposable tensors, i.e. the set of tensors φ in X such that φ = ψ ◦ l where l ∈ E ∗ is a linear form. Prove that Y has a desingularition which is a total space of a vector bundle over the Grassmannian Grass(n − 1, F). Identify ξ = Sd R and η = Q ⊗ Sd−1 E. Prove that Y is not normal but its normalization has rational singularities. Identify the normalization of Y .

Exercises for Chapter 7

249

Higher Rank Varieties for Orthogonal and Symplectic Groups 13. Let F be an orthogonal space of dimension m = 2n + 1 or m = 2n. We denote the nondegenerate symmetric form on F by ( , ). We choose a standard hyperbolic basis of F, denoted {e1 , . . . , en , e, e¯ n , . . . , e¯ 1 } in the odd case and {e1 , . . . , en , e¯ n , . . . , e¯ 1 } in the even case. The only nonzero values of the form ( , ) are (ei , e¯ i ) = 1, (e, e) = 1. Let λ be a partition, and let Vλ F be a representation of SO(F) of highest weight λ. This is a space of tensors from L λ F modulo the tensors containing a trace element (compare exercise 14 of chapter 6). For λ1 ≤ n we deﬁne the rank variety Yrλ as the set of tensors that (after the change of basis) can be written only in terms of e1 , . . . , er . Construct the desingularization Z rλ of the variety Yrλ using the isotropic Grassmannian IGrass(r, F). Prove that the variety Yrλ is normal and has rational singularities. 14. Let F be an symplectic space of dimension 2n. We denote the skew symetric nondegenerate form on F by ( , ). We choose a standard symplectic basis of F, denoted {e1 , . . . , en , e¯ n , . . . , e¯ 1 }. The only nonzero values of the form ( , ) are (ei , e¯ i ) = 1, (e, e) = 1. Let λ be a partition, and let Vλ F be a representation of Sp(F) of highest weight λ. These are tensors from L λ F modulo the tensors containing a trace element (compare exercise 4 of chapter 6). For λ1 ≤ n we deﬁne the rank variety Yrλ as the set of tensors that (after the change of basis) can be written only in terms of e1 , . . . , er . Construct the desingularization Z rλ of the variety Yrλ using the isotropic Grassmannian IGrass(r, F). Prove that the variety Yrλ is normal and has rational singularities.

The Isotropic Grassmannian IGrass(3, 6) 15. Let F be a symplectic space of dimension 6. Consider the representation V13 F. It ﬁts into the exact sequence t

0→F →

3

F → V13 F → 0,

where the map t is the multiplication by the element t = 1≤i≤n ei ∧ e¯ 1 2 from F given by the form. Prove that the representation V13 F has four Sp(F)-orbits (except the zero orbit). There is a general orbit, a hypersurface given by the vanishing of (the unique up to scalar) invariant of degree 4, the orbit X given by the tensors where the partial derivatives of (forming a representation V13 F in degree 3) vanish, and the orbit Y which is the cone over IGrass(3, F). Prove that codim X = 4, codim Y = 7.

250

Higher Rank Varieties

16. Find the desingularization of X which is a homogeneous bundle over some isotropic ﬂag variety. Conclude that the minimal rsolution of K[X ] has the following terms: F0 = A,

F1 = V13 F ⊗ A(−3),

F3 = V12 F ⊗ A(−6),

F2 = V2 F ⊗ A(−4),

F4 = V1 F ⊗ A(−7).

17. Calculate the minimal resolution of K[Y ]. Show that its terms are F0 = A,

F1 = V2 F ⊗ A(−2),

F2 = V2,1 F ⊗ A(−3),

F3 = V2,1,1 F ⊗ A(−4), F4 = V2,1,1 F ⊗ A(−6), F6 = V2 F ⊗ A(−8),

F5 = V2,1 F ⊗ A(−7), F7 = A(−10).

This representation is one of the so-called subexceptional series, corresponding to the entries in the row of the Freudenthal magic square. These representations are (using Bourbaki’s notation for fundamental weights) (a) G = SL(2), V = V (3ω1 ), (b) G = SL(2) × SL(2) × SL(2) × 3 , V = V (ω1 ) ⊗ V (ω1 ) ⊗ V (ω1 ), where 3 denotes a permutation group, (c) G = SP(6), V = V (ω3 ), (d) G = SL(6), V = V (ω3 ), (e) G = (spinor group.)(12), V = V (ω5 ) (highest weight ( 12 , 12 , 12 , 12 , 1 1 , )), 2 2 (f) G = E 7 , V = V (ω6 ). The uniformity, described by Landsberg and Manivel in [LM], is with respect to the parameter m, which in the above cases takes values − 23 , 0, 1, 2, 4, 8. The dimension of the representation V is 6m + 8. Again there are four orbits: the general one, the hypersurface deﬁned by the vanishing of a (unique up to scalar) invariant of degree 4, X , and Y . The codimension of X is m + 3; the codimension of Y is 3m + 4. Notice that item (d) on the list is the representation 3 F for dim F = 3. The resolution of X in that case is described in (7.3.8). The resolution of Y is calculated in exercises 4, 5, 6.

8 The Nilpotent Orbit Closures

In this chapter we deal with another important class of varieties – the nilpotent orbit closures of the adjoint action of a simple algebraic group on its Lie algebra. These varieties play an important role in representation theory. All such orbit closures have desingularizations which are total spaces of vector bundles over homogeneous spaces. We describe the applications of the geometric method. The vector bundles involved in the construction of these desingularizations are more complicated than in the case of determinantal varieties. The explicit formula for the terms of complexes F(L)• is not known in general. Still, one can prove some interesting results. The ﬁrst two sections of the chapter are devoted to the nilpotent orbit closures for the general linear group. In section 8.1 we describe the desingularizations of these orbit closures explicitly. We apply theorems from chapter 5 to prove that all orbit closures are normal, are Gorenstein, and have rational singularities. We also describe the combinatorial way of estimating the terms of the complexes F• in this case. This method is then used in section 8.2 to describe the generators of the deﬁning ideals of nilpotent orbit closures. In section 8.3 we treat the case of general simple groups. We prove a theorem of Hinich and Panyushev saying that the normalization of every nilpotent orbit closure is Gorenstein and has rational singularities. Finally, in sections 8.4 and 8.5 we look at the case of classical groups. We mainly work with examples showing how geometric method can be applied in special cases. We give examples of nonnormal orbit closures and discuss some special cases. In the case of the symplectic groups, for orbits corresponding to partitions with even parts, we prove the estimate on the weights of representations generating the deﬁning ideals. We give conjectures for such estimates for all nilpotent orbits for classical groups.

251

252

The Nilpotent Orbit Closures

8.1. The Closures of Conjugacy Classes of Nilpotent Matrices Let E be a vector space of dimension n over a ﬁeld K. We consider the afﬁne space X = HomK (E, E) of n × n matrices. We identify the space X with E ∗ ⊗ E. The general linear group GL(E) acts on X by conjugation; the element g ∈ GL(E) sends the matrix φ ∈ Hom(E, E) to g −1 φg. The coordinate ring of X can be identiﬁed with the symmetric algebra A = Sym(E ⊗ E ∗ ). We start with the brief analysis of the ring of invariants AGL(E) . Let us denote by vi (i = 1, . . . , n ) the unique (up to scalar) GL(E) invariant in i E ⊗ i E ∗ . To ﬁx the scalar we can choose φ I,I vi = I ⊂[1,n], |I |=i

to be the sum of principal minors of the generic matrix φ. Notice that vi is the coefﬁcient of s n−i in the characteristic polynomial χ (φ, s) := det(φ − s Id) of φ. The following proposition is a special case of Chevalley’s theorem. (8.1.1) Proposition. The ring AGL(E) is a polynomial ring in v1 , . . . , vn . Proof. We notice that by Cauchy decomposition A= L α E ⊗ L α E ∗. α

Moreover, by the Littlewood–Richardson rule (2.3.4) we see that for each α the tensor product L α E ⊗ L α E ∗ contains precisely one trivial representation. This means that the dimension of the d-th graded component AdGL(E) is equal to the number of partitions of d with at most n parts. This in turn means that the Poincar´e series of AGL(E) is given by the formula 1 (dim AGL(E) )t d = PAGL(E) (t) := . d (1 − t) . . . (1 − t n ) d≥0 This suggests that AGL(E) is a polynomial ring in n variables, with generators of degree 1, . . . , n. To conclude the proof of (8.1.1) it is enough to show that the polynomials v1 , . . . , vn are algebraically independent. This is however clear: after substituting φi, j = 0 for i = j and φi,i = xi , we see that the polynomial vi specializes to the elementary symmetric function ei (x1 , . . . , xn ). The embedding AGL(E) → A induces the orbit map χ : X → X/GL(E) = Kn sending the matrix φ to its characteristic polynomial χ(φ, s).

8.1. The Closures of Conjugacy Classes of Nilpotent Matrices

253

This map has been analyzed in many contexts. We will take the point of view of geometric invariant theory and analyze the nullcone of X – the ﬁber χ −1 (0). This is the set of matrices with all eigenvalues equal to 0, i.e. the set of nilpotent matrices. By the Jordan canonical form we know that the set χ −1 (0) has ﬁnitely many GL(E)-orbits. They correspond to the partitions µ of n. For a partition µ = (µ1 , . . . , µr ) we denote by O(µ) the set of nilpotent matrices with Jordan blocks of dimensions µ1 , . . . , µr . If µ = (µ1 , . . . , µs ) is the partition conjugate to µ, we can describe the orbit O(µ) as a set of endomorphisms φ of E for which dim Ker φ i = µ1 + . . . + µi for i = 1, . . . , s. We denote by Yµ the closure of the orbit O(µ) in X . (8.1.2) Examples. (a) µ = (n). In this case the set Yµ is the set of all nilpotent matrices. (b) µ = (1n ). In this case Yµ is just a point 0. (c) More generally, let µ = ( p, 1n− p ). The variety Yµ is a set of nilpotent matrices of rank < p. (d) Let µ = (2i , 1 j ) where 2i + j = n. In this case Yµ is a set of matrices φ such that φ 2 = 0 and rank φ ≤ i. (8.1.3) Proposition. The closure Yµ is deﬁned set-theoretically by the conditions dim Ker φ i ≥ µ1 + . . . + µi for i = 1, . . . , s. Proof. The conditions of the Proposition are algebraic and satisﬁed on O(µ), so they have to be satisﬁed on Yµ . Let us denote by Yµ the closed subset of X of endomorphisms φ for which all the above inequalities are equalities. In order to show that Yµ = Yµ it will be enough to show that Yµ is irreducible, of the same dimension as O(µ). In order to do that we will construct a desingularization of Yµ . Let us consider the ﬂag variety Vµ = Flag(µ1 , µ1 + µ2 , . . . , µ1 + . . . + µs−1 ; E). We denote the typical point in Vµ by (Rµ1 , . . . , Rµ1 +...+µs−1 ), and we also write in that setup that Rµ1 +...+µs = E.

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The Nilpotent Orbit Closures

Consider the incidence variety Z µ = {(φ, (Rµ1 , . . . , Rµ1 +...+µs−1 )) ∈ X × Vµ | φ(Rµ1 ) }. = 0, ∀2≤i≤s φ(Rµ1 +...+µi ) ⊂ Rµ1 +...+µi−1

Notice that the last conditions imply Rµ1 +...+µi ⊂ Ker (φ i ). Now we can consider the diagram Zµ ⊂ ↓ qµ Yµ ⊂

X × Vµ ↓ qµ X

We denote by pµ the projection of X × Vµ onto Vµ and its restriction to Z µ . It is clear that this makes Z µ a vector bundle on Vµ . Thus we have the exact sequence of vector bundles on Vµ 0 −→ Sµ −→ E ⊗ E ∗ −→ Tµ −→ 0, where E ⊗ E ∗ denotes a trivial bundle on Vµ with the ﬁber E ⊗ E ∗ , and Z µ is a total space of Sµ . The ﬁber of qµ over a point from O(µ) consists of one point. Indeed, if the pair (φ, (Rµ1 , . . . , Rµ1 +...+µs−1 )) ∈ Z µ and φ ∈ O(µ), we are forced to have Rµ1 +...+µi = Ker φ i . Since the variety Z µ is irreducible, we see that Yµ is irreducible and of the same dimension that O(µ). This completes the proof of Proposition (8.1.3). The construction of the desingularization Z µ places us in the situation of section 5.1. Thus we get a Koszul complex K(ξµ )• of O X ×Vµ -modules which is a Koszul complex resolving the structure sheaf O Z µ . The terms of K(ξµ )• are given by the formula K(ξµ )• : 0 →

t

( p ∗ ξµ ) → . . . →

2

( p ∗ ξµ ) → p ∗ (ξµ ) → O X ×Vµ

where ξµ = Tµ∗ . We also denote ηµ = Sµ∗ . For a vector bundle V on Vµ we denote by M µ (V) the sheaf O Z µ ⊗ p ∗ (V) of O X ×Vµ -modules. We also recall that by A we denote the polynomial ring Sym(E ⊗ E ∗ ) of regular functions on X . Applying Theorem (5.1.2) to our situation, we get (8.1.4) Basic Theorem for Nilpotent Orbits. For a vector bundle V on Vµ we deﬁne free graded A-modules i+ j µ j ξµ ⊗ V ⊗k A(−i − j). F (V)i = H Vµ , j≥0

8.1. The Closures of Conjugacy Classes of Nilpotent Matrices

255

(a) There exist minimal differentials µ

di (V) : F µ (V)i → F µ (V)i−1 of degree 0 such that F•µ (V) is a complex of graded free A-modules with H−i (F µ (V)• ) = Ri q∗ M(V). In particular the complex F µ (V)• is exact in positive degrees. (b) The sheaf Ri q∗ M µ (V) is equal to H i (Z , M µ (V)), and it can be also identiﬁed with the graded A-module H i (V, Sym(ηµ ) ⊗ V). (c) If φ : M µ (V) → M µ (V )(n) is a morphism of graded sheaves, then there exists a morphism of complexes f • (φ) : F µ (V)• → F µ (V )• (n). Its induced map H−i ( f • (φ)) can be identiﬁed with the induced map H i (Z , M µ (V)) → H i (Z , M µ (V ))(n). The calculation of cohomology groups H j (Vµ , i+ j ξµ ⊗ V) and H i (V, Sym(ηµ ) ⊗ V) is much more difﬁcult than in the case of determinantal varieties, or even in the case of hyperdeterminants, considered in chapter 9. The reason is that the bundle ξµ cannot be expressed conveniently as a tensor product of tautological bundles. In fact this calculation has not been done explicitly even for the case where V is trivial, i.e. for the case of syzygies of the coordinate ring of Yµ . This is a very interesting problem that should be a subject of further research. I believe the full solution is possible and should lead to very interesting combinatorics. In the remainder of this section we will describe the inductive procedure which allows to give the estimates on the terms of complexes F•µ in the case of nilpotent orbits. This will be the main tool in the next section when we describe the generators of their deﬁning ideals. Let us ﬁrst mention that the constructions of Z µ , ξµ , and ηµ can be done in a relative setting, i.e. when we replace the vector space E by the vector bundle E over some scheme. Therefore we can talk about the bundles ξµ and ηµ associated to the bundle E of dimension n. They are respectively a subbundle and a factorbundle of E ⊗ E ∗ . Since the bundles ξµ and ηµ do not change when we replace E by E ∗ , we will denote them by ξµ (E, E ∗ ) and ηµ (E, E ∗ ) respectively. ˆ the Now we ﬁx n and a partition µ = (µ1 , . . . , µr ). We denote by µ partition (µ1 − 1, . . . , µr − 1). The conjugate partition µˆ is the partition (µ2 , . . . , µs ).

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The Nilpotent Orbit Closures

On the flag variety Vµ we have the tautological subbundles Rµ1 +...+µi and the corresponding factorbundles Qµi+1 +...+µs . The indices here denote the dimensions. We will denote Rµ1 by R and Qµ2 +...+µs by Q. In this setting we have an exact sequence of vector bundles on Vµ : 0 −→ R ⊗ E ∗ −→ ξµ −→ ξµˆ (Q, Q∗ ) −→ 0, where ξµˆ (Q, Q∗ ) denotes the bundle ξµˆ in the relative setting described above. For each t ≥ 0 this sequence induces a ﬁltration Fit , 0 ≤ i ≤ t, on t t t (ξµ ) such that for each i ≤ t − 1 Fit ⊂ Fi+1 and Fi+1 /Fit = t−i (R ⊗ E ∗ ) ⊗ i ξµˆ (Q, Q∗ ). Indeed, we can deﬁne F0 = 0 and, for each i ≤ t − 1, F i+1 to be the image of the map t−i

(R ⊗ E ∗ ) ⊗

i

ξµ →

s

ξµ

induced by exterior multiplication. This allows us to consider for 0 ≤ i ≤ t − 1 the exact sequences t 0 → Fit → Fi+1 →

t−i

(R ⊗ E ∗ ) ⊗

i

ξµˆ (Q, Q∗ ) → 0

(∗)

0≤i≤t

and the associated long sequences of cohomology groups. We will use these sequences to estimate the terms in the complex F•µ . We start with the introduction of the bundle ξµ = Rµ1 ⊗ E ∗ ⊕ (Rµ1 +µ2 /Rµ1 ) ⊗ Q∗n−µ ⊕ . . . ⊕ 1

(E/Rµ1 +...+µs−1 ) ⊗ Q∗µs The consecutive sequences of type (∗) deﬁne a ﬁltration on the bundle ξµ whose associated graded is the bundle ξµ . We will need a recursive form of (∗), which is ξµ = R ⊗ E ∗ ⊕ ξµˆ (Q, Q∗ ),

(∗∗)

where ξµˆ (Q, Q∗ ) is a bundle of type ξ constructed in a relative situation. We introduce GL(E)-modules µ

G i :=

i+ j H j Vµ , ξµ .

j≥0

It is clear from the long exact sequences of cohomology associated to se quences (∗) that the groups H j (Vµ , i+ j ξµ ) are smaller than the groups

8.1. The Closures of Conjugacy Classes of Nilpotent Matrices

257

H j (Vµ , i+ j ξµ ). Therefore our ﬁrst step will be to give a procedure to µ calculate the terms G i . It is based on the formula (∗∗), which implies •

ξµ =

• • (R ⊗ E ∗ ) ⊗ ξµˆ (Q, Q∗ ).

µ

The terms of G i can be calculated in the following way. Let us choose the term of G µ•ˆ (Q, Q∗ ) which is a representation of the type K α Q where α is an integral dominant weight for the group GL(n − µ1 ). Let us assume that this term occurs in homogeneous degree t and in homological degree u. For this term we calculate G(K α Q) =

H j (Grass(µ1 , E), K α Q) ⊗

i+ j

(R ⊗ E ∗ ),

(?)

i, j≥0

where the terms corresponding to a given i, j will appear in homogeneous degree t + i + j and in homological degree u + i. Notice that the terms of the collection G(K α Q) will appear as tensor products K β E ⊗ K γ E ∗ , but we can decompose them into irreducible representations to make the next step. Now all the collections G(K α Q) give us the terms of G µ• . Notice that the collection (?) comprises the terms of the complex F(V)• associated to the bundle ξ = R ⊗ E ∗ with a twist V = K α Q. These are the twisted complexes with the support in the determinantal variety of n × n matrices of rank ≤ n − µ1 . Such complexes were considered in section 6.5. These complexes are GL(E) × GL(E ∗ )-equivariant. Now we will use our procedure to estimate the weights of the terms of µ G • . First we notice that all terms in G µ• are the representations K β E where the sum of all entries in the weight β is equal to 0. To separate positive, zero, and negative entries of β, let us write β = (σ, 0c , \τ ) where σ and τ are two partitions. Here for τ = (τ1 , . . . , τt ) we let \τ = (−τt , . . . , −τ1 ). (8.1.5) Lemma. Let n and µ be as above. Let us consider the term K α Q of G µ•ˆ (Q, Q∗ ) where α is such that the sum of its entries equals 0 and its last entry is ≥ −µ1 . (a) We consider the twisted complex F(V)• associated to the bundle ξ = R ⊗ E ∗ , with the twist V = K α Q. Then F(V)i = 0 for i < 0. Moreover, F(V)0 = K σ E ⊗ K τ E ∗ ⊗ A(−|σ |) and F(V)1 = K σ,1c+1 E ⊗ K τ,1c+1 E ∗ ⊗ A(−|σ | − c − 1).

258

The Nilpotent Orbit Closures

All terms of F(V)• are of the form K β E ⊗ K γ E ∗ where β and γ are partitions of the same number, and γ1 ≤ µ1 . µ ˆ (b) Let us assume that the term K α Q occurs in G i (Q, Q∗ ). Then all terms µ from G(K α Q) occur in G j with j ≥ i. Proof. First of all, let us notice that part (b) follows instantly from part (a). We will prove part (a). Let us write in this proof t = µ1 for short. By the Cauchy decomposition (3.2.5) we have •

(R ⊗ E ∗ ) =

Kβ R ⊗ Kβ E ∗.

β

This means that for each partition β we have to calculate the cohomology groups of K α Q ⊗ K β R ⊗β E ∗ , which are the cohomology groups of K α Q ⊗ K β R tensored with K β E ∗ . To calculate this cohomology for each β = (β1 , . . . , βt ) we have to apply Bott’s theorem (4.1.4) to the sequence z(β) = (α, β) = (σ, 0c , \τ, β). Let us consider the weight z(β) + ρ. This is a sequence (σ1 + n − 1, . . . , σu + n − u, t + v + c − 1, . . . , t + v, t + v − 1 − τv , . . . , t − τ1 , β1 + t − 1, . . . βt ), where we write σ = (σ1 , . . . , σu ), τ = (τ1 . . . , τv ) and keep in mind that n = u + c + v + t. By (4.1.9) the partitions β giving nonzero contributions to cohomology are these for which the sequence z(β) + ρ has no repetitions. For such β we have to reorder our sequence to make it decreasing, and subtract ρ from it. We get a weight γ (β). The resulting cohomology group will then be K γ (β) E ⊗ K β E ∗ occurring in the complex F(V)• in the place p(β) = |β| − l(w) where w is a permutation needed to reorder z(β) + ρ. First we notice that since t ≥ τ1 then all the entries in z(β) + ρ are positive which means that all the entries of γ (β) are nonnegative. Let us consider two partitions: β and γ = (β1 , . . . , βs + j, βs+1 , . . . , βt ), both giving nonzero contributions to the terms of F• (V). We will show that p(γ ) > p(β). Indeed, the sequences z(β) + ρ and z(γ ) + ρ differ only in one place, and the term βs + j + t − s in z(γ ) + ρ can be exchanged with at most j − 1 additional numbers compared to the corresponding term βs + t − s in z(β) + ρ. This will account for an increase of at most j − 1 in l(w).

8.1. The Closures of Conjugacy Classes of Nilpotent Matrices

259

Starting with an arbitrary β we can now use the steps described above to produce terms with smaller β that lie in smaller homological degree. Continuing like this we can make β satisfy the condition c + v ≥ β1 . On the other hand if c + v ≥ β1 , then {t + v + c + 1, . . . , t + v, t + v − 1 − τv , . . . , t − τ1 , β1 + t − 1, . . . , βt } are t + v + c numbers belonging to {0, 1, . . . , t + v + c − 1}. They can be distinct for a unique partition β and it is easy to see that β = τ . It is also clear that β = τ is the smallest partition giving a nonzero contribution the terms of F(V)• . By the previous argument p(β) is the smallest for β = τ . It is very easy to check that in fact p(τ ) = 0. Similarly we can identify γ = (τ, 1c ) as the only partition with p(γ ) = 1. It remains to prove the last statement in (a). But this follows if we take into account that the term of F• (V) corresponding to β is K γ (β) E ⊗ K β E ∗ and that β1 ≤ t. (8.1.6) Theorem. Let µ be a partition of n. Let F•µ be a complex of A-modules described in (8.1.4). (a) (b) (c) (d) (e)

µ

The terms Fi are zero for i < 0. µ The term F0 equals A(0). For any representation K α E occurring in F•µ we have αn ≥ −µ1 . The varieties Yµ are normal and they have rational singularities. The coordinate rings of varieties Yµ are Gorenstein.

Proof. First we notice Theorem (5.1.3) (c) implies that (d) follows from (a) and (b). Also we notice that (e) follows from Theorem (5.1.4). The point is that the maximal exterior power of ξµ is isomorphic to the canonical bundle on Vµ , by Exercise 13, chapter 3. Therefore it is enough to prove the ﬁrst three statements of the theorem. We will actually deal with the spaces G µ• . We will prove the following statements: µ

(a ) The terms G i are zero for i < 0. µ (b ) The term G 0 equals k. (c ) For any representation K α E occurring in G µ• we have αn ≥ −µ1 . The statements for F•µ follow because the cohomology groups of exterior powers of ξµ are smaller than those of exterior powers of ξµ . We argue by induction on the number of parts in µ .

260

The Nilpotent Orbit Closures

If µ has only one part, i.e. µ = (n), then Yµ is the origin and ξµ = ξµ , µ so G i = i (E ⊗ E ∗ ). Then obviously (a ) and (b ) are satisﬁed, and (c ) follows from the Cauchy formula (2.3.3) (b) and the Littlewood–Richardson rule (2.3.4). Let us therefore assume that µ = (µ1 , . . . , µs ) and that all three statements are true for partitions with at most s − 1 parts. Let µˆ = (µ2 , . . . , µs ). We consider the terms of the complex G µ•ˆ (Q, Q∗ ). By the inductive assumption, all such terms K α Q occur in nonnegative homological degrees and they satisfy the assumption αn−µ1 ≥ −µ2 . This means each term satisﬁes the assumptions of Lemma (8.1.5). Now statement (a ) for µ follows from (8.1.5) (b). We also µ see that the contribution to G 0 can come only from the 0th term of G µ•ˆ (Q, Q∗ ). But this consists of one copy of the trivial representation. Therefore by (8.1.5) (a) statement (b ) is satisﬁed for µ. Finally (c ) also follows from the last part of (8.1.4) (a) and from the Littlewood–Richardson rule (2.3.4). In the sequel we will denote by Aµ the coordinate ring of Yµ . Theorem (8.1.6) implies that the complex F•µ is a minimal free resolution of Aµ treated as an A-module. It also says that the rings Aµ are normal, are Gorenstein, and have rational singularities. We will denote by Jµ the deﬁning ideal of Yµ . This means that Aµ = A/Jµ . Thus the ﬁrst term of F•µ gives us the information about the minimal generators of the ideal Jµ . Let us note the following corollary from our calculation of G µ• . µ

(8.1.7) Corollary. Let µ be a partition of n. The term G 1 contains only the representations K (1 j ,0n−2 j ,(−1) j ) E for 0 ≤ j ≤ n2 . Proof. Again we use the induction on the number of parts of µ . For µ having one part, again the result is true, because G i(n) = i (E ⊗ E ∗ ). To make an inductive step from µ ˆ to µ, we again use Lemma (8.1.5). We ﬁnish this section with some examples where the complete calculation of the complex F•µ is possible. (8.1.8) Examples. (a) Let µ = (n). Then Yµ = {0}. The bundle ξµ = E ⊗ E ∗ , and therefore Fi(n) =

i

(E ⊗ E ∗ ) ⊗ A(−i)

8.1. The Closures of Conjugacy Classes of Nilpotent Matrices

261

and F•(n) is the Koszul complex associated to the ideal generated by all variables φi, j . (b) Let µ = (1n ). The variety Y(1n ) is the set of all nilpotent matrices. We n will show that the complex F•(1 ) is the Koszul complex associated to the ideal generated by the basic invariants v1 , . . . , vn . This means that n (1n ) A − j j Fi = {(1 ,...,n ) | j ∈{0,1},

n

j=1

j =i}

j=1 n

) We show ﬁrst that the cohomology groups G (1 • consist of trivial representations only. We proceed by induction on n. For n = 1, the statement is obviously true. Let us assume the statement is true for µ = (1n−1 ). Using the formula for G(K α Q), we see that each trivial representation n−1 n which is a term of G •(1 ) (Q1 , Q∗1 ) contributes to G •(1 ) the terms

H

j

Grass(1, E),

i+ j

∗

(R ⊗ E ) ,

i, j≥0

where R is the tautological subbundle on Grass(1, E), i.e. the bundle O(−1). Now it follows from Serre’s theorem ([H1, chapter III, Theorem 5.1]) that only two cohomology groups in the above formula are nonzero. These are 0 0 ∗ H Grass(1, E), (R ⊗ E ) = K (0) E and

n H n−1 Grass(1, E), (R ⊗ E ∗ ) = K (0) E.

Therefore we get two copies of the trivial representation. Our staten ment is proved. It follows that the terms of the complex F•(1 ) consist entirely of trivial representations. This implies that the ideal of functions vanishing on Y(1n ) is generated by GL(E)-invariants, i.e. by v1 , . . . , vn (in view of the fact that B = AGL(E) is a polynomial ring in v1 , . . . , vn ). It remains to show that v1 , . . . , vn form a regular sequence in A. This follows from dimensional considerations because dim Y(1n ) = dim Z (1n ) = n 2 − n. (c) Let µ = (n − p, 1 p ) be a hook. The variety Yµ is the set of nilpotent matrices of rank ≤ p. We again use the formulas for G(K α Q). p ) ∗ Each term of G (1 • (Q, Q ) (which has to be a trivial representation)

262

The Nilpotent Orbit Closures

contributes

i+ j H j Grass(n − p, E), (R ⊗ E ∗ ) ,

i, j≥0

where R and Q denote respectively the tautological subbundle and factorbundle on Grass(n − p, E). This gives us the terms of the Lascoux complex resolving p + 1 order minors of φ, described in (6.1.3). Lemma (8.1.5) now shows that the generators of the deﬁning ideal of Yµ are a subrepresentation of a direct sum of p copies of trivial representations (in homogeneous degrees 1, . . . , p) and of the copy of p+1 E ⊗ p+1 E ∗ in homogeneous degree p + 1. On the other hand, we have some obvious polynomials vanishing on Yµ : the minors of degree p + 1 of the matrix φ, and the invariants v1 , . . . , v p . Moreover, these equations obviously deﬁne Yµ set-theoretically. If we consider the determinantal variety Y p of matrices of rank ≤ p, the dimension count shows that dim Y(n− p,1 p ) = dim Y p − p. This means that the invariants v1 , . . . , v p form a regular sequence in the coordinate ring of the determinantal variety Y p . Now we identify the generators of the deﬁning ideal of Yµ by induction on the homogeneous degree. In degrees 1, . . . , p the only generators are v1 , . . . , v p , because they have to come from trivial representations. In degree p + 1 the minors of degree p + 1 of φ have to be among the minimal generators (because no linear combinations of these minors with coefﬁcients in K can be in the ideal generated by v1 , . . . , v p ). Our estimate from above of the p ﬁrst term of F•(n− p,1 ) shows now that the deﬁning ideal of Y(n− p,1 p ) is generated by v1 , . . . , v p and by p + 1 order minors of φ. The complex p F•(n− p,1 ) being a minimal resolution of this ideal, has to be the tensor product of the Lascoux’s resolution (6.1.3) of the ideal of p + 1 order minors, and of the Koszul complex on v1 , . . . , v p . (d) Let µ = (n − p, p) be the partition with two columns. The variety Yµ consists of matrices φ such that rank φ ≤ p and φ 2 = 0. These varieties were ﬁrst considered by Strickland in [S], where they were called the projector varieties, because of the property φ 2 = 0. Here we just identify the homogeneous components of the coordinate ring of Yµ . We notice that the ﬂag variety Vµ we are using is in this case just Grass(n − p, E). It is easy to identify the bundle ηµ with Q ⊗ R∗ , where as usual R and Q denote respectively the tautological subbundle and factorbundle on Grass(n − p, E). Applying Theorem (8.1.4)(b) and Theorem (8.1.6), we see that F•µ is a minimal resolution of the coordinate ring of Yµ which can be identiﬁed with

8.2. The Equations of the Conjugacy Classes of Nilpotent Matrices 263

H 0 (Grass(n − p, E), Sym• (Q ⊗ R∗ )). Thus Cauchy’s formula (3.2.5) gives Symd (Q ⊗ R∗ ) = K α Q ⊗ K α R∗ . |α|=d

Using Corollary (4.1.9), we get that for α = (α1 , . . . , α p ) H 0 (Grass(n − p, E), K α Q ⊗ K α R∗ ) = K (α1 ,...,α p ,0n−2 p ,−α p ,...,−α1 ) E. The result is that (A(n− p, p) )d =

K (α1 ,...,α p ,0n−2 p ,−α p ,...,−α1 ) E.

|α|=d

This allows us to see that A(n− p, p) is a factor of A. Cauchy’s formula for A gives Ad = Kα E ⊗ Kα E ∗. |α|=d

We see now that if α = (α1 , . . . , αs ) with s ≥ p + 1, then the summand K α E ⊗ K α E ∗ is contained in J(n− p, p) . If s ≤ p, the only surviving part of K α E ⊗ K α E ∗ in A(n− p, p) is the Cartan representation K (α1 ,...,α p ,0n−2 p ,−α p ,...,−α1 ) E from K α E ⊗ K α E ∗ . The kernel of the epimorphism K α E ⊗ K α E ∗ → K (α1 ,...,α p ,0n−2 p ,−α p ,...,−α1 ) E consists of all polynomials having a trace component, i.e., combi n nations of polynomials divisible by ws,t = i=1 φi,s φt,i or by v1 =

n i=1 φi,i . Therefore the ideal J(n− p, p) is spanned by the quadratic polynomials of that kind, by the invariant v1 , and by the p + 1 order minors of φ.

8.2. The Equations of the Conjugacy Classes of Nilpotent Matrices We use the inductive procedure from the previous section to get the information about the generators of the deﬁning ideals Jµ of the varieties Yµ . We preserve the notation from the previous section. In particular, µ = (µ1 , . . . , µr ) is a partition of n, and µ ˆ = (µ1 − 1, . . . , µr − 1). For such a partition µ we denote the coordinate ring of Yµ by Aµ . Thus Aµ = A/Jµ . We start by exhibiting some explicit polynomials vanishing on Yµ . By Corollary (8.1.7) we know that the generators of the ideals Jµ consist of the representations of type K (1 j 0n−2 j ,(−1) j ) E for 0 ≤ j ≤ n2 . The representations of this type are exactly the ones occurring as composition factors of repre sentations p E ⊗ p E ∗ which are linear spans of p × p minors of the

264

The Nilpotent Orbit Closures

matrix φ. The straightening law (3.2.5) tells us that there is a basis of the polynomial ring A = Sym(E ⊗ E ∗ ) consisting of the products of minors of the generic n × n matrix φ = (φi, j )1≤i, j≤n . This gives the idea of looking for the polynomials vanishing on Yµ which are linear combinations of the minors of various sizes of φ. Let p be a number such that 1 ≤ p ≤ n, and let I = (i 1 , . . . , i p ) , J = ( j1 , . . . , j p ) be two multiindices with entries in the set [1, n]. We denote by φ I,J the p × p minor of φ corresponding to rows i 1 , . . . , i p and columns j1 , . . . , j p . The polynomial φ(I |J ) is obviously antisymmetric in the entries i 1 , . . . , i p and in the entries j1 , . . . , j p . We consider the linear span of p × p minors of φ. This is a linear supspace of A p , which can be identiﬁed with p E ⊗ p E ∗ . If {e1 , . . . , en } is a basis of E, and {e1∗ , . . . , en∗ } is a dual basis of E ∗ , the tensor ei1 ∧ . . . ∧ ei p ⊗ e∗j1 ∧ . . . ∧ e∗j p is identiﬁed with φ I,J . Using the Littlewood–Richardson rule (2.3.4), we see the following decomposition: p

E⊗

p

E∗ =

K (1i ,0n−2i ,(−1)i ) E.

0≤i≤ min( p,n− p)

We will denote the copy of the representation K (1i ,0n−2i ,(−1)i ) E inside the span p E ⊗ p E ∗ of p × p minors by Ui, p . If i > min( p, n − p), we set Ui, p = 0. This means we can rewrite the previous decomposition as p

p

E⊗

E∗ =

Ui, p .

0≤i≤ min( p,n− p)

Next, we denote by Vi, p the subspace of the span minors deﬁned as

p

E⊗

p

E ∗ of p × p

Vi, p = U0, p ⊕ U1, p ⊕ . . . ⊕ Ui, p . The point of introducing the spaces Vi, p is that they have a simple description in terms of minors of φ. The space Vi, p is isomorphic as a GL(E)-module to i E ⊗ i E ∗ , and it can be identiﬁed with the image of the map h i, p :

i

E⊗

m⊗m

E ∗ −→

p

i

1⊗1⊗t p−i

E ∗ −→

E⊗

p

E ∗, p−i

i

E⊗

i

E∗ ⊗

p−i

E⊗

p−i

where t p−i : K → p−i E ⊗ E ∗ is the map sending 1 to the GL(E)invariant t p−i , and m ⊗ m denotes exterior multiplication of the ﬁrst component with 3rd, and of the second with 4-th.

8.2. The Equations of the Conjugacy Classes of Nilpotent Matrices 265

Let us ﬁx two multiindices P = ( p1 , . . . , pi ) and Q = (q1 , . . . , qi ). We have φ(P,J |Q,J ) , h i, p (e p1 ∧ . . . ∧ e pi ⊗ eq∗1 ∧ . . . ∧ eq∗i ) = |J |= p−i

and so Vi, p is the span of such elements for different choices of P and Q. (8.2.1) Lemma. Let µ be a partition of n. The elements of the space Vi, p vanish identically on the variety Yµ if and only if p > µ1 + . . . + µi − i. Proof. Let us ﬁrst assume that the condition of Lemma (8.2.1) is satisﬁed. We will show that Vi, p vanishes on Yµ . Let us consider the typical generator φ(P,J |Q,J ) | j|= p−i

for ﬁxed subsets P and Q of cardinality i. Since these elements span a GL(E)stable subspace in A p , it is enough to show that they vanish on a single matrix from O(µ). Choosing a matrix from O(µ) in a canonical Jordan form, it is easy to see that in fact all summands φ(P,J |Q,J ) are zero when evaluated on that matrix. To prove the other implication, let us assume that the condition of (8.2.1) is not satisﬁed. This means that p ≤ µ1 + . . . + µi − i. Let us choose j for which µ j > 1 and µ j+1 = 1 (if µi > 1, we choose j = i). Now we set P = (1, µ1 + 1, . . . , µ1 + . . . + µ j + 1). We can choose the numbers w1 , . . . , w j in such way that 1 ≤ wm ≤ µm for m = 1, . . . , j, and w1 + . . . + w j = p + i. We set Q = (w1 , µ1 + w2 , . . . , µ1 + . . . + µ j−1 + w j ).

We consider the polynomial |J |= p− j φ(P,J |Q,J ) . Clearly its value on a matrix from O(µ) in Jordan canonical form is not zero. Moreover, our element is in V j, p , which is contained in Vi, p by deﬁnition. This completes the proof of the lemma. Our goal in this section is to prove that the representations Vi, p satisfying the condition (8.2.1) are the generators of the ideal Jµ . It is worthwhile to point out right away that they do not deﬁne a minimal set of generators of

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the ideals Jµ . For example, the following result is a simple consequence of Laplace expansion. (8.2.2) Lemma. For i ≥ 1 the representation Vi, p+1 is contained in the ideal generated by the representation Vi, p . Let us denote by Jµ the ideal generated by the representations Vi, p satisfying the condition in (8.2.1). Lemma (8.2.2) immediately implies (8.2.3) Proposition. The ideal Jµ is generated by the spaces Ui,µ(i) (1 ≤ i ≤ n) where µ(i) = µ1 + . . . + µi − i + 1 (which are zero if i > min(µ(i), n − µ(i))), and by the spaces U0, p (for 1 ≤ p ≤ n) which correspond to the invariants t p . We will use the following graphical representation of the representations Ui, p . We will represent them by an (n + 1) × ([ n2 ] + 1) matrix whose rows are indexed by 0, 1, . . . , n and columns are indexed by 0, 1, . . . , [ n2 ]. The ( p, i)th entry corresponds to Ui, p . We will treat the entries corresponding to the spaces Ui, p which are zero as empty. For a given partition µ we will represent the generators of Jµ by a matrix M(µ) whose entry equals 1 if the corresponding Ui, p vanishes on Yµ , equals 0 if the corresponding Ui, p does not vanish on Yµ , and is empty if the corresponding Ui, p = 0. (8.2.4) Example. Let us take n = 12, µ = (3, 3, 2, 2, 1, 1). We have µ(1) = 3, µ(2) = 5, µ(3) = 6, µ(4) = µ(5) = µ(6) = 7. This means we have 0 1 1 1 1 1 M(µ) = 1 1 1 1 1 1 1

0 0 1 1 1 1 1 1 1 1 1

0 0 0 1 1 1 1 1 1

0 0 0 1 1 1 1

0 0 0 0 0 0. 1 1 1

8.2. The Equations of the Conjugacy Classes of Nilpotent Matrices 267

The fourth row of the matrix tells us that U0,3 and U1,3 are in Jµ and that U2,3 and U3,3 are not. Lemma (8.2.2) tells us that Jµ is generated by the Ui, p ’s corresponding to the 1’s in the ﬁrst column of M(µ) and by the ones corresponding to the highest 1’s in each other column. In our exam is generated by U0, p (1 ≤ p ≤ 12) and by ple we would conclude that J(6,4,2) U1,3 , U2,5 , U3,6 , U4,7 , U5,7 . Now we state the main result of this section. (8.2.5) Theorem. For each partition µ the ideals Jµ and Jµ are equal, i.e., Jµ is generated by the spaces Ui,µ(i) (for 1 ≤ i ≤ n ) and by the spaces U0, p (for 1 ≤ p ≤ n). (8.2.6) Remark. The set of generators given in (8.2.5) is not claimed to be minimal. We will discuss the minimal sets of generators at the end of this section. Proof of Theorem (8.2.5). We proceed by induction on the number s of parts in µ . If s = 1, we have µ = (n) and Y(1n ) = 0. Therefore the ideal Jµ is generated by the entries φi, j , i.e., by the representations U0,1 and U1,1 . The , so we combinatorial condition in (8.2.1) tells us that U0,1 and U1,1 are in J(n) are done. ˆ = Let us consider the partition µ = (µ1 , . . . , µr ). As before we denote µ (µ1 − 1, . . . , µr − 1). By induction we know that the generators of Jµˆ are (1 ≤ i ≤ n − µ1 ) and by the spaces U0, p (1 ≤ p ≤ given by the spaces Ui,µ(i) ˆ n − µ1 ). Let us consider the complex F•µˆ . We can construct this complex in a relative situation, taking the bundle Q on the Grassmannian Grass(µ1 , E) instead of a vector space E. We get a complex F•µˆ (Q, Q∗ ) of locally free sheaves over the sheaf of algebras A := Sym(Q ⊗ Q∗ ) deﬁned over the Grassmannian Grass(µ1 , E). The terms of this complex are the sheaves of type K α Q ⊗ A where α is a dominant integral weight for the group GL(n − µ1 ). Moreover, by (8.1.6)(c) we see that only the weights satisfying α1 ≤ n − µ1 occur. Let us ﬁx two terms of the complex F•µˆ (Q, Q∗ ): the term K α Q ⊗ A occurring in homological dimension t and the term K β Q ⊗ A occurring in homological degree t − 1. The component dα,β of the differential from the term K α Q ⊗ A to the term K β Q ⊗ A comes from the natural map K α Q → K β Q ⊗ S j (Q ⊗ Q∗ ). Let B denote the sheaf of algebras Sym(Q ⊗ E ∗ ) on Grass(µ1 , E). We can deﬁne the complex F˜ µˆ (Q, Q∗ )• of B-modules to be the complex with the same

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The Nilpotent Orbit Closures

terms as F•µˆ (Q, Q∗ ) but replacing the differential dα,β with its composition with the natural embedding of K β Q ⊗ S j (Q ⊗ Q∗ ) into K β Q ⊗ S j (Q ⊗ E ∗ ). The complex F˜ µˆ (Q, Q∗ )• is acyclic. Indeed, the complex F•µˆ (Q, Q∗ ) is, and locally (on Grass(µ1 , E)) the complex F˜ µˆ (Q, Q∗ )• has the same differentials as F•µˆ (Q, Q∗ ), though it is a complex over a polynomial ring with some additional irrelevant variables. Let us identify the only homology group of F˜ µˆ (Q, Q∗ )• . We deﬁne the subvariety Wµ of X × Grass(µ1 , E) as follows. First of all Wµ ⊂ {(φ, R) ∈ X × Grass(µ1 , E) | φ| R = 0}. An endomorphism φ such that φ| R = 0 induces the morphism of bundles φ : Q → E over X × Grass(µ1 , E). We deﬁne φˆ : Q → Q to be the composition of φ with the natural epimorphism E → Q. Now we set Wµ = {(φ, R) ∈ X × Grass(µ1 , E) | φ| R = 0, φˆ ∈ Yµˆ (Q, Q∗ )}, where Yµˆ (Q, Q∗ ) is the variety Yµˆ constructed in relative situation. For the remainder of this section we will denote by p (by q) the projection from X × Grass(µ1 , E) onto Grass(µ1 , E) (onto X ). (8.2.7) Proposition. (a) The homology sheaf H0 ( F˜ µˆ (Q, Q∗ )• ) is equal to the direct image p∗ OWµ of the structure sheaf OWµ . (b) The higher direct images Ri q∗ OWµ are 0 for i > 0 and q∗ OWµ = OYµ . Proof. It is clear that Wµ is a reduced variety in X × Grass(µ1 , E). The calculation can be done locally on Grass(µ1 , E). The term F˜ µˆ (Q, Q∗ )0 equals B, which accounts for the condition φ R = 0. The reduced equations giving the condition φ ∈ Y µˆ (Q, Q∗ ) are given by images of the elements in the term F˜ µˆ (Q, Q∗ )1 . This proves part (a). The map p is afﬁne, so to prove part (b) it is enough to show that the cohomology groups H i (Grass(µ1 , E), p∗ OWµ ) are 0 for i > 0 and H 0 (Grass(µ1 , E), p∗ OWµ ) = OYµ . The complex F˜ µˆ (Q, Q∗ )• is an acyclic complex of locally free B-modules with the terms being direct sums of sheaves of the form K α ⊗ BQ. Let us consider the subvariety Z = Z µ1 ⊂ X × Grass(µ1 , E), which is a total space of the vector bundle Q∗ ⊗ E. In other words, Z = {(φ, R) ∈ X × Grass(µ1 , E) | φ| R = 0}.

8.2. The Equations of the Conjugacy Classes of Nilpotent Matrices 269

This is a desingularization of the determinantal variety of matrices φ of rank ≤ n − µ1 . We considered such varieties in section 6.1. Notice that by Proposition (5.1.1)(b) the sheaf of algebras B is just the direct image p∗ O Z . Therefore, each term K α Q ⊗ B is the direct image p∗ M(K α Q) of the corresponding twisted module M(K α Q) associated to the variety Z . Such modules were considered in section 6.5. We also notice that Lemma (8.1.5) says that for all modules M(K α Q) occurring in the complex F˜ µˆ (Q, Q∗ )• we have Ri q∗ (M(K α Q)) = 0 for i > 0. This proves the ﬁrst part of (b). To prove the second part of (b), let us calculate the graded Euler characteristic of F˜ µˆ (Q, Q∗ )• . By part (a) it is equal to the graded Hilbert function of q∗ OWµ . On the other hand, the graded Euler characteristic of F˜ µˆ (Q, Q∗ )• can be calculated in another way. By (8.1.6) we know that the graded Euler characteristic of OYµˆ can be calculated as the graded Euler characteristic of the complex F µˆ (Q, Q∗ ), i.e. as the Euler characteristic of the Koszul complex on the bundle p ∗ (ξµˆ ) on X × Vµ . Thus the Euler characteristic of OWµ can be calculated as the Euler characteristic of the Koszul complex on the bundle p ∗ (ξµˆ ⊕ (R ⊗ E ∗ )) on X × Vµ . But by the exact sequence 0 −→ R ⊗ E ∗ −→ ξµ −→ ξµˆ (Q, Q∗ ) −→ 0, this is the same as the graded Euler characteristic of F•µ , i.e., by (8.1.6), the graded Hilbert function of OYµ . This concludes the proof of part (b) of the lemma. Let us concentrate on two ﬁrst terms of the complex F˜ µˆ (Q, Q∗ )• . The term F˜ µˆ (Q, Q∗ )0 is just B. To identify the term F˜ µˆ (Q, Q∗ )1 , we notice that by the inductive assumption we can assume that the ideal Jµˆ is generated by the representations U0, p (1 ≤ p ≤ n ) and by the representations Ui,µ(i) (1 ≤ i ≤ n ), where we denote ˆ n := n − µ1 . This means that Jµˆ is minimally generated by a subset of these representations. Let us denote this subset by C. This means that F˜ µˆ (Q, Q∗ )1 = K (1i ,0n −2i ,(−1)i ) Q ⊗ B(− p). (i, p)∈C

The map ∂(i, p) , from the summand corresponding to Ui, p to F˜ µˆ (Q, Q∗ )0 , is induced by Ui, p . We consider the map q∗ (∂(i, p) ) : q∗ (M(K (1i ,0n −2i ,(−1)i ) Q)) → q∗ (M(K (0n ) Q)). To make our notation more transparent we will denote N0 = q∗ (M(K (0n ) Q)) and N(i, p) = q∗ (M(K (1i ,0n −2i ,(−1)i ) Q)).

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By Lemma (8.1.5) we know that the higher direct images Ri q∗ applied to the modules N0 and N(i, p) are 0 for i > 0. Moreover, we know that the module N0 is a coordinate ring of determinantal variety of matrices φ of rank ≤ n − µ1 . The module N(i, p) has the presentation i+c+1

E⊗

i+c+1

E ∗ ⊗ A(−i − c − 1) →

i

E⊗

i

E ∗ ⊗ A(−i)

→ N(i, p) → 0, where c = n − 2i. Since N0 is the coordinate ring of the determinantal variety, and its deﬁning ideal (generated by the (n + 1) × (n + 1) minors of φ) is clearly contained in Jµ . We denote the image of Jµ in N0 by Jµ . Theorem (8.2.5) clearly follows from the following lemma. (8.2.8) Lemma. Let (i, p) ∈ C. The image of the map q∗ (∂(i, p) ) is contained in the ideal Jµ . Proof. Let us consider two cases. Either (i, p) = (0, p) or (i, p) = (i, µ(i)). ˆ In the ﬁrst case the map ∂(0, p) is a GL(E)-equivariant map from N0 to itself. This means it has to be a multiplication by a GL(E)-invariant. Therefore its image is contained in Jµ . Let us assume that (i, p) = (i, µ(i)). ˆ According to the Theorem (5.1.2)(b) the modules N0 and N(i, p) can be identiﬁed as follows: H 0 (Grass(µ1 , E), Sd (Q ⊗ E ∗ )), N0 = d≥0

N(i, p) =

H 0 (Grass(µ1 , E), K (1i ,0n −2i ,(−1)i ) Q ⊗ Sd (Q ⊗ E ∗ )).

d≥0

We know that the module N(i, p) is generated by its component in homogeneous degree i. We calculate the action of ∂(i, p) on the generators of N(i, p) . The key statement is (8.2.9) Lemma. The map ∂(i, p) factors as follows: i E ⊗ i E∗ ↓ t p ⊗1 p E ⊗ p E∗ ⊗ i E ⊗ i E∗ ↓w p p ∗ i E⊗ E ⊗ E ⊗ i E∗ ↓ ζ p ⊗ζi Si+ p (E ⊗ E ∗ )/(Iµ1 +1 )i+ p ,

8.2. The Equations of the Conjugacy Classes of Nilpotent Matrices 271

where w is the identity on the components involving E ∗ tensored with a GL(E)-equivariant map on the components involving E, and ζ p ⊗ ζi is the product of maps coming from straightening formula. The composition of the last two maps in the composition (8.2.9) is a GL(E) × GL(E ∗ )-equivariant map. The rest of the proof of Lemma (8.2.8) (and thus of Theorem (8.2.5)) is based on the following idea. We exhibit an explicit set of generators of the group p p i i HomGL(E)×GL(E ∗ ) E⊗ E∗ ⊗ E⊗ E ∗, Si+ p (E ⊗ E ∗ )/(Iµ1 +1 )i+ p , and we show that the image of each of them composed with the map t p ⊗ 1 from (8.2.9) is in Jµ . The precise statements we need are (8.2.10) Lemma. The vector space p p i i E⊗ E∗ ⊗ E⊗ E ∗, HomGL(E)×GL(E ∗ ) ∗ Si+ p (E ⊗ E )/(Iµ1 +1 )i+ p is generated by the elements w j deﬁned as compositions p E ⊗ p E∗ ⊗ i E ⊗ i E∗ ↓ 1⊗1⊗ˆ p ∗ j p E⊗ E ⊗ E ⊗ j E ∗ ⊗ i− j E ⊗ i− j E ∗ ˆ ↓ m⊗1⊗1 p+ j E ⊗ p+ j E ∗ ⊗ i− j E ⊗ i− j E ∗ ↓ ζ p+ j ⊗ζi− j Si+ p (E ⊗ E ∗ )/(Iµ1 +1 )i+ p ˆ denotes the product of two for j satisfying 0 ≤ j ≤ i and p + j ≤ n. Here diagonal maps composed with the appropriate permutation of the factors. Similarly mˆ is the permutation of the factors composed with the product of two exterior multiplications. (8.2.11) Lemma. Let t p ⊗ 1 be the ﬁrst map in the composition (8.2.9). Let w j be the maps deﬁned in the statement of (8.2.10). Then for each j satisfying 0 ≤ j ≤ i, p + j ≤ n, the image of the composition w j (t p ⊗ 1) is contained in the ideal Jµ .

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To conclude the proof of Theorem (8.2.5) it remains to show the statements (8.2.9), (8.2.10), (8.2.11). Proof of (8.2.10). We will show that the morphisms w j are a basis of p p i i ∗ ∗ ∗ ∗ HomGL(E)×GL(E ) E⊗ E ⊗ E⊗ E , Si+ p (E ⊗ E ) . Indeed, decomposing two sides into irreducibles by Pieri’s formula and by Cauchy’s formula, we see that the common irreducibles are L ( p+ j,i− j) E ⊗ L ( p+ j,i− j) E ∗ for each j satisfying 0 ≤ j ≤ i, p + j ≤ n. Each of these irreducibles occurs with multiplicity 1 in both modules. Therefore the basis of p p i i ∗ ∗ ∗ ∗ E⊗ E ⊗ E⊗ E , Si+ p (E ⊗ E ) HomGL(E)×GL(E ) is given by the following compositions v j : p

E⊗

p

E∗ ⊗

i

E⊗

i

pr

E ∗ →L ( p+ j,i− j) E ⊗ L ( p+ j,i− j)

incl

E ∗ →Si+ p (E ⊗ E ∗ ), where pr and incl denote respectively the GL(E) × GL(E ∗ )-equivariant projection and inclusion. Now it is a direct calculation that the matrix expressing the elements w j as combinations of v j ’s is triangular with nonzero diagonal entries. The lemma follows. Proof of (8.2.11). Let us consider the composition w j (t p ⊗ 1). The image is a combination of products of ( p + j) × ( p + j) and (i − j) × (i − j) minors of φ, with ( p + j) × ( p + j) minors containing the traces on p components. It is enough to show that each such combination of ( p + j) × ( p + j) minors is in Jµ . In order to establish this we need to check the condition of (8.2.1), which in this case reads p + j > µ1 + . . . + µ j − j. However, we know that our pair (i, p) corresponds to the generators of Jµˆ , so p = µ(i) ˆ =µ ˆ1 + ... + µ ˆ i − i + 1 and by deﬁnition of U(i, p) we have p ≥ i. We know also that µ ˆ k = µk − 1 as long as k ≤ µ1 . This has to happen for all i in question, because otherwise p + j ≥ i + j > µ1 and we are in Jµ anyway. Therefore we have p+ j =µ ˆ1 + ... + µ ˆi −i +1+ j > µ ˆ1 + ... + µ ˆi −i + j = µ1 + . . . + µi + j ≥ µ1 + . . . + µ j + j, and the condition above is proved.

8.2. The Equations of the Conjugacy Classes of Nilpotent Matrices 273

Proof of (8.2.9). It follows from the deﬁnition that the map ∂(i, p) is the map induced on the sections by the following composition of maps: K (1i ,0n −2i ,(−1)i ) Q ↓ i Q ⊗ i Q∗ ↓ i Q ⊗ i Q∗ ⊗ p−i Q ⊗ p−i Q∗ ↓ p Q ⊗ p Q∗ ↓ p Q ⊗ p E∗ ↓ B p = S p (Q ⊗ E ∗ ). Therefore the action of ∂(i, p) on the generators of N(i, p) is given by a composition i E ⊗ i E ∗ = H 0 (K (1i ,0n −2i ,(−1)i ) Q ⊗ i Q ⊗ i E ∗ ) ↓ H 0 (U ⊗1) p p ∗ i,ip 0 H ( Q⊗ Q ⊗ Q ⊗ i E ∗) ↓ H 0 (1⊗i⊗1⊗1) p p ∗ i 0 H ( Q⊗ E ⊗ Q ⊗ i E ∗) (∗) ↓ H 0 (ζ p ⊗ζi ) H 0 (Si+ p (Q ⊗ E ∗ )) ↓ Si+ p (E ⊗ E ∗ )/(Iµ1 +1 )i+ p . Here we denote by i the canonical inclusion of Q∗ into E ∗ , and by ζ p the embedding p F ⊗ p G → S p (F ⊗ G) (with F := E, G := E ∗ ) deﬁned in section 3.2. Let us notice that the last two maps in (∗) are GL(E) × GL(E ∗ )-equivariant. Let us describe more precisely the composition of the ﬁrst two maps in (∗). It comes from applying the functor H 0 to the composition i Q ⊗ i E∗ ↓ j1 ⊗1 K (1i ,0n −2i ,(−1)i ) Q ⊗ i Q ⊗ i E ∗ ↓ j2 ⊗1⊗1 i Q ⊗ i Q∗ ⊗ i Q ⊗ i E ∗ (∗∗) ∧t ⊗1⊗1 ↓ p−i p Q ⊗ p Q∗ ⊗ i Q ⊗ i E ∗ ↓ 1⊗i⊗1⊗1 p Q ⊗ p E ∗ ⊗ i Q ⊗ i E ∗, where j1 denotes and j2 denote the canonical inclusions.

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Let us look at the composition of all maps in (∗∗). We notice that all of them are identities on the factor i E ∗ appearing on the right hand side. This means our composition is a tensor product of a composition i ↓

Q j1

K (1i ,0n −2i ,(−1)i ) Q ⊗ i Q ↓ j ⊗1 i 2 Q ⊗ i Q∗ ⊗ i Q ↓ ∧t p−i ⊗1 p ∗ i p Q⊗ Q ⊗ Q ↓ 1⊗i⊗1 p Q ⊗ p E∗ ⊗ i Q tensored with the identity on i E ∗ . The map induced by the last composition by applying a functor H 0 is a GL(E)-equivariant map from i E = H 0 ( i Q) to H 0 ( p Q ⊗ p E ∗ ⊗ i Q). If p + i ≤ n , the latter group is p E ⊗ p E ∗ ⊗ i E. If p + i > n , the latter group is a factor of p E ⊗ p E ∗ ⊗ i E by the image of the map n +1

E⊗

p

∗

E ⊗

p+i−n −1

E→

p

E⊗

p

E∗ ⊗

i

E,

which is the identity on the second factor, and the composition n +1

E⊗ 1⊗m

E→ i

p+i−n −1

p

E⊗

p ⊗1

E→

i

E⊗

n +1− p

E⊗

p+i−n −1

E.

Let us consider the vector space HomGL(E) ( i E, p E ⊗ p E ∗ ⊗ E). We will exhibit an explicit basis of this vector space.

(8.2.12) Lemma. For all j satisfying 0 ≤ j ≤ i, p + j ≤ n we deﬁne the morphisms h j : i E → p E ⊗ p E ∗ ⊗ i E as the compositions

hj :

i

p t p ⊗1

E→

E⊗

p

E∗ ⊗

i

gj

E→

i

E⊗

p

E∗ ⊗

p

E,

8.2. The Equations of the Conjugacy Classes of Nilpotent Matrices 275

where g j is the identity on the second factor tensored with the map g j deﬁned as a composition p E⊗ iE ↓ 1⊗ p E ⊗ j E ⊗ i− j E ↓ m⊗1 p+ j E ⊗ i− j E ↓ ⊗1 p E ⊗ j E ⊗ i− j E ↓ 1⊗m p E ⊗ i E. Then the maps h j form a basis of HomGL(E) ( i E, p E ⊗ p E ∗ ⊗ i E). Proof. The vector space HomGL(E) ( i E, p E ⊗ p E ∗ ⊗ i E) is canonp i p i E⊗ E, E⊗ E). The identiically isomorphic with HomGL(E) ( p ﬁcation is done by associating to a morphism f ∈ HomGL(E) ( E ⊗ i E, i p E⊗ E) the composition h:

i

p t p ⊗1

E→

E⊗

p

E∗ ⊗

i

g

E→

p

E⊗

p

E∗ ⊗

i

E,

where g is the identity on the second factor tensored with the map f . It is therefore enough to show that the maps g j given in (8.2.12) form a basis of HomGL(E) ( p E ⊗ i E, p E ⊗ i E). This is however clear. p i Indeed, decomposing E⊗ E into irreducibles L ( p+ j,i− j) E, we see that the natural basis of p p i i HomGL(E) ( E ⊗ E, E⊗ E)

consists of the morphisms uj :

p

E⊗

i

pr

p incl

E →L ( p+ j,i− j) E →

E⊗

i

E

where pr and incl denote respectively the GL(E)-equivariant projection and inclusion, and j satisﬁes conditions 0 ≤ j ≤ i, p + j ≤ n. Now it is clear that the transition matrix expressing g j ’s as combinations of u j ’s is triangular with the nonzero entries on the diagonal. This concludes the proof of (8.2.12). Now we come back to the map h we get when applying the functor H 0 to the composition of the maps in (∗∗). Lemma (8.2.12) shows that the map h

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can be written as a linear combination of maps h j . This means that it can be written as a composition h:

i

p t p ⊗1

E→

E⊗

p

E∗ ⊗

i

h

E→

p

E⊗

p

E∗ ⊗

i

E,

where h is the identity on the second factor tensored with the GL(E)equivariant map on the remaining two factors. Since the composition of the ﬁrst two maps in (∗) is the map h tensored with the identity on the remaining factor i E ∗ , we can write it in the form (8.2.9) as required. This concludes the proof of (8.2.9) and therefore the proof of Theorem (8.2.5).• Let us illustrate the inductive step in the proof of Theorem (8.2.5) with the following example. (8.2.13) Example. Let us consider µ = (6, 4, 2). Then µˆ = (4, 2). We obviously have dim E = 12. The inductive step consists of pushing down from the Grassmannian Grass(6, E) the complex of sheaves F˜ µˆ (Q, Q∗ )• where Q is a tautological factorbundle on Grass(6, E). The generators of J(4,2) can be described as follows: 0 1 1 M((4, 2)) = 1 1 1 1

0 1 0 1 1 1. 1 1 1

This means that J(4,2) is generated by the invariants U0, p (1 ≤ p ≤ 3) and by the representations U1,2 , U2,3 , and U3,3 . The ﬁrst two terms of F˜ µˆ (Q, Q∗ )• are easy to describe. The term F˜ µˆ (Q, Q∗ )0 equals B = Sym(Q ⊗ E ∗ ), and F˜ µˆ (Q, Q∗ )1 is a direct sum of the six terms B ⊗ K (1i ,06−2i ,(−1)i ) Q corresponding to the six representations Ui, p listed. We see ﬁrst of all that the direct image q∗ (B) = A/I7 . This tells us that 7 × 7 minors of the generic matrix φ are in the ideal J(6,4,2) . The invariants U0, p lead to the generators that are the invariants of degrees 1, 2, 3 in A/I7 . The remaining three Ui, p ’s give the following representations. The term U1,2 gives a representation E ⊗ E ∗ in degree 3. The term U2,3 gives a rep resentation 2 E ⊗ 2 E ∗ in degree 5. The term U3,3 gives a representation 3 ∗ 3 E⊗ E in degree 6. The proof above shows that these three representations have to be contained in the ideal generated by I7 , the invariants U0, p

8.2. The Equations of the Conjugacy Classes of Nilpotent Matrices 277

(1 ≤ p ≤ 6), and the representations U1,3 , U2,5 , and U3,6 . In fact one would expect them to be respectively V1,3 , V2,5 , and V3,6 . The conclusion in any case . This follows from Example (8.2.4). is that all generators have to be in J(6,4,2) (8.2.14) Remark. In the above proof of Theorem (8.2.5) we did not identify precisely the generators coming from the terms Ui, p occurring in F˜ µˆ (Q, Q∗ )1 . The reader might worry that we did not show that some of them are dependent on others. However, this is not a concern. We are assured by the fact that the only homology of F˜ µˆ (Q, Q∗ )• is OWµ and by Proposition (8.2.7) that the factor of A/Iµ1 +1 by the images of the terms coming from F˜ µˆ (Q, Q∗ )1 is A/Jµ . Therefore all elements of Jµ have to be contained in that image. The only concern is whether we get some generators not contained in Jµ . We ﬁnish this section with an example related to rectangular partitions. (8.2.15) Proposition. Let n = r e and let us consider the rectangular partition µ = (er ). Then the ideal J(er ) is generated by the invariants v1 , . . . , ve−1 and by the entries of the matrix φ e . These polynomials form a minimal set of generators of J(er ) . Proof. Let us use the inductive procedure used in the proof of (8.2.5) for the ˆ is just ((e − 1)r ). Thus we can family of partitions µ = (er ). The partition µ assume by induction that (8.2.15) is true for µ. ˆ The inductive assumption means that the ﬁrst term of F•µˆ consists of the trivial representations in homogeneous degrees 1, . . . , e − 1 and the representation K 1,0,...,0,−1 Q in homogeneous degree e − 1. This means that the µ possible terms in F1 are the trivial representations in homogeneous degrees 1, . . . , e − 1, the representation E ∗ ⊗ E in the degree e and the terms coming from n − µ1 + 1 = n − r + 1 size minors of the matrix φ. The generators in degrees ≤ e have to consist of invariants v1 , . . . , ve−1 and the vanishing of µ entries of φ e , because these terms have to occur in F1 and they match the terms we described. Thus the induction implies that the minimal generators of J (µ) are those listed in (8.2.15) plus possibly some linear combinations of (n − r + 1) × (n − r + 1) minors of φ. However recall that by (8.2.5) the ideal J (µ) is generated (nonminimally) by the invariants U0, p (1 ≤ p ≤ n) and by the representations Ui,ie−i+1 (for 1 ≤ i ≤ r ). The only possible generator in degree n − r + 1 on the above list is the representation Ur,n−r +1 which does not occur in the span of n − r + 1 minors of . This concludes the proof.

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The Nilpotent Orbit Closures

A slightly more general result is proved in [W7]. Let e < n and let us consider the division of n by e with the remainder, n = r e + f with 0 ≤ f ≤ e − 1. We deﬁne the partition µ(n, e) = (er , f ). Then the ideal Jµ(n,e) is generated by the invariants v1 , . . . , ve−1 and by the entries of the matrix φ e . These polynomials form a minimal set of generators of Jµ(n,e) .

8.3. The Nilpotent Orbits for Other Simple Groups In this section we investigate the closures of conjugacy classes for other simple groups. We give their explicit desingularization by the collapsing of a homogeneous bundle. We sketch the proof that the normalizations of the closures of conjugacy classes have rational singularities. We use the approach of Broer [Br5]. Let G be a simple algebraic group with Lie algebra g. The group G acts on g by conjugation. We will still denote this action as a left action. Let e be a nilpotent element in g. We denote by Ge its orbit under the conjugation action, and by Ge its closure in g. By the Jacobson–Morozov lemma ([Bou, chapter VIII, section 11]) there exist elements h, f ∈ g such that {e, h, f } forms an sl2 -triple in g, i.e. [h, e] = 2e,

[h, f ] = −2 f,

[e, f ] = h.

If {e, h , f } forms another sl2 -triple, then there exists g ∈ G (and even in the connected component of the identity, CG (e)0 , of the centralizer of e), centralizing e, such that g f = f , gh = h . Let us ﬁx e and the sl2 -triple {e, h, f }. Let V be a ﬁnite dimensional g-module. The action of h induces a grading V = Vi , Vi = {v ∈ V | h . v = iv}. i∈Z

The associated ﬁltration . . . ⊂ V≥i+1 ⊂ V≥i ⊂ . . . does not depend on the choice of h, f . Let us consider V = g. Then e ∈ g 2 , h ∈ g 0 , f ∈ g −2 . We also clearly have [g i , g j ] ⊂ g i+ j . This means that g ≥0 is a Lie algebra of a parabolic group P ⊂ G with a nilpotent radical n = g >0 and the Levi factor g 0 . The subgroup P depends only on the ﬁltration and therefore does not depend on the choice of f, h. We consider the subspace g ≥2 of g. This is clearly a P-submodule of g. We can consider an induced homogeneous vector bundle Z = G ×P g ≥2 over

8.3. The Nilpotent Orbits for Other Simple Groups

279

G/P. Let us take X = g, V = G/P. Then Z ⊂ X × V . We can consider the diagram Z ⊂ ↓ q Y ⊂

X×V ↓q X

where Y = Ge and q is the restriction of the ﬁrst projection q. The map q sends the coset of the pair (g, x) to gx. In this situation we have (8.3.1) Proposition ([KP2], section 7.4). The map q makes Z a resolution of singularities of Y . Proof. Let us recall that the irreducible sl2 -modules are just the symmetric powers Sd F where F = K2 . The Lie algebra sl2 = sl(F) acts by the derivative action of the conjugation action, i.e. the commutator action. The tangent space at e to Pe is therefore the commutator space [g ≥0 , e] = g ≥2 . It follows that Pe is an open set in g ≥2 . Since the group P is uniqely determined by e, we have P = gPg −1 for each g in the stabilizer Ge . Since each parabolic subgroup equals its normalizer ([Hu2], Corollary B, section 23), we have Ge = Pe . This means that q restricted to the open orbit G(1, e) is an isomorphism, and therefore q is a birational map. The variety Z is nonsingular, because it is a vector bundle over G/P. (8.3.2) Example. Consider the special linear group G = SL(n) = SL(E), for E = Kn . The Lie algebra g of G is the set of n × n matrices of trace 0. We identify the set of n × n matrices with HomK (E, E) = E ∗ ⊗ E. Let us start with the nilpotent element e, which in canonical Jordan form has one Jordan block of size n. Then we can choose a basis e1 , . . . , en of E such that e(e1 ) = 0 and e(ei ) = ei−1 for i = 2, . . . , n. Then the element f can be chosen as follows: f (ei ) = ei+1 for i = 1, . . . , n − 1, f (en ) = 0. The element h can be chosen to be the diagonal matrix h(ei ) = (n + 1 − 2i)ei . We introduce the grading on E by letting deg(ei ) = n + 1 − 2i. This induces a grading on HomK (E, E) = E ∗ ⊗ E deﬁned by letting deg(ei∗ ⊗ e j ) = deg(e j ) − deg(ei ). The module g ≥2 consists then of all upper triangular matrices, and therefore the desingularization constructed above is the same as one constructed in section 8.1. Let µ = (µ1 , . . . , µr ) be a partition of n. Let us consider the nilpotent element e from the nilpotent orbit O(µ). We choose a basis e1 , . . . , en of E in such way that e(ei ) = 0 if i ∈ {1, µ1 + 1, µ1 + µ2 + 1, . . .}, e(ei ) = ei−1

otherwise.

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The Nilpotent Orbit Closures

This means that for each j = 1, . . . , r the span of eµ1 +...+µ j−1 +1 , . . . , eµ1 +...+µ j is the Jordan block of size µ j . We can choose f to act as follows: f (ei ) = 0 if i = µ1 + . . . + µ j ,

f (ei ) = ei+1

otherwise.

The element h is a diagonal matrix acting in the j-th block by h(eµ1 +...+µ j−1 +i ) = (µ j + 1 − 2i)ei

for i = 1, . . . , µ j .

Let us introduce the grading on E by letting deg(eµ1 +...+µ j−1 +i ) = µ j + 1 − 2i. The grading induced on HomK (E, E) is then given by identifying HomK (E, E) with E ∗ ⊗ E and letting deg(ei∗ ⊗ e j ) = deg(e j ) − deg(ei ). This induces a grading on g. This allows us to deﬁne the module g ≥2 . Notice that the desigularization given by g ≥2 is almost never the same as the one considered in the section 8.1. To see this we consider n = 3 and µ = (2, 1). Then the degrees of ei are as follows: deg(e1 ) = 1, deg(e2 ) = −1, deg(e3 ) = 0. Reordering our basis, we can assume that deg(e1 ) = 1, deg(e2 ) = 0, deg(e3 ) = −1. We see that the module g ≥2 is spanned by e3∗ ⊗ e1 . This is a B-submodule for the group B of upper triangiular matrices. The parabolic subgroup P deﬁned above is obviously equal to B. The desingularization G ×P g ≥2 we deﬁned above is a line bundle over G/B. Let us denote this desingularization by Z 1 . Identifying G/B with the set of flags R1 ⊂ R2 ⊂ E, we can identify our desingularization Z 1 with the set of pairs Z 1 = {(φ, R1 , R2 ) ∈ HomK (E, E) × G/B | φ(R2 ) = 0, φ(E) ⊂ R1 }. The desingularization Z 2 constructed in section 8.1 was a two dimensional bundle over the Grassmannian Grass(2, E). It was deﬁned as the set of pairs Z 2 = {(φ, R2 ) ∈ HomK (E, E) × Grass(2, E) | φ(R2 ) = 0, φ(E) ⊂ R2 }. To see that these two desingularizations are different, let us observe that the map that forgets R1 deﬁnes a regular map Z 1 → Z 2 . This map is not an isomorphism, because its ﬁber over a point (φ, R2 ) such that φ = 0 is clearly isomorphic to P1 . Therefore the desingularizations are different.• The main result of this section is the following theorem: (8.3.3) Theorem (Hinich [Hi], Panyushev [Pa2]). Let G be a simple group with the Lie algebra g. Let e be a nilpotent element in g. We consider the desingularization Z of the closure Y of the orbit Ge constructed above. (a) For all i > 0 we have H i (Z , O Z ) = 0. (b) The normalization of Y is a Gorenstein variety with rational singularities.

8.3. The Nilpotent Orbits for Other Simple Groups

281

Before we prove (8.3.3) we need some preparatory statements. (8.3.4) Proposition. Let g be a simple Lie algebra with the bracket [ , ]. Then all adjoint orbits in g have even dimensions. Proof. We denote by ( , ) the Killing form on g. Every element z ∈ g deﬁnes an antisymmetric form ( , )z on g given by (x, y)z := (z, [x, y]). If x is in the radical of ( , )z , then for all y ∈ g we have (z, [y, x]) = ([z, x], y) = 0. Since the Killing form is nondegenerate, we have [z, x] = 0. This means that the radical of ( , )z equals the centralizer g z . The induced antisymmetric form on g/g z is nondegenerate, and therefore dim g/g z is even. However, dim Gz = dim g/g z . (8.3.5) Lemma. Let g and e be as above. Let us consider the grading g = i∈Z g i induced by e, and the associated parabolic subgroup P. We denote b = dim g ≥2 , c := dim g 1 , d = dim G/P. Then: (a) The number c is even. (b) There is a nonzero P-equivariant map s : g ≥2 →

d

(g/ p)∗ ⊗

b

g ∗≥2 .

(c) s(ux) = u c/2 s(x) for all u ∈ K, x ∈ g ≥2 . Proof. Let z ∈ g 2 . The form ( , )z deﬁned in the proof of (8.3.4) restricts to a form on g −1 , which can be identiﬁed with an element wz ∈ 2 g ∗−1 . Consider the Levi decomposition P = LPu of the parabolic group P. The linear map w• : g 2 →

2

g ∗−1

sending z to wz is L-equivariant. For any z ∈ Le ⊂ g 2 we have g z ⊂ p = g ≥0 . From the proof of (8.3.4) it follows that wz is nondegenerate. This means that ∧c/2 c is even and the top exterior power wz ∈ c g ∗−1 does not vanish. The map s : g2 → ∧c/2

c

g ∗−1

sending z into wz is an L-equivariant polynomial map of homogeneous degree 2c . It does not vanish on the open orbit Le. We extend the L-modules

282

The Nilpotent Orbit Closures

c ∗ g 2 and c g ∗−1 to P-modules g˜ 2 and % g −1 by the trivial action of Pu . We have the obvious isomorphisms of P-modules g ≥2 /g >2 / g˜2 ,

c% d b g ∗−1 / (g/ p)∗ ⊗ g ∗≥2 .

We deﬁne s as a composition s

s : g ≥2 → g ≥2 /g >2 / g˜ 2 −→

c% d b g ∗−1 / (g/ p)∗ ⊗ g ∗≥2 .

Then s clearly satisﬁes (b) and (c). We need one more statement. (8.3.6) Proposition. Let P = LPu be a Levi decomposition of a parabolic subgroup in G. We consider a P-submodule m of n := Lie(Pu ). We can construct as above a homogeneous bundle Z = G ×P m over G/P, which projects onto Y = Gm. Let us assume that dim Z = dim Y . We consider a one dimensional P-module u and the associated line bundle L := G ×P u on G/P. Let p denote as usual the projection of Z onto G/P. Assume that the twisted sheaf O Z ⊗ p ∗ (L) has a global G-invariant section s0 . The section s0 induces the morphism of s˜0 : O Z -modules O Z → O Z ⊗ p ∗ (L) given locally by sending a section f to f s0 . The morphism s˜0 then induces the isomorphism *(O Z ) / *(O Z ⊗ p ∗ (L)). Proof. Let s be a section of O Z ⊗ p ∗ (L). Then s = f s0 where f is a rational function on Z with poles only along the zeros of s0 . We know that Y := Spec K[Z ] is normal because it has an open orbit whose complement has codimension ≥ 2, and the morphism Z → Y is birational. Therefore f can be considered as a rational function on Y without poles on the open orbit. This means f has no poles on Y , because on the normal variety the set of poles of a function has codimension 1. Thus f is a regular function on Y . Since by deﬁnition of Y the regular functions on Y and Z are the same, f can be treated as a regular function on Z . We conclude that the map *(˜s0 ) : *(O Z ) → *(O Z ⊗ p ∗ (L)) is surjective. This map is obviously injective, which ﬁnishes the proof. Proof of Theorem (8.3.3). Let G, e, P be as in the statement of the theorem. We denote d = dim G/P, c = dim g −1 and b = dim g ≥2 . We take

8.4. Conjugacy Classes for the Orthogonal Group

283

u := d (g/ p)∗ ⊗ b g ∗≥2 . Consider the associated line bundle L = G ×P u on G/P. There exists m ∈ Z such that the canonical sheaf ω Z is isomorphic to the line bundle O Z ⊗ p ∗ (L)(m) with the grading shifted by m. By Lemma (8.3.5) there exists a P-equivariant map s : g ≥2 → u of degree 2c . This means that the sheaf O Z ⊗ p ∗ (L) has a nonzero global G-invariant section of degree 2c (the image of a constant section of g ≥2 ). This implies that the sheaf ω Z has a global G-invariant section s0 . Now Proposition (8.3.6) gives a morphism of sheaves s˜0 : O Z → ω Z (−m + 2c ) which induces an isomorphism on global sections. Applying Proposition (1.2.32), we get that Spec K[Z ] is a Gorenstein variety with rational singularities. We know however that Spec K[Z ] is the normalization of the closure Y of the orbit Ge. This concludes the proof of the theorem. (8.3.7) Remarks. Broer in a series of beautiful papers [Br1, Br2, Br3] applied the geometric method to deal with the twisted modules supported in nilpotent orbit closures. He also applied the method to decide normality of several nilpotent orbit closures for the exceptional groups. Kraft ([Kr2]) classiﬁed normal orbit closures for the groups of type G 2 . Recently the normal nilpotent orbit closures were classiﬁed for groups of type F4 (Broer, [Br6]) and of type E 6 (Sommers).

8.4. Conjugacy Classes for the Orthogonal Group In this section F denotes a vector space of dimension n with a nondegenerate symmetric form ( , ). The special orthogonal group SO(F) is the set of linear automorphisms of F preserving ( , ), i.e. φ ∈ SO(F) if and only if for each x, y ∈ F we have (φ(x), φ(y)) = (x, y). By deﬁnition SO(F) is a subgroup of GL(F). The corresponding Lie algebra so(F) is a subalgebra of the Lie algebra gl(F). The morphism φ ∈ HomK (F, F) is in so(F) if and only if for each x, y ∈ F we have (φ(x), y) + (x, φ(y)) = 0. Let us choose the hyperbolic basis e1 , . . . , em , e¯ m , . . . , e¯ 1 of F (in the case of n = 2m), and e1 , . . . , em , e0 , e¯ m , . . . , e¯ 1 . This means (ei , e j ) = (¯ei , e¯ j ) = 0

(ei , e¯ j ) = δi, j ,

(e0 , ei ) = (e0 , e¯ i ) = 0,

(e0 , e0 ) = 1,

where δi, j denotes the Kronecker delta.

284

The Nilpotent Orbit Closures

We can write φ as a matrix, writing in consecutive columns the images of vectors −¯e1 , . . . , −¯em , e0 , em , . . . , e1 expanded in the basis e1 , . . . , em , e0 , e¯ m , . . . , e¯ 1 . Then φ ∈ so(F) if and only if the matrix of φ is skew symmetric. This allows us to identify the adjoint representation of so(F) with 2 F. Since so(F) is a Lie subalgebra of gl(F), we might expect that the nilpotent conjugacy classes in so(F) will be related to intersections of the nilpotent conjugacy classes in gl(F) with so(F). The following result is proved in [SS; IV. 2.15]. (8.4.1) Proposition. Let µ be a partition of n. Let us consider the nilpotent conjugacy class O(µ) of gl(F) corresponding to µ. Then O(µ) intersects so(F) if and only if every even part of µ occurs even number of times. In such case the intersection O(µ) ∩ so(F) consists of a single conjugacy class of SO(F). For the remainder of this section we denote Po (n) the set of partitions µ of n in which every even part occurs even number of times. For µ ∈ Po (n) we denote by C(µ) the corresponding conjugacy class in so(F), and by Yµ its closure. In order to list the representatives of conjugacy classes for the orthogonal group, we need to exhibit the blocks corresponding to odd rows in our partition and the blocks corresponding to the pairs of even rows. The canonical forms of nilpotent orthogonal endomorphisms corresponding to both kinds of blocks are as follows. The nilpotent e corresponding to the partition (n) = (2m + 1) has the form e(¯e1 ) = 0,

e(¯ei ) = e¯ i−1

for i > 0,

e(en ) = e0 ,

e(e0 ) = −¯en , e(ei ) = −ei+1

for i < n

We can express the action of e by the sequence of arrows ±e1 → ∓e2 → . . . → en−1 → −en → e0 → e¯ n → e¯ n−1 → . . . → e¯ 2 → e¯ 1 → 0. If we extend e to an sl2 triple {e, h, f }, the grading induced by e described in section 8.3 is as follows. For the endomorphism e corresponding to the partition 2n we have h(ei ) = (2n + 1 − 2i)ei , h(¯ei ) = (2i − 1 − 2n)¯ei . Therefore the degrees of the basis vectors are deg ei = 2n + 1 − 2i, deg e¯ i = 2i − 1 − 2n.

8.4. Conjugacy Classes for the Orthogonal Group

285

If n = 2t is even, then the nilpotent e corresponding to the partition (n, n) has the form e(ei ) = ei+2

for i < 2t,

e(¯ei ) = −¯ei−2

e(e2t ) = e¯ 2t+1 ,

for i > 2,

e(e2t+1 ) = e¯ 2t ,

e(¯e2 ) = e(¯e1 ) = 0.

We can express the action of e by two sequences of arrows e1 → e3 → . . . → e2t+1 → e¯ 2t → −¯e2t−2 → . . . → ±¯e2 → 0, e2 → e4 → . . . → e2t → e¯ 2t+1 → −¯e2t−1 → . . . → ∓¯e1 → 0. Extending e to the sl2 -triple {e, h, f }, we see that the element h acts as follows: h(e2i+1 ) = 2(t − i)e2i+1 , h(e2i ) = 2(t + 1 − i)e2i , and h(¯e2i+1 ) = −2(t − i)¯e2i+1 , h(¯e2i ) = −2(t + 1 − i)¯e2i . This gives deg e2i+1 = 2(t − i), deg e2i = 2(t + 1 − i), and deg e¯ 2i+1 = − 2(t − i), deg (¯e2i ) = − 2(t + 1 − i). In this setup the degrees of vectors ei are always nonnegative and we always have deg ei = −deg e¯ i , because h ∈ so(F). If a partition µ corresponds to the conjugacy class in so(F), then we can assign the grading separately in each block, as in Example (8.3.2). We also use the convention that when dealing with several blocks, after assigning the grading, we order ei ’s in such a way that deg ei ≥ deg ei+1 . (8.4.2) Example. Let us take n = 8, µ = (3, 2, 2, 1). The grading of basis elements is as follows: 2 0 −2 1 −1 1 −1 0 We order the elements e1 , . . . e4 , e¯ 4 , . . . , e¯ 1 so their grading is nonincreasing. We get deg e1 = deg e2 = 2, deg e3 = 1, deg e4 = deg e5 = 0, with deg e¯ i = −deg ei for i = 1, . . . , 5. This allows us to determine the grading on so(F) in all cases. Identifying the adjoint representation with 2 F, we can arrange the weight vectors in it in a triangular grid. In order to describe it, let us introduce the involution ()¯ of our symplectic basis by requiring that e¯i = ei . The elements of the grid correspond to the entries above the diagonal of our matrix representation of φ. If u, v are the elements of our symplectic basis, then the entry corresponding to the row u and the column ±v will correspond to the weight vector u v¯ .

286

The Nilpotent Orbit Closures

For a given conjugacy class C(µ) we will mark the element of the grid with X if the corresponding weight vector is in g ≥2 , and with O otherwise. We will denote this grid by GC(µ). (8.4.3) Example. Let n = 5, µ = (3, 2, 2, 1). Then the degrees of the elements ei are given in Example (8.4.2) and we have X

X X

X O O

GC(µ) =

X O O O

O O O O O

O O O O O O

O O O O. O O O

We continue with several examples of conjugacy classes. (8.4.4) Example. Consider the class C((2m )) for even n = 2m. This is a class analogous to the previous example. The associated grading gives deg ei = 1, deg e¯ i = −1 for i = 1, . . . , m. This means that in the grid GC((2m )) the entries marked by X correspond to the vectors ei e j for i ≤ j. For example, if n = 10 we get X

GC((25 )) =

X X

X X X

X X X X

O O O O O

O O O O O O

O O O O O O O

O O O O O O O O

O O O O O. O O O O

The parabolic subgroup P is the set of elements ﬁxing a given isotropic subspace of dimension n. The homogeneous space G/P is the connected component of the isotropic Grassmannian IGrass+ (m, F). The desingularization Z constructed in section 8.3 can be identiﬁed as Z = {(φ, R) ∈ so(F) × IGrass+ (m, F) | φ(F) ⊂ R, φ(R) = 0}. Let R be a tautological subbundle (of dimension m ) on IGrass+ (m, F). We apply the results of section 5.1 to Z . The vector bundles ξ and η are easily identiﬁed: η = 2 (F/R) and ξ = Ker( 2 F → 2 (F/R)).

8.4. Conjugacy Classes for the Orthogonal Group

287

Theorem (5.1.2)(b) implies now that 2 (F/R) . R i q∗ O Z = H i IGrass+ (m, F), Sym Using the Cauchy formula (2.3.8)(a), we see that if we denote by EC(d) the set of partitions of 2d with every part occurring even number of times, Sd

2

(F/R) =

K β (F/R).

β∈EC (d), β1 ≤n

Theorem (4.3.1) implies that R i q∗ O Z = 0 for i > 0 and that K[Y((2m )) ] = Vβ (F), d≥0 β∈EC (d), β1 ≤m

where we identify the partition with at most m parts with the dominant weight for the group SO(F). We have an exact sequence describing the representation Vβ (F) as a cokernel of a map of Schur functors K β/(2) F → K β F → Vβ (F) → 0 (compare Exercise 14 of chapter 6) with the left map being induced by the

trace element tr = 1≤i≤m ei e¯ i ∈ S2 F. This means that the deﬁning ideal of Y(2m ) consists of all polynomials which can be expressed as (x1 ∧ ei )(x2 ∧ x3 ) . . . (x2t−1 ∧ e¯ i ) + (x1 ∧ e¯ i )(x2 ∧ x3 ) . . . (x2t−1 ∧ ei ), i

i

where x 1 , . . . , x2t−1 ∈ F. Since any such polynomial is clearly a product of a polynomial of degree 2 and t − 2 polynomials of degree 1, we see that the deﬁning ideal of Y(2m ) is generated by elements of degree 2 of type (x1 ∧ ei )(x2 ∧ e¯ i ) + (x1 ∧ e¯ i )(x2 ∧ ei ). i

i

(8.4.5) The Nonnormal Orbit. An important phenomenon, discovered by Kraft and Procesi, is that some of the closures of nilpotent conjugacy classes for symplectic and orthogonal groups are not normal. The smallest example is the orbit Y(3,2,2) . Let us consider this case. The grid GC((3, 2, 2)) looks as

288

The Nilpotent Orbit Closures

follows: X GC((3, 2, 2)) =

X X

X O O

O O O O

O O O O O

O O O . O O O

Let us consider the second symmetric power S1 (η). By the Hinich– Panyushev theorem (8.3.3) we know that the only cohomology group that does not vanish is H 0 (G/P, S1 (η)). To calculate this group, it is enough to calculate the Euler characteristic of the bundle S1 (η). To do this one can replace the bundle S1 (η) with the direct sum of its composition factors of dimension 1. The whole matter becomes an exercise of using Bott’s theorem. The result is H 0 (G/P, S2 (η)) = V(1,1,0,0) (F) ⊕ V(1,0,0,0) (F). Applying (5.1.3)(b), we see that the group H 0 (G/P, S2 (η)) is the ﬁrst graded component of the normalization of the coordinate ring of Y(3,2,2) . If this variety were normal, H 0 (G/P, S1 (η)) would be a factor of 2 F). Since 2 F = V(1,1,0,0) (F), we see that our orbit closure is not normal. The normal orbit closures for the orthogonal groups were determined by Kraft and Procesi in [KP2]. They used the method of minimal degenerations. Their result in this case is (8.4.6) Theorem (Kraft–Procesi). Let µ ∈ Po (2n). The orbit closure Yµ is normal if and only if for every i < j such that µi and µ j are even, with µi > µ j , at least one of the parts µi+1 , . . . , µ j−1 is odd. All of the above describes the results for the orbits of the orthogonal group. However one can also consider the nilpotent conjugacy classes of the special orthogonal group. Kraft and Procesi prove in [KP2] (8.4.7) Proposition (Kraft–Procesi). The conjugacy class C(µ) for the orthogonal group is also a conjugacy class for the special orthogonal group unless µ is very even, i.e., all parts of µ and µ are even. For a very even partition µ the conjugacy class C(µ) for the orthogonal group is a disjoint union of two conjugacy classes C(µ)(1) and C(µ)(2) for the special orthogonal group. We ﬁnish by presenting a conjecture on the generators of the deﬁning ideals of orbit closures Yµ .

8.4. Conjugacy Classes for the Orthogonal Group

289

(8.4.8) Conjecture. Let µ ∈ Po (n). The deﬁning ideal of Yµ is generated by the representations V(β1 ,...,βn ) (F) with β1 ≤ 2. The generators can be chosen to be subrepresentations of the Schur functors K γ F with γ1 ≤ 2. For the remainder of this section we look at the very even conjugacy classes, i.e., we assume that the partitions µ and µ have only even parts. These orbit closures they have another desingularization which is more convenient for explicit calculations. Let µ = (µ1 , . . . , µr ) be a very even partition of n = 2m, m even, µi = 2νi for i = 1, . . . , r . We denote ν = (ν1 , . . . , νr ). Note that the parts of ν are even. Take V = IFlag(ν1 , ν1 + ν2 , . . . , n; F). The “desingularization” Zˆ ν of Yµ is deﬁned as follows Zˆ ν = {(φ, (Rν1 , Rν1 +ν2 , . . . , Rn )) ∈ sp(F) × IFlag(ν1 , ν1 + ν2 , . . . , n; F) | for i = 2, . . . , r, φ(Rν1 ) = 0, φ(Rν1 +...+νi ) ⊂ Rν1 +...+νi−1

φ(Rν∨ +...+ν ) ⊂ Rν∨ +...+ν for i = 2, . . . , r, φ(F) ⊂ Rν∨ } 1

i−1

i

1

1

The word desingularization is given above in quotation marks because the variety IFlag(ν1 , ν1 + ν2 , . . . , n; F) has two connected components IFlag+ (ν1 , ν1 + ν2 , . . . , n; F) and IFlag− (ν1 , ν1 + ν2 , . . . , n; F), depending on whether Rn is in IGrass+ (m, F) or in IGrass− (m, F). The corresponding components Zˆ ν+ and Zˆ µ− give desingularizations of the closures of conjugacy classes Cµ(1) and Cµ(2) for the special orthogonal groups. In the sequel we work with Zˆ ν+ and with Cµ(1) . (8.4.9) Example. n = 5, µ = (4, 4, 2, 2). The partition ν = (4, 2). The grid corresponding to Zˆ ν is X

& GC((4, 4, 2, 2)) =

X X

X X X

X X X X

X X X X X

X X O O O O

X X O O O O O

X X O O O O O O

X X O O O O O O O

O O O O O O O O O O

O O O O O O. O O O O O

Notice that in this grid all entries corresponding to ei ∧ e j are marked with X , all entries corresponding to e¯ i ∧ e¯ j are marked with O, and in the region

290

The Nilpotent Orbit Closures

corresponding to ei ∧ e¯ j we have the same pattern as for the desingularization of the conjugacy class O(ν) for GL(n). This rule is true for a general very even partition. The variety Zˆ ν+ again ﬁts all the assumptions of the setup from chapter 5. Denote the corresponding bundles by ξ := Sν , η := Tν . Deﬁne the partition ˆ = 2ˆν is again very even. νˆ = (ν1 − 1, . . . , ν j − 1). Notice that the partition µ The idea of the inductive procedure for very even conjugacy classes is to look at the bundle corresponding to the weights in our grid in the columns entirely ﬁlled by circles. There are j = ν1 such columns. The corresponding bundle S j lives on IGrass( j, F), and it can be best described by two exact sequences it ﬁts: 0 → S2 R → R ⊗ F → S j → 0, 0→

(∗)

2

(∗∗) R → S j → R ⊗ (F/R) → 0. Notice that the factor T j := 2 F/S j can be identiﬁed with 2 (F/R). The point now is that we have an exact sequence 0 → Sν1 → Sν → Sνˆ (R∨ /R) → 0, where Sνˆ (R∨ /R) is the bundle Sνˆ for the partition νˆ in the relative situation where the orthogonal space of dimension 2(n − j) is replaced by the bundle R∨ /R. Our strategy is again to take a complex F•νˆ in a relative situation and for each term of that complex (which is a special orthogonal irreducible V(α1 ,...,αn− j ) (R∨ /R)) to estimate the terms resulting in the cohomology of V(α1 ,...,αn− j ) (R∨ /R) ⊗ • S j . Notice that these cohomology groups are the terms of a twisted complex F(V(α1 ,...,αn− j ) (R∨ /R))• supported in the variety Y j which is the image of the incidence variety Z j = {(φ, R) ∈ X × IGrass( j, F) | φ(R) = 0}. We start with the results we need about the twisted complexes of that kind. (8.4.10) Proposition. Assume that α1 ≤ j. (a) The complex F(V(α1 ,...,αn− j ) (R∨ /R))• has terms in nonnegative degree, i.e., m i ∨ Sj = 0 H IGrass( j, F), V(α1 ,...,αn− j ) (R /R) ⊗ for i > m. (b) The terms of the complex F(V(α1 ,...,αn− j ) (R∨ /R))• contain only the representations V(β1 ,...,βn ) F with β1 ≤ j.

8.4. Conjugacy Classes for the Orthogonal Group

291

Proof. We start with property (a). We use the exact sequence (∗). The m-th exterior power of the two term complex S2 R → R ⊗ F gives by (2.4.7) an acyclic complex 0 → Sm (S2 R) → Sm−1 (S2 R) ⊗ (R ⊗ F) m−1 m → . . . → S2 R ⊗ (R ⊗ F) → (R ⊗ F)

(?)

resolving m S j . The complex (?) ⊗ V(α1 ,...,αn− j ) (R∨ /R) provides a resolum S j ⊗ V(α1 ,...,αn− j ) (R∨ /R). We split that complex into short exact tion of sequences and use the induced long sequences of cohomology groups. It follows that in order to show (a) it is enough to prove that H i (IGrass( j, F),

m−u

(R ⊗ F) ⊗ Su (S2 R) ⊗ V(α1 ,...,αn− j ) (R∨ /R)) = 0

for i > m + u. Decomposing into representations K γ R using the formula (2.3.8)(b), the Cauchy fomula (2.3.3), and the Littlewood–Richardson rule (2.3.4), we see that in order to prove part (a) of Proposition (8.4.10) we need to show (8.4.11) Lemma. Let α = (α1 , . . . , αn− j ) be a partition such that α1 ≤ j. Let γ be a partition of m. Then H i (IGrass( j, F), K γ R ⊗ V(α1 ,...,αn− j ) (R∨ /R)) = 0 for all i > m. Proof. We ﬁx α and look at the set of all γ such that the cohomology H l(α,γ ) (IGrass( j, F), K γ R ⊗ V(α1 ,...,αn− j ) (R∨ /R)) = 0 for a unique number l(α, γ ). We deﬁne N (α, γ ) = |γ | − l(α, γ ). We want to show that N (α, γ ) ≥ 0. We recall that by Corollaries (4.3.7) and (4.3.9) the cohomology of the vector bundle K γ R ⊗ V(α1 ,...,αn− j ) (R∨ /R) is calculated by looking at the sequence (−γ j + n − 1, −γ j − 1 + n − 2, . . . , −γ1 + n − j, α1 + n − j − 1, . . . , αn− j ) and trying to make it decreasing, with the sum of last two entries positive, using the Weyl group action. Since we have −γ j + n − 1 > −γ j−1 + n − 2 > . . . > −γ1 + n − j, there exists the smallest s for which −γs + n − j + s − 1 < 0. The number l(α, γ ) is calculated as follows. We move −γ1 + n − j to the end by a sequence of exchanges with numbers following it, change its value

292

The Nilpotent Orbit Closures

to its negative, and do the same with −γ2 + n − j + 1, . . . , −γs + n − j + s − 1. We get a sequence of positive numbers P(α, γ ), which we reorder to get a decreasing sequence R(α, γ ). Each exchange of neighboring elements in this process and change of the sign of the last number contributes 1 to N (α, γ ). Notice that in the sequence P(α, γ ) the numbers from the positions of γ1 , . . . , γs occupy the last s spots. Also notice that α1 + n − j, . . . , αn− j + 1 as well as −γs+1 + (n − j) + s + 1, . . . , −γ j + n are all numbers ≤ n. Therefore there is a unique γ1 , . . . , γs for which the resulting sequence is (n, n − 1, . . . , 1). We claim that the minimum of N (α, γ ) is achieved for such γ . Indeed, assume that some of the numbers in R(α, γ ) are bigger than n. Let u be the smallest number not occurring in R(α, γ ). We have u ≤ n. Let m be the smallest number ≥ u that occurs in R(α, γ ) and comes from among the positions corresponding to γ1 , . . . , γs , say from γt . Let u = m − l. Then there exists a unique partition δ such that δ ⊂ γ , |δ| = |γ | − l, R(α, δ) = R(α, γ ) ∪ {u} \ {m}. Now δ is obtained from γ in at most l steps. Each step consists of decreasing a part of γ by 1 (which forces some of the following parts to also decrease so we again get a partition), starting with γt . At each stage the new repetition in the reordered sequence appears. This repetition has to involve a position coming from γ1 , . . . , γs , so we have to decrease this part of γ next, and so on. We know that |δ| = |γ | − l. Comparing the reordering processes for (α, γ ) and (α, δ), we see that l(α, δ) ≥ l(α, γ ) − l + 1 and the decrease in the number of exchanges is equal to the number of entries {m − 1, . . . , u + 1} in R(α, γ ) that did not come from positions corresponding to −γs + n − j + s, . . . , −γ1 + n − j + 1. Thus N (α, δ) ≤ N (α, γ ) − 1, as desired. Assume that (α, γ ) is such that the reordered sequence is (n, n − 1, . . . , 1). We shall prove that for ﬁxed α and for any γ for which R(α, γ ) = (n, n − 1, . . . , 1), the partition γ with minimal N (α, γ ) is the one with s = 0. Indeed, let us assume that s > 0. Consider the sequence (−γ j + n, −γ j − 1 + n − 1, . . . , −γ1 + n − j + 1, α1 + n − j, . . . , αn− j + 1). We construct a new partition δ as follows. We modify the above sequence by changing the negative entry −γs + n − j + s to its negative x = γs − n + j − s > 0. Then we order the positive part of this sequence and see that it can be written as (−δ j + n, −δ j−1 + n − 1, . . . , −δ1 + n − j + 1, α1 + n − j, . . . , αn− j + 1) for a partition δ with |δ| = |γ | − 2x. The partition δ has smaller invariant s. We count the number l(α, γ ) − l(α, δ). The difference is that the number −γs + n − j + s gets exchanged twice with numbers from the set α1 + n − j, . . . , αn− j + 1 that are in [1, x − 1], then it is reﬂected to its negative and

8.4. Conjugacy Classes for the Orthogonal Group

293

then gets exchanged additionally with the numbers from the interval [1, x − 1] that preceed it. Therefore we have l(α, γ ) − l(α, δ) ≤ 2x − 1 and thus N (α, γ ) > N (α, δ). It remains to show that for ﬁxed α and for partitions γ such that R(α, γ ) = (n, n − 1, . . . , 1) and s = 0 we have N (α, γ ) ≥ 0. But now we see that the last reﬂection does not occur in exchanging entries. Therefore we are in the situation of Lemma (8.1.5). We see that the choice of γ minimizing N (α, γ ) is γ = α and that N (α, α ) = 0. This proves Lemma (8.4.11) and therefore part (a) of Proposition (8.4.10). We prove part (b) of Proposition (8.4.10). We use the exact sequence (∗∗). We will prove that the cohomology of •

(S2 R) ⊗

•

(R ⊗ (F/R)) ⊗ Vα (R∨ /R)

does not contain the representations Vβ F with β1 > j. Using the exact sequence 0 → R∨ /R → F/R → F/R∨ → 0, we see that it is enough to show that the cohomology of • • • (S2 R) ⊗ (R ⊗ (R∨ /R)) ⊗ (R ⊗ (F/R∨ )) ⊗ Vα (R∨ /R)

does not contain the representations Vβ F with β1 > j. Let us look at a typical term K β R ⊗ K γ R ⊗ K γ (R∨ /R) ⊗ K δ R ⊗ K δ (F/R∨ ) ⊗ Vα (R∨ /R). After decomposing the Weyl functors to the symplectic irreducibles and calculating tensor products, we see that the term above will decompose to the summands of the form K λ (F/R∨ ) ⊗ Vθ (R∨ /R). Such a bundle has a representation Vβ F with β1 > j in cohomology if a sequence (λ1 + n, λ2 + n − 1, . . . , λ j + n − j + 1, θ1 + n − j, . . . , θn− j + 1) contains either the number > n + j or a number < −n − j. Since θ is an integral dominant weight for Sp(2n − 2 j), i.e. a partition, this can happen if

294

The Nilpotent Orbit Closures

one of the following cases occurs: (a) λ1 + n > n + j, i.e. λ1 > j, (b) θ1 + n − j > n + j, i.e. θ1 > 2 j, (c) λ j + n − j + 1 < −n − j, i.e. −λ j > 2n + 1. Case (a) cannot occur because K λ (F/R∨ ) is a summand in a tensor product K β R ⊗ K γ R ⊗ K δ R ⊗ K δ (F/R∨ ), and the only possible positive weights come from δ . However, δ1 ≤ j, since dim R = j. Case (b) cannot occur because the weight θ comes from a summand in K γ (R∨ /R) ⊗ Vα (R∨ /R) and α1 ≤ j and γ1 ≤ j (because the corresponding functor K γ R is nonzero). Thus K γ (R∨ /R) decomposes to the representations V (R∨ /R), and in the tensor product V (R∨ /R) ⊗ Vα (R∨ /R) all weights have to be ≤ α + by Klimyk’s formula (exercise 12 of chapter 4), and therefore the ﬁrst entry of each occurring weight is ≤ 2 j. Case (c) cannot occur because in the tensor product K β R ⊗ K γ R ⊗ K δ R ⊗ K δ (F/R∨ ) the last entries are −β1 ≥ − j − 1, −γ1 ≥ 2 j − 2n, and −δ1 ≥ − j (because K δ (F/R∨ ) has to be nonzero). This concludes the proof of part (b) of Proposition (8.4.10).• (8.4.12) Corollary. (a) The term F0 consists of one copy of the trivial representation in homogeneous degree 0. In particular the variety Z j is normal, with rational singularities. (b) The term F1 contains only the representations Vβ (F) with β1 ≤ 1. In particular the deﬁning ideal of Y j := q(Z j ) is generated by such representations. More precisely, the minimal set of generators of the deﬁning ideal of Y j consists of the vectors in the representation 2(n− j+1)

F ⊂ K (22(n− j+1) ) F ⊂ S2(n− j+1) (S2 F).

Proof. The ﬁrst claim follows from the proof of Lemma (8.4.11). Indeed, the only term contributing to F0 corresponds to γ = 0 which, looking at the sequences (?), can appear only for m = 0.

8.4. Conjugacy Classes for the Orthogonal Group

295

The proof of the second part is based on the reﬁnement of the Lemma (8.4.11). Claim. The only partition γ with |γ | = m such that H m−1 (IGrass( j, F), K γ R) = 0 is γ = (2(n − j + 1)), and the resulting cohomology group is a trivial representation of Sp(F). Proof. We repeat the proof of Lemma (8.4.11) with α = (0). Our partition γ has to lead in one step to the partition γ = 0, which gives the only partition with N (γ , (0)) = 0. The sequence R(γ , (0)) has to be (n, n − 1, . . . , 1), because if not, then s > 0 and the ﬁrst step in the procedure does decrease s. We see now that s = 1; otherwise we need s steps to reduce to γ = 0. Now we can do a case by case analysis, because the number of possibilities for γ corresponds to possibilities for the number γ1 − n + j − 1 ∈ {n, n − 1, . . . , n − j + 1}. The only viable possibility turns out to be γ1 − n + j − 1 = n − j + 1 which implies γt = 0 for t ≥ 2. Now we see, using the exact sequence (?) resolving m S j , that the only 2(n− j+1) possible contribution to F1 is F in homogeneous degree 2(n − j + 1). We easily identify this set of equations with the ones given in (8.4.12) (b) because they vanish on Y j . We can ﬁnally state the results of our induction. (8.4.13) Theorem. Let µ be an even partition, µ = 2ν. (a) The orbit closure Yµ is normal, with rational singularities, (b) The irreducible representations occurring in the terms of the complex F•µ have highest weights (β1 , . . . , βn ) with β1 ≤ ν1 . µ (c) The term F1 contains only the representations Vβ (F) with β1 ≤ 1. Therefore the deﬁning ideal of Yµ is generated by the representations of this kind. Proof. We use the induction described above, connecting the relative complex F•µˆ to the complex F•µ . We use the exact sequence 0 → Sν1 → Sν → Sνˆ (R∨ /R) → 0 to estimate the terms in the cohomology of Sν . They come from the terms of the cohomology in Vα (R∨ /R) ⊗ • (Sν1 ). By induction the terms Vα (R∨ /R) satisfy the assumption of Proposition (8.4.10), which shows that the i-th term of the complex F µˆ (R∨ /R)• can produce only terms in the homological degree ≥ i. Thus, by induction, only the terms in nonnegative homological

296

The Nilpotent Orbit Closures µ

degree occur, and by Corollary (8.4.12) the term F0 consists of one copy of the trivial representation in homogeneous degree 0. Parts (a) and (b) of the theorem follow. µ To prove part (c) let us analyze the term F1 . The terms there can come from the term F• , which has the required form by Corollary (8.4.12)(b), and from µ ˆ the terms from F1 (R∨ /R), which by induction are the terms of homological degree zero in F(V(α1 ,...,αn− j ) (R∨ /R))• , with weight α for which α1 ≤ 1. By the proof of Lemma (8.4.11) the only such term comes from γ = α , so (by the use of the complexes Vα (R∨ /R) ⊗ (?)) the resulting contribution to the possible terms of F(V(α1 ,...,αn− j ) (R∨ /R))• is K α F ⊗ V(0) F, which decomposes to the representations of Sp(F) of the required form. 8.5. Conjugacy Classes for the Symplectic Group In this section, F denotes a vector space of dimension 2n with a nondegenerate antisymmetric form ( , ). The symplectic group Sp(F) is the set of linear automorphisms of F preserving ( , ), i.e., φ ∈ Sp(F) if and only if for each x, y ∈ F we have (φ(x), φ(y)) = (x, y). By deﬁnition Sp(F) is a subgroup of GL(F). The corresponding Lie algebra sp(F) is a subalgebra of the Lie algebra gl(F). The morphism φ ∈ HomK (F, F) is in sp(F) if and only if for each x, y ∈ F we have (φ(x), y) + (x, φ(y)) = 0. Let us choose the symplectic basis e1 , . . . , en , e¯ n , . . . , e¯ 1 of F. This means (ei , e j ) = (¯ei , e¯ j ) = 0,

(ei , e¯ j ) = δi, j ,

where δi, j denotes the Kronecker delta. We can write φ as a matrix, writing in consecutive columns the images of vectors −¯e1 , . . . , −¯en , en , . . . , e1 expanded in the basis e1 , . . . , en , e¯ n , . . . , e¯ 1 . Then φ ∈ sp(F) if and only if the matrix of φ is symmetric. This allows us to identify the adjoint representation of sp(F) with S2 F. Since sp(F) is a Lie subalgebra of gl(F), we might expect that the nilpotent conjugacy classes in sp(F) will be related to intersections of the nilpotent conjugacy classes in gl(F) with sp(F). The following result is proved in [SS; IV.2.15]. (8.5.1) Proposition. Let µ be a partition of 2n. Let us consider the nilpotent conjugacy class O(µ) of gl(F) corresponding to µ. Then O(µ) intersects sp(F) if and only if every odd part of µ occurs an even number of times. In this case the intersection O(µ) ∩ sp(F) consists of a single conjugacy class of Sp(F).

8.5. Conjugacy Classes for the Symplectic Group

297

For the remainder of this section we denote by Ps (n) the set of partitions µ of 2n in which every odd part occurs even number of times. For µ ∈ Ps (n) we denote C(µ) the corresponding conjugacy class in sp(F), and by Yµ its closure. One might say that the Jordan blocks for the symplectic group are of two kinds: the blocks corresponding to even rows in our partition and the blocks corresponding to the pairs of odd rows. The canonical forms of nilpotent symplectic endomorphisms corresponding to both kinds of blocks are as follows. The nilpotent e corresponding to the partition (2n) has the form e(¯e1 ) = 0,

e(¯ei ) = e¯ i−1

e(en ) = −¯en ,

e(ei ) = −ei+1

for i > 0, for i < n.

We can express the action of e by the sequence of arrows ±e1 → ∓e2 → . . . → en−1 → −en → e¯ n → e¯ n−1 → . . . → e¯ 2 → e¯ 1 → 0. If we extend e to an sl2 triple {e, h, f }, the grading induced by e described in section 8.3 is as follows. For the endomorphism e corresponding to the partition 2n we have h(ei ) = (2n + 1 − 2i)ei , h(¯ei ) = (2i − 1 − 2n)¯ei . Therefore the degrees of the basis vectors are deg ei = 2n + 1 − 2i, deg e¯ i = 2i − 1 − 2n. If n = 2t + 1 is odd, then the nilpotent e corresponding to the partition (n, n) has the form e(ei ) = ei+2

for i < 2t,

e(¯ei ) = −¯ei−2

e(e2t ) = e¯ 2t+1 ,

for i > 2,

e(e2t+1 ) = e¯ 2t ,

e(¯e2 ) = e(¯e1 ) = 0.

We can express the action of e by two sequences of arrows e1 → e3 → . . . → e2t+1 → e¯ 2t → −¯e2t−2 → . . . → ±¯e2 → 0, e2 → e4 → . . . → e2t → e¯ 2t+1 → −¯e2t−1 → . . . → ∓¯e1 → 0. Extending e to the sl2 -triple {e, h, f }, we see that the element h acts as follows: h(e2i+1 ) = 2(t − i)e2i+1 , h(e2i ) = 2(t + 1 − i)e2i , and h(¯e2i+1 ) = −2(t − i)¯e2i+1 , h(¯e2i ) = −2(t + 1 − i)¯e2i . This gives deg e2i+1 = 2(t − i), deg e2i = 2(t + 1 − i), and deg e¯ 2i+1 = −2(t − i), deg e¯ 2i = −2(t + 1 − i). In this setup the degrees of vectors ei are always nonnegative and we always have deg ei = −deg e¯ i , because h ∈ sp(F). If a partition µ corresponds to the conjugacy class in sp(F), then we can assign the grading separately in each block, as in Example (8.3.2). We also use the convention that when dealing with several blocks, after assigning the grading, we order ei ’s in such a way that deg ei ≥ deg ei+1 .

298

The Nilpotent Orbit Closures

(8.5.2) Example. Let us take n = 5, µ = (3, 3, 2, 1, 1). The grading of basis elements is as follows: 2 0 −2 2 0 −2 1 −1 . 0 0 We order the elements e1 , . . . e5 , e¯ 5 , . . . , e¯ 1 so their grading is nonincreasing. We get deg e1 = deg e2 = 2, deg e3 = 1, deg e4 = deg e5 = 0, with deg e¯ i = −deg ei for i = 1, . . . , 5. This allows us to determine the grading on sp(F) in all cases. Identifying the adjoint representation with S2 F, we can arrange the weight vectors in it in a triangular grid. In order to describe it, let us introduce the involution ()¯ of our symplectic basis by requiring that e¯i = ei . The elements of the grid correspond to the entries on or above the diagonal of our matrix representation of φ. If u, v are the elements of our symplectic basis, then the entry corresponding to the row u and the column ±v will correspond to the weight vector u v¯ . For a given conjugacy class C(µ) we will mark the element of the grid with X if the corresponding weight vector is in g ≥2 , and with O otherwise. We will denote this grid by GC(µ). (8.5.3) Example. Let n = 5, µ = (3, 3, 2, 1, 1). Then the degrees of elements ei are given in Example (8.5.2), and we have X

GC(µ) =

X X

X X X

X X O O

X X O O O

X X O O O O

X X O O O O O

O O O O O O O O

O O O O O O O O O

O O O O O . O O O O O

We continue with several examples of conjugacy classes. (8.5.4) Example. Consider the class C((2n )). This is a very interesting conjugacy class. The associated grading gives deg(ei ) = 1, deg(¯ei ) = −1 for

8.5. Conjugacy Classes for the Symplectic Group

299

i = 1, . . . , n. This means that in the grid GC((2n )) the entries marked by X correspond to the vectors ei e j for i ≤ j. For example, if n = 5 we get X

GC((25 )) =

X X

X X X

X X X X

X X X X X

O O O O O O

O O O O O O O

O O O O O O O O

O O O O O O O O O

O O O O O . O O O O O

The parabolic subgroup P is the set of elements ﬁxing a given isotropic subspace of dimension n. The homogeneous space G/P is the isotropic Grassmannian IGrass(n, F). The desingularization Z constructed in section 8.3 can be identiﬁed as Z = {(φ, R) ∈ sp(F) × IGrass(n, F) | φ(F) ⊂ R, φ(R) = 0}. Let R be a tautological subbundle (of dimension n ) on IGrass(n, F). We apply the results of section 5.1 to Z . The vector bundles ξ and η are easily identiﬁed: η = S2 (F/R) and ξ = Ker(S2 F → S2 R). Theorem (5.1.2)(b) implies now that R i q∗ O Z = H i (IGrass(n, F), Sym(S2 (F/R))). Using the Cauchy formula (2.3.8)(a), we see that if we denote by EP(d) the set of partitions of 2d with even parts, then Sd (S2 (F/R)) = K β (F/R). β∈EP(d), β1 ≤n

Theorem (4.3.1) implies that R i q∗ O Z = 0 for i > 0 and that ¯ n ))] = K[C((2 Vβ (F), d≥0 β∈EP(d), β1 ≤n

where we identify the partition with at most n parts with the dominant weight for the group SP(F). We have an exact sequence describing the representation Vβ (F) as a cokernel of a map of Schur functors K β/(12 ) F → K β F → Vβ (F) → 0 (compare Exercise 4 of chapter 6) with the left map being induced by the trace

element tr = 1≤i≤n ei ∧ e¯ i ∈ 2 F.

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The Nilpotent Orbit Closures

¯ n )) consists of all polynomials This means that the deﬁning ideal of C((2 which can be expressed as (x1 ei )(x2 x3 ) . . . (x2t−1 e¯ i ) − (x1 e¯ i )(x2 x3 ) . . . (x2t−1 ei ) i

i

where x1 , . . . , x2t−1 ∈ F. Since any such polynomial is clearly a product of a polynomial of degree 2 and t − 2 polynomials of degree 1, we see that the deﬁning ideal of C((2n )) is generated by elements of degree 2 of type (x1 ei )(x2 e¯ i ) − (x1 e¯ i )(x2 ei ). i

i

(8.5.5) The Nonnormal Orbit. We give the example of the smallest nonnormal orbit closure for the symplectic group. It is Y(3,3,1,1) . Let us analyze this case. The grid G R(3, 3, 1, 1) looks as follows: X

GC((3, 3, 1, 1)) =

X X

X X O

X X O O

X X O O O

X X O O O O

O O O O O O O

O O O O . O O O O

Let us consider the second symmetric power S2 (η). By the Hinich– Panyushev theorem (8.3.3) we know that the only cohomology group that does not vanish is H 0 (G/P, S2 (η)). To calculate this group, it is enough to calculate the Euler characteristic of the bundle S2 (η). To do this one can replace the bundle S2 (η) with the direct sum of its composition factors of dimension 1. The whole matter becomes an exercise of using Bott’s theorem 55 times. The result is H 0 (G/P, S2 (η)) = V(4,0,0,0) (F) ⊕ V(2,2,0,0) (F) ⊕ V(1,1,0,0) (F) ⊕ V(1,1,1,1) (F). Applying (5.1.3)(b), we see that the group H 0 (G/P, S2 (η)) is the second graded component of the normalization of the coordinate ring of Y(3,3,1,1) . If this variety were normal, H 0 (G/P, S2 (η)) would be a factor of S2 (S2 F). However using the sequences of exercise 4 of chapter 6, we can easily see that S2 (S2 F) = V(4,0,0,0) (F) ⊕ V(2,2,0,0) (F) ⊕ V(1,1,0,0) (F) ⊕ V(0,0,0,0) (F). This proves that our closure is not normal.

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301

The normal orbit closures for the symplectic Lie algebra were determined by Kraft and Procesi in [KP2]. They used the method of minimal degenerations. We state their result. (8.5.6) Theorem (Kraft–Procesi). Let µ ∈ Ps (2n). The orbit closure Yµ is normal if and only if for every i < j such that µi and µ j are odd, with µi > µ j , at least one of the parts µi+1 , . . . , µ j−1 has to be even. Even though the method of minimal degenerations is more effective in describing normality, calculations such as the one above are still useful, as they allow to describe the decomposition to irreducibles of the normalization of the coordinate ring of an orbit closure. For the remainder of this section we look at the even conjugacy classes, i.e., we assume that the partition µ has only even parts. These classes show certain similarities with the conjugacy classes for the general linear group. In particular, they have another desingularization which is more convenient for explicit calculations. Let µ = (µ1 , . . . , µr ) be a partition of 2n with only even parts, µi = 2νi for i = 1, . . . , r . We denote ν = (ν1 , . . . , νr ). Take V = IFlag(ν1 , ν1 + ν2 , . . . , n; F). The desingularization Zˆ ν of C(µ) is deﬁned as follows: Zˆ ν = {(φ, (Rν1 , Rν1 +ν2 , . . . , Rn )) ∈ sp(F) × IFlag(ν1 , ν1 + ν2 , . . . , n; F) | for i = 2, . . . , r, φ(Rν1 ) = 0, φ(Rν1 +...+νi ) ⊂ Rν1 +...+νi−1

φ(Rν∨ +...+ν ) ⊂ Rν∨ +...+ν for i = 2, . . . , r, φ(F) ⊂ Rν∨ }. 1

i−1

i

1

1

(8.5.7) Example. Let n = 5, µ = (6, 4). The partition ν = (2, 2, 1). The grid corresponding to Zˆ ν is X

& GC((4, 4, 2)) =

X X

X X X

X X X X

X X X X X

X X X X O O

X X O O O O O

X X O O O O O O

O O O O O O O O O

O O O O O . O O O O O

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The Nilpotent Orbit Closures

Notice that in this grid all entries corresponding to ei e j are marked with X , all entries corresponding to e¯ i e¯ j are marked with O, and in the region corresponding to ei e¯ j we have the same pattern as for the desingularization of the conjugacy class O(ν) for GL(n). This rule is true for a general even partition. The variety Zˆ ν again ﬁts all the assumptions of the setup from chapter 5. Denote the corresponding bundles ξ := Sν , η := Tν . Deﬁne the partition νˆ = (ν1 − 1, . . . , ν j − 1). The idea of the inductive procedure for even conjugacy classes is to look at the bundle corresponding to the weights in our grid in the columns entirely ﬁlled by circles. There are j = ν1 such columns. The corresponding bundle S j lives on IGrass( j, F), and it can be best described by two exact sequences it ﬁts: 0→

2

R → R ⊗ F → S j → 0,

0 → S2 R → S j → R ⊗ (F/R) → 0.

(∗) (∗∗)

Notice that the factor T j := S2 F/S j can be identiﬁed with S2 (F/R). The point now is that we have an exact sequence 0 → Sν1 → Sν → Sνˆ (R∨ /R) → 0, where Sνˆ (R∨ /R) is the bundle Sνˆ for the partition νˆ in the relative situation where the symplectic space of dimension 2(n − j) is replaced by the bundle R∨ /R. Our strategy is again to take a complex F•νˆ in a relative situation and for each term of that complex (which is a symplectic irreducible V(α1 ,...,αn− j ) (R∨ /R)) to estimate the terms resulting in the cohomology of V(α1 ,...,αn− j ) (R∨ /R) ⊗ • S j . Notice that these cohomology groups are the terms of a twisted complex F(V(α1 ,...,αn− j ) (R∨ /R))• supported in the variety Y j which is the image of the incidence variety Z j = {(φ, R) ∈ X × IGrass( j, F) | φ(R) = 0}. We start with the results we need about the twisted complexes of that kind. (8.5.8) Proposition. Assume that α1 ≤ j. (a) The complex F(V(α1 ,...,αn− j ) (R∨ /R))• has terms in nonnegative degree, i.e., m H i IGrass( j, F), V(α1 ,...,αn− j ) (R∨ /R) ⊗ Sj = 0 for i > m.

8.5. Conjugacy Classes for the Symplectic Group

303

(b) The terms of the complex F(V(α1 ,...,αn− j ) (R∨ /R))• contain only the representations V(β1 ,...,βn ) F with β1 ≤ j. Proof. We start with property (a). We use the exact sequence (∗). The m-th exterior power of the two term complex 2 R → R ⊗ F gives by (2.4.7) an acyclic complex 2 2 0 → Sm R → Sm−1 R ⊗ (R ⊗ F) → . . . 2 m−1 m R⊗ (R ⊗ F) → (R ⊗ F) (?) → m S j . The complex (?) ⊗ V(α1 ,...,αn− j ) (R∨ /R) provides a resoluresolving m S j ⊗ V(α1 ,...,αn− j ) (R∨ /R). We split that complex into short exact tion of sequences and use the induced long sequences of cohomology groups. It follows that in order to show (a) it is enough to prove that

H

i

IGrass( j, F),

m−u

(R ⊗ F) ⊗ Su

2

∨ R ⊗ V(α1 ,...,αn− j ) (R /R) = 0

for i > m + u. Decomposing into representations K γ R using the formula (2.3.8)(a), the Cauchy formula (2.3.3), and the Littlewood–Richardson rule (2.3.4), we see that in order to prove part (a) of Proposition (8.5.8) we need to show (8.5.9) Lemma. Let α = (α1 , . . . , αn− j ) be a partition such that α1 ≤ j. Let γ be a partition of m. Then H i (IGrass( j, F), K γ R ⊗ V(α1 ,...,αn− j ) (R∨ /R)) = 0 for all i > m. Proof. We ﬁx α and look at the set of all γ such that the cohomology H l(α,γ ) (IGrass( j, F), K γ R ⊗ V(α1 ,...,αn− j ) (R∨ /R)) = 0 for a unique number l(α, γ ). We deﬁne N (α, γ ) = |γ | − l(α, γ ). We want to show that N (α, γ ) ≥ 0. We recall that by Corollary (4.3.4) the cohomology of the vector bundle K γ R ⊗ V(α1 ,...,αn− j ) (R∨ /R) is calculated by looking at the sequence (−γ j + n, −γ j − 1 + n − 1, . . . , −γ1 + n − j + 1, α1 + n − j, . . . , αn− j + 1)

304

The Nilpotent Orbit Closures

and trying to make it decreasing using the Weyl group action. Since we have −γ j + n > −γ j−1 + n − 1 > . . . > −γ1 + n − j + 1, there exists the smallest s for which −γs + n − j + s < 0. The number l(α, γ ) is calculated as follows. We move −γ1 + n − j + 1 to the end by a sequence of exchanges with numbers following it, change its value to its negative, and do the same with −γ2 + n − j + 2, . . . , −γs + n − j + s. We get a sequence of positive numbers P(α, γ ), which we reorder to get a decreasing sequence R(α, γ ). Each exchange of neighboring elements in this process and changing of the sign of the last number contribute 1 to N (α, γ ). Notice that in the sequence P(α, γ ) the numbers from the positions of γ1 , . . . , γs occupy the last s spots. We also notice that α1 + n − j, . . . , αn− j + 1 as well as −γs+1 + (n − j) + s + 1, . . . , −γ j + n are all numbers ≤ n. Therefore there is a unique γ1 , . . . , γs for which the resulting sequence is (n, n − 1, . . . , 1). We claim that the minimum of N (α, γ ) is achieved for such γ . Indeed, assume that some of the numbers in R(α, γ ) are bigger than n. Let u be the smallest number not occurring in R(α, γ ). We have u ≤ n. Let m be the smallest number ≥ u that occurs in R(α, γ ) and comes from the positions corresponding to γ1 , . . . , γs , say from γt . Let u = m − l. Then there exists a unique partition δ such that δ ⊂ γ , |δ| = |γ | − l, R(α, δ) = R(α, γ ) ∪ {u} \ {m}. Here δ is obtained from γ in at most l steps. Each step consists of decreasing a part of γ by 1 (which forces some of the following parts to also decrease so we again get a partition), starting with γt . At each stage the new repetition in the reordered sequence appears. This repetition has to involve a position coming from γ1 , . . . , γs , so we have to decrease this part of γ next, and so on. We know that |δ| = |γ | − l. Comparing the reordering processes for (α, γ ) and (α, δ), we see that l(α, δ) ≥ l(α, γ ) − l + 1, and the decrease in the number of exchanges is equal to the number of entries {m − 1, . . . , u + 1} in R(α, γ ) that did not come from positions corresponding to −γs + n − j + s, . . . , −γ1 + n − j + 1. Thus N (α, δ) ≤ N (α, γ ) − 1, as desired. Assume that (α, γ ) is such that the reordered sequence is (n, n − 1, . . . , 1). We shall prove that for ﬁxed α and for such γ for which R(α, γ ) = (n, n − 1, . . . , 1), the partition γ with minimal N (α, γ ) is the one with s = 0. Indeed, let us assume that s > 0. Consider the sequence (−γ j + n, −γ j − 1 + n − 1, . . . , −γ1 + n − j + 1, α1 + n − j, . . . , αn− j + 1). We construct a new partition δ as follows. We modify the above sequence by changing the negative entry −γs + n − j + s to its negative x = γs − n + j − s > 0. Then we order the positive part of this sequence and see that it

8.5. Conjugacy Classes for the Symplectic Group

305

can be written as (−δ j + n, −δ j−1 + n − 1, . . . , −δ1 + n − j + 1, α1 + n − j, . . . , αn− j + 1) for a partition δ with |δ| = |γ | − 2x. The partition δ has smaller invariant s than γ has. We count the number l(α, γ ) − l(α, δ). The difference is that the number −γs + n − j + s gets exchanged twice with numbers from the set α1 + n − j, . . . , αn− j + 1 that are in [1, x − 1], then it is reﬂected to its negative, and then it gets exchanged additionally with the numbers from the interval [1, x − 1] that preceed it. Therefore we have l(α, γ ) − l(α, δ) ≤ 2x − 1 and thus N (α, γ ) > N (α, δ). It remains to show that for ﬁxed α and for partitions γ such that R(α, γ ) = (n, n − 1, . . . , 1) and s = 0 we have N (α, γ ) ≥ 0. But now we see that the last reﬂection does not occur in exchanging entries. Therefore we are in the situation of Lemma (8.1.5). We see that the choice of γ minimizing N (α, γ ) is γ = α and that N (α, α ) = 0. This proves Lemma (8.5.9) and therefore part (a) of Proposition (8.5.8). We prove part (b) of Proposition (8.5.8). We use the exact sequence (∗∗). We will prove that the cohomology of •

(S2 R) ⊗

•

(R ⊗ (F/R)) ⊗ Vα (R∨ /R)

does not contain the representations Vβ F with β1 > j. Using the exact sequence 0 → R∨ /R → F/R → F/R∨ → 0, we see that it is enough to show that the cohomology of • • • (S2 R) ⊗ (R ⊗ (R∨ /R)) ⊗ (R ⊗ (F/R∨ )) ⊗ Vα (R∨ /R)

does not contain the representations Vβ F with β1 > j. Let us look at a typical term K β R ⊗ K γ R ⊗ K γ (R∨ /R) ⊗ K δ R ⊗ K δ (F/R∨ ) ⊗ Vα (R∨ /R). After decomposing the Weyl functors to the symplectic irreducibles and calculating tensor products, we see that the term above will decompose to the

306

The Nilpotent Orbit Closures

summands of the form K λ (F/R∨ ) ⊗ Vθ (R∨ /R). Such a bundle has a representation Vβ F with β1 > j in cohomology if a sequence (λ1 + n, λ2 + n − 1, . . . , λ j + n − j + 1, θ1 + n − j, . . . , θn− j + 1) contains either a number > n + j or a number < −n − j. Since θ is an integral dominant weight for Sp(2n − 2 j), i.e. a partition, this can happen if one of the following cases occurs: (a) λ1 + n > n + j, i.e. λ1 > j, (b) θ1 + n − j > n + j, i.e. θ1 > 2 j, (c) λ j + n − j + 1 < −n − j, i.e. −λ j > 2n + 1. Case (a) cannot occur because K λ (F/R∨ ) is a summand in a tensor product K β R ⊗ K γ R ⊗ K δ R ⊗ K δ (F/R∨ ) and the only possible positive weights come from δ . However, δ1 ≤ j, since dim R = j. Case (b) cannot occur because the weight θ comes from a summand in K γ (R∨ /R) ⊗ Vα (R∨ /R), and α1 ≤ j and γ1 ≤ j (because the corresponding functor K γ R is nonzero). Thus K γ (R∨ /R) decomposes to the representations V (R∨ /R), and in the tensor product V (R∨ /R) ⊗ Vα (R∨ /R) all weights have to be ≤ α + by Klimyk’s formula (exercise 12 of chapter 4), and therefore the ﬁrst entry of each occurring weight is ≤ 2 j. Case (c) cannot occur because in the tensor product K β R ⊗ K γ R ⊗ K δ R ⊗ K δ (F/R∨ ) the last entries are −β1 ≥ − j − 1, −γ1 ≥ 2 j − 2n, and −δ1 ≥ − j (because K δ (F/R∨ ) has to be nonzero). This concludes the proof of part (b) of Proposition (8.5.8).•

8.5. Conjugacy Classes for the Symplectic Group

307

(8.5.10) Corollary. (a) The term F• consists of one copy of the trivial representation in homogeneous degree 0. In particular the variety Z j is normal, with rational singularities. (b) The terms F1 contains only the representations Vβ (F) with β1 ≤ 1. In particular the deﬁning ideal of Y j := q(Z j ) is generated by such representations. More precisely, the minimal set of generators of the deﬁning ideal of Y j consists of the vectors in the representation 2(n− j+1)

F ⊂ K (22(n− j+1) ) F ⊂ S2(n− j+1) (S2 F).

Proof. The ﬁrst claim follows from the proof of Lemma (8.5.9). Indeed, the only term contributing to F0 corresponds to γ = 0, which, as seen from the sequences (?), can appear only for m = 0. The proof of the second part is based on the reﬁnement of Lemma (8.5.9). Claim. The only partition γ with |γ | = m such that H m−1 (IGrass( j, F), K γ R) = 0 is γ = (2(n − j + 1)), and the resulting cohomology group is a trivial representation of Sp(F). Proof. We repeat the proof of Lemma (8.5.9) with α = (0). Our partition γ has to lead in one step to the partition γ = 0, which gives the only partition with N (γ , (0)) = 0. The sequence R(γ , (0)) has to be (n, n − 1, . . . , 1), because if not, then s > 0 and the ﬁrst step in the procedure does decrease s. We see now that s = 1; otherwise we need s steps to reduce to γ = 0. Now we can do a case by case analysis, because the number of possibilities for γ corresponds to the possibilities for the number γ1 − n + j − 1 ∈ {n, n − 1, . . . , n − j + 1}. The only viable possibility turns out to be γ1 − n + j − 1 = n − j + 1, which implies γt = 0 for t ≥ 2. Now we see using the exact sequence (?) resolving m S j that the only 2(n− j+1) F in homogeneous degree 2(n − j + possible contribution to F1 is 1). We easily identify this set of equations with the ones given in (8.5.10)(b), because they vanish on Y j . We can ﬁnally state the results of our induction. (8.5.11) Theorem. Let µ be an even partition, µ = 2ν. (a) The orbit closure Yµ is normal, with rational singularities,

308

The Nilpotent Orbit Closures

(b) The irreducible representations occurring in the terms of the complex F•µ have highest weights (β1 , . . . , βn ) with β1 ≤ ν1 . µ (c) The term F1 contains only the representations Vβ (F) with β1 ≤ 1. Therefore the deﬁning ideal of Yµ is generated by the representations of this kind. Proof. We use the induction described above, connecting the relative complex F•µˆ to the complex F•µ . We use the exact sequence 0 → Sν1 → Sν → Sνˆ (R∨ /R) → 0 to estimate the terms in the cohomology of Sν . They come from the terms of the cohomology in Vα (R∨ /R) ⊗ • (Sν1 ). By induction the terms Vα (R∨ /R) satisfy the assumption of Proposition (8.5.8), which shows that the i-th term of the complex F µˆ (R∨ /R)• can produce only the terms in the homological degree ≥ i. Thus, by induction, only the terms in nonnegative homological µ degree occur, and by Corollary (8.5.10) the term F0 consists of one copy of the trivial representation in homogeneous degree 0. Parts (a) and (b) of the theorem follow. µ To prove part (c) let us analyze the term F1 . The terms in this complex come from the term F• , which has the required form by Corollary µ ˆ (8.5.10)(b), and from the terms F1 (R∨ /R), which by induction are the terms of homological degree zero in F(V(α1 ,...,αn− j ) (R∨ /R))• , with weight α for which α1 ≤ 1. By the proof of Lemma (8.5.9) the only such term comes from γ = α , so (by the use of the complexes Vα (R∨ /R) ⊗ (?)) the resulting contribution to the possible terms of F(V(α1 ,...,αn− j ) (R∨ /R))• is K α F ⊗ V(0) F, which decomposes to the representations of Sp(F) of the required form. Let me ﬁnish this section with a general conjecture regarding the equations of the nilpotent orbit closures for the symplectic group. (8.5.12) Conjecture. Let µ ∈ Ps (2n). The deﬁning ideal of Yµ is generated by representations V(β1 ,...,βn ) (F) with β1 ≤ 2. More precisely, the generators of the deﬁning ideal of Yµ can be chosen as subrepresentations of Schur functors K (2i ) F inside Si (S2 F) for 1 ≤ i ≤ 2n. (8.5.13) Remark. Klimek has proved that Conjecture (8.5.12) is true up to a radical.

Exercises for Chapter 8

309

Exercises for Chapter 8 Orbits Corresponding to Special Partitions for Classical Groups Type Bn . We assume that F is a vector space of dimension 2n + 1 and that ( , ) is a nondegenerate symmetric form. We identify F with F ∗ by means of the morphism induced by the form. 1. Consider the adjoint representation X = 2 F. We identify A with 2 F). Show that the space K (22i ) F (1 ≤ i ≤ n) contains a unique Sym( (up to a scalar) SO(F)-invariant v2i . Prove that the ring of invariants ASO(F) is isomorphic to the polynomial ring generated by the invariants v2 , . . . , v2n . This is a special case of Chevalley’s theorem. 2. Let F be a vector space of dimension 2n + 1, and let ( , ) be a nondegene rate symmetric form. We take g = so(2n + 1) = 2 F. The subregular orbit in g is the only orbit of codimension 2 in the nilpotent cone. It corresponds to the partition µ = (2n − 1, 1, 1). Let {e, h, f } be an sl2 -triple where e is an element from the subregular orbit. We denote by g i the component of weight i in g considered as a representation of sl2 . (a) Let ξn be the vector bundle occurring in the resolution (5.1.1). We saw in the section 8.4 that we have p ∗ (ξn ) = g ∗

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