Standard Monomial Theory: Invariant Theoretic Approach

Encyclopaedia of Mathematical Sciences Volume 137 Invariant Theory and Algebraic Transformation Groups VIII Subseries Editors: Revaz V. Gamkrelidze Vladimir L. Popov

Venkatramani Lakshmibai Komaranapuram N. Raghavan

Standard Monomial Theory Invariant Theoretic Approach

Venkatramani Lakshmibai Department of Mathematics Northeastern University, Boston 02115 USA e-mail: [email protected]

Komaranapuram N. Raghavan Institute of Mathematical Sciences C. I. T. Campus, Taramani Chennai, 600 113 INDIA e-mail: [email protected]

Founding editor of the Encyclopaedia of Mathematical Sciences: Revaz V. Gamkrelidze

ISBN 978-3-540-76756-5

e-ISBN 978-3-540-76757-2

DOI 10.1007/978-3-540-76757-2 Encyclopaedia of Mathematical Sciences ISSN 0938-0396 Library of Congress Control Number: 2007939889 Mathematics Subject Classification (2000): 13F50, 14M12, 14M15, 14M17, 14L35 c 2008 Springer-Verlag Berlin Heidelberg  This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH, Heidelberg, Germany Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com

To the memory of Professor C. Musili

Preface

The goal of this book is to present the results of Classical Invariant Theory (abbreviated CIT) and those of Standard Monomial Theory (abbreviated SMT) in such a way as to bring out the connection between the two theories. Even though there are many recent books on CIT, e.g., [25, 35, 53, 97, 99], none of them discusses SMT: there is but only a passing mention of the main papers of SMT towards the end of [53]. Details about the connection are also not to be found in the comprehensive treatment of SMT [59] that is in preparation. Hence the need was felt for a book that describes in some detail this natural and beautiful connection. After presenting SMT for Schubert varieties—especially, for those in the ordinary, orthogonal, and symplectic Grassmannians—it is shown (using SMT) that the categorical quotients appearing in CIT may be identified as “suitable” open subsets of certain Schubert varieties. Similar results are presented for certain canonical actions of the special linear and special orthogonal groups. We have also included some important applications of SMT: to the determination of singular loci of Schubert varieties, to the study of some affine varieties related to Schubert varieties—ladder determinantal varieties, quiver varieties, variety of complexes, etc.—and to toric degenerations of Schubert varieties. Prerequisite for this book is some familiarity with commutative algebra, algebraic geometry and algebraic groups. A basic reference for commutative algebra is [27], for algebraic geometry [37], and for algebraic groups [7]. We have also included a brief review of GIT (geometric invariant theory), a reference for which is [87] (and also [96]). We have mostly used standard notation and terminology, and have tried to keep notation to a minimum. Throughout the book, we have numbered Theorems, Lemmas, Propositions etc., in order according to their section and subsection; for example, 3.2.4 refers to fourth item in the second subsection of third section of the present chapter. The chapter number is also mentioned if the item appears in another chapter. This book may be used for a year long course on invariant theory and Schubert varieties. The material covered in this book should provide adequate preparation for

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graduate students and researchers in the areas of algebraic geometry and algebraic groups to work on open problems in these areas. A Homage & an acknowledgement: In the original plan for this book, Musili was one of the co-authors. Unfortunately, Musili passed away suddenly on Oct 9, 2005. We dedicate this book to the memory of Musili. We also would like to thank Ms. Bhagyavati (Musili’s wife) and Ms. Lata (Musili’s daughter) for providing us with the files that Musili had prepared. We wish to thank the referees for their comments.

Boston, Trieste October 2007

V. Lakshmibai K. N. Raghavan

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The subject matter in a nutshell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 What is CIT? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 What is SMT? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 The SMT approach to CIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The subject matter in detail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Proof by the SMT approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 SLn (K), SOn (K) actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Why this book? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 A brief history of SMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Some features of the SMT approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 The organization of the book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 2 2 2 3 5 6 7 7 9

2

Generalities on algebraic varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Some basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Algebraic varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Affine varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 12 12

3

Generalities on algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Abstract root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Root systems of algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Linear algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Parabolic subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Schubert varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Weyl and Demazure modules . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Line bundles on G/Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Equations defining a Schubert variety . . . . . . . . . . . . . . . . . . .

17 17 19 19 21 22 24 25 27

4

Grassmannian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Plücker embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 The partially ordered set Id,n . . . . . . . . . . . . . . . . . . . . . . . . . .

29 29 30

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4.1.2 Plücker embedding and Plücker coordinates . . . . . . . . . . . . . . 4.1.3 Plücker quadratic relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 More general quadratic relations . . . . . . . . . . . . . . . . . . . . . . .  4.1.5 The cone G d,n over Gd,n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.6 Identification of G/Pd with Gd,n . . . . . . . . . . . . . . . . . . . . . . . Schubert varieties of Gd,n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Bruhat decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Dimension of Xi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Further results on Schubert varieties . . . . . . . . . . . . . . . . . . . . Standard monomial theory for Schubert varieties in Gd,n . . . . . . . . . 4.3.1 Standard monomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Linear independence of standard monomials . . . . . . . . . . . . . 4.3.3 Generation by standard monomials . . . . . . . . . . . . . . . . . . . . . 4.3.4 Equations defining Schubert varieties . . . . . . . . . . . . . . . . . . . Standard monomial theory for a union of Schubert varieties . . . . . . . 4.4.1 Linear independence of standard monomials . . . . . . . . . . . . . 4.4.2 Standard monomial basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vanishing theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Arithmetic Cohen-Macaulayness, normality and factoriality . . . . . . . 4.6.1 Factorial Schubert varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30 31 32 33 33 34 34 35 35 36 36 36 37 38 39 39 39 40 41 44 46

Determinantal varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Recollection of facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Equations defining Schubert varieties in the Grassmannian . 5.1.2 Evaluation of Plücker coordinates on the opposite big cell in Gd,n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Ideal of the opposite cell in X(w) . . . . . . . . . . . . . . . . . . . . . . 5.2 Determinantal varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 The variety Dt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Identification of Dt with Yφ . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 The bijection θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 The partial order  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.5 Cogeneration of an ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.6 The monomial order ≺ and Gröbner bases . . . . . . . . . . . . . . .

47 47 48 48 49 49 49 49 51 51 52 53

Symplectic Grassmannian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Some basic facts on Sp(V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Schubert varieties in G/BG . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The variety G/Pn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . − 6.2.1 Identification of Sym Mn with OG ...................... 6.2.2 Canonical dual pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 The bijection θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 The dual Weyl G-module with highest weight ωn . . . . . . . . . 6.2.5 Identification of Dt (Sym Mn ) with YPn (ϕ) . . . . . . . . . . . . . . .

55 56 59 60 61 62 62 63 64

4.2

4.3

4.4

4.5 4.6

5

6

Contents

6.2.6 6.2.7 6.2.8 6.2.9 6.2.10

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Admissible pairs and canonical pairs . . . . . . . . . . . . . . . . . . . . Canonical pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The inclusion η : In,2n → W Pn × W Pn . . . . . . . . . . . . . . . . . . A standard monomial basis for Dt (Sym Mn ) . . . . . . . . . . . . . De Concini-Procesi’s basis for Dt (Sym Mn ) . . . . . . . . . . . . .

65 65 66 67 68

7

Orthogonal Grassmannian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 The even orthogonal group SO(2n) . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Schubert varieties in G/BG . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The variety G/Pn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . − ....................... 7.2.1 Identification of Sk Mn with OG 7.2.2 Canonical dual pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 The bijection θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 The dual Weyl G-module with highest weight ωn . . . . . . . . . 7.2.5 Identification of Dt (Sk Mn ) with YG (ϕ) . . . . . . . . . . . . . . . . . 7.2.6 A standard monomial basis for Dt (Sk Mn ) . . . . . . . . . . . . . . .

71 71 74 77 78 78 79 79 80 82

8

The standard monomial theoretic basis . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 SMT for the even orthogonal Grassmannian . . . . . . . . . . . . . . . . . . . . 8.2 SMT for the symplectic Grassmannian . . . . . . . . . . . . . . . . . . . . . . . . .

85 86 89

9

Review of GIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 G-spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Reductive groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Affine quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Affine actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Categorical quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Good quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Some results on good quotients . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Stable and semi-stable points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 Stable, semistable, and polystable points . . . . . . . . . . . . . . . . . 9.5.2 Other characterizations of stability, semistability . . . . . . . . . . 9.6 Projective quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 L-linear actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8 Hilbert-Mumford criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95 95 95 98 99 101 102 103 105 108 108 110 114 117 117

10 Invariant theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Preliminary lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 SLd (K)-action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 The functions fτ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 The first and second fundamental theorems . . . . . . . . . . . . . . . 10.3 GLn (K)-action: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 The first and second fundamental theorems . . . . . . . . . . . . . . . 10.4 On (K)-action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

121 121 124 124 126 128 129 132

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10.5 Sp2 (K)-action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 11 SLn (K)-action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Quadratic relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 The partially ordered set Hr,d . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The K-algebra S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 The SLn (K)-action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Standard monomials in the K-algebra S . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Quadratic relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 Linear independence of standard monomials . . . . . . . . . . . . . 11.3.3 The algebra S(D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.4 A standard monomial basis for R(D) . . . . . . . . . . . . . . . . . . . 11.3.5 Standard monomial bases for M(D), S(D) . . . . . . . . . . . . . . . 11.4 Normality and Cohen-Macaulayness of the K-algebra S . . . . . . . . . . 11.4.1 The algebra associated to a distributive lattice . . . . . . . . . . . . 11.4.2 Flat degenerations of certain K-algebras . . . . . . . . . . . . . . . . . 11.4.3 The distributive lattice D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.4 Flat degeneration of Spec R(D) to the toric variety Spec A(D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 The ring of invariants K[X]SLn (K) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

137 138 139 140 141 142 143 145 146 148 149 150 150 151 152

12 SOn (K)-action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.1 The Lagrangian Grassmannian variety . . . . . . . . . . . . . . . . . . . 12.1.2 Schubert varieties in Lm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.3 The opposite big cell in Lm . . . . . . . . . . . . . . . . . . . . . . . . . . . . − ............................. 12.1.4 The functions fτ,ϕ on OG 12.1.5 The opposite cell in X(w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.6 Symmetric determinantal varieties . . . . . . . . . . . . . . . . . . . . . . 12.1.7 The set Hm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 The algebra S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Standard monomials and their linear independence . . . . . . . . 12.2.2 Linear independence of standard monomials . . . . . . . . . . . . . 12.3 The algebra S(D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Quadratic relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 A standard monomial basis for R(D) . . . . . . . . . . . . . . . . . . . 12.3.3 Standard monomial bases for S(D) . . . . . . . . . . . . . . . . . . . . . 12.4 Cohen-Macaulayness of S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1 A doset algebra structure for R(D) . . . . . . . . . . . . . . . . . . . . . 12.5 The equality R SOn (K) = S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 Application to moduli problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7 Results for the adjoint action of SL2 (K) . . . . . . . . . . . . . . . . . . . . . . .

159 160 161 161 162 163 164 164 165 167 168 169 169 170 171 172 173 175 176 180 181

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13 Applications of standard monomial theory . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Tangent space and smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.1 The Zariski tangent space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.2 Smooth and non-smooth points . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.3 The space T (w, τ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.4 A canonical affine neighborhood of a T -fixed point . . . . . . . 13.1.5 The affine variety Y (w, τ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.6 Equations defining Y (w, τ ) in Oτ− . . . . . . . . . . . . . . . . . . . . . . 13.1.7 Jacobian criterion for smoothness . . . . . . . . . . . . . . . . . . . . . . . 13.1.8 T -stable curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Singularities of Schubert varieties in the flag variety . . . . . . . . . . . . . 13.2.1 Ideal of Y (w, τ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.2 A criterion for smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.3 Components of the singular locus . . . . . . . . . . . . . . . . . . . . . . . 13.3 Singular loci of Schubert varieties in the Grassmannian . . . . . . . . . . . 13.3.1 Multiplicity at a singular point . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Results for Schubert varieties in a minuscule G/P . . . . . . . . . . . . . . . 13.4.1 Homogeneity of IP (w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.2 A basis for (Mτ,w )r /(Mτ,w )r+1 . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Applications to other varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.1 Ladder determinantal varieties . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.2 The varieties Vi , 1 ≤ i ≤ l . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.3 Quiver varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6 Variety of complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6.1 A partial order on {(k1 , k2 , . . . , kh )} . . . . . . . . . . . . . . . . . . . . . 13.7 Degenerations of Schubert varieties to toric varieties . . . . . . . . . . . . . 13.7.1 Generalities on distributive lattices . . . . . . . . . . . . . . . . . . . . . . 13.7.2 An important example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.7.3 Generalities on toric varieties . . . . . . . . . . . . . . . . . . . . . . . . . . 13.7.4 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.7.5 The algebra associated to a distributive lattice . . . . . . . . . . . . 13.7.6 Varieties defined by binomials . . . . . . . . . . . . . . . . . . . . . . . . . 13.7.7 Degenerations of Schubert varieties in the Grassmannian to toric varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

187 187 187 188 188 188 189 189 189 190 190 191 193 193 195 196 200 201 202 202 202 204 205 209 210 210 210 211 212 213 213 214 215

Appendix: Proof of the main theorem of SMT . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Admissible pairs and the first basis theorem . . . . . . . . . . . . . . . . . . . . A.2.1 More notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.2 Chevalley multiplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.3 Minuscule and classical type parabolics . . . . . . . . . . . . . . . . . A.2.4 Admissible pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 The three examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.1 Example A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.2 Example B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

219 219 220 220 220 220 221 221 222 222

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A.3.3 Example C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Tableaux and the statement of the main theorem . . . . . . . . . . . . . . . . . A.5 Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.6 The tableau character formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.7 The structure of admissible pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.8 The procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.9 The basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.10 The first basis theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.11 Linear independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.12 Arithmetic Cohen-Macaulayness and arithmetic normality . . . . . . . . A.12.1 Arithmetic Cohen-Macaulayness . . . . . . . . . . . . . . . . . . . . . . .

223 224 225 226 226 227 229 230 232 235 236

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Index of notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 Author index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

1 Introduction

1.1 The subject matter in a nutshell This book aims to describe the beautiful connection between Schubert varieties and their S TANDARD M ONOMIAL T HEORY (SMT for short) on the one hand and C LASSICAL I NVARIANT T HEORY (CIT for short) on the other. This connection was discovered by Lakshmibai and Seshadri [67]. Here is the opening paragraph of [67] in a slightly edited form: The main aim of this paper is to show how the work of De Concini and Procesi [22] on classical invariant theory can be interpreted to suggest a generalization of the Hodge-Young theory of standard monomials (cf. Hodge [41] and Hodge and Pedoe [42]). This generalization is given as a set of conjectures (which have now been proved in collaboration with C. Musili [61]). On the other hand, we also show that the results of De Concini and Procesi follow as consequences of the generalization.

SMT is the name given to this generalization of the Hodge-Young theory. And the term CIT in this book refers, for the most part, to the results of De Concini and Procesi mentioned in the quote above. These results, when the characteristic of the ground field is zero, are classical—cf. Weyl’s book [115]. As we will see, determinantal varieties form the bridge connecting CIT and SMT, and Schubert varieties become relevant because they are natural compactifications of determinantal varieties. 1.1.1 What is CIT? CIT concerns certain canonical actions of classical groups G on affine spaces, namely, cases A, B, C, D, and E to be discussed in §1.2 below. It describes, in each case, a presentation for the ring of G-invariant polynomial functions on the affine space. This description comprises of two theorems known as the first and second fundamental theorems. The first fundamental theorem specifies a finite set of algebra generators, over the ground field, for the ring of invariants. Note that for the action of

2

1 Introduction

a reductive group G on an affine variety, the ring of G-invariant polynomial functions is a finitely generated algebra (classically, i.e., in characteristic zero, this result goes back to Weyl [115]; in positive characteristic, this follows from well-known results of Nagata [93, 94] and Haboush [36]: see Chapter 9 for details). The second fundamental theorem specifies a finite set of generators for the ideal of relations among the algebra generators: that there always exists a finite set of generators for the ideal follows from Hilbert’s basis theorem [79]. 1.1.2 What is SMT? The roots of SMT are to be found in the work of Hodge [41, 42], who described nice bases for the homogeneous co-ordinate rings of Schubert varieties of the Grassmannian in the Plücker embedding (over a field of characteristic 0). Grassmannians being precisely the homogeneous spaces that arise as quotients of special linear groups by maximal parabolic subgroups, it is natural to try to generalize Hodge’s work to natural projective embeddings of other quotients G/Q where G is a semisimple algebraic group and Q a parabolic subgroup. In the early ’70s Seshadri initiated this generalization and called it SMT. 1.1.3 The SMT approach to CIT The main idea in this approach is to relate a certain subring of the ring of invariants (which will turn out to be in fact the ring of invariants) as the ring of functions on an affine variety related to a Schubert variety. This allows the use of SMT to prove the first and second fundamental theorems.

1.2 The subject matter in detail In this section, we shall explain the SMT-approach [67] as well as the approach of De Concini-Procesi [22] for the actions of general linear, symplectic, and the orthogonal groups. In both approaches, one makes a guess on the ring of invariants; to be more precise, there are some obvious invariants (we call these the basic invariants), and one shows (in both approaches) that the ring of invariants is in fact generated (as an algebra over the base field) by these invariants. Fix a field K, algebraically closed of arbitrary characteristic, and V a finite dimensional vector space over K. A. The general linear group G := GL(V ) of invertible linear transformations acts naturally on both V and its dual V ∗ . Consider G acting diagonally on Z := V ⊕m ⊕ V ∗ ⊕q ; here V ⊕m denotes the direct sum of m copies of V . Let R be the ring of polynomial functions on Z. There is a natural action of G on R. Let R G be the subring of R consisting of those polynomial functions that are invariant under G. The “scalar products” (v1 , . . . , vm ; f1 , . . . , fq ) → ϕij := fj (vi ), for 1 ≤ i ≤ m and 1 ≤ j ≤ q, are evidently invariant—these are the “basic invariants.”

1.2 The subject matter in detail

3

B. Let now V be equipped with an alternating non-degenerate bilinear form ( , )— note that this forces V to be even dimensional. Let Sp(V ) be the symplectic group, namely, the subgroup of GL(V ) consisting of those linear automorphisms of V that preserve the form. Consider the diagonal action of the symplectic group Sp(V ) on Z := V ⊕m . Let R be the ring of polynomial functions on Z. The “basic invariants” in this case are the bilinear products (v1 , . . . , vm ) → ϕij := (vi , vj ) for 1 ≤ i, j ≤ m. Now suppose that instead of an alternating form we have a symmetric form. More precisely: C. Let the characteristic of K be different from 2 and let V be equipped with a nondegenerate symmetric bilinear form ( , ). Let O(V ) be the orthogonal group, namely, the sub group of GL(V ) consisting of those linear automorphisms of V that preserve the form. Consider the diagonal action of the orthogonal group O(V ) on Z := V ⊕m . Let R be the ring of polynomial functions on Z. The “basic invariants” are still the bilinear products ϕij := (vi , vj ). In all of the three cases A, B, C above, let us denote by S the subalgebra of R G generated by the basic invariants. The goal is (1) to show S = R G which will yield immediately the First Fundamental Theorem. (2) to construct a “nice” basis for S(= R G ) which will yield the “straightening relations” (see §1.5) and hence the Second Fundamental Theorem. To explain the proof of De Concini-Procesi, it suffices to quote in a slightly edited form from [22]: The line of the proof is the following: we have an algebraic group G acting on an affine variety E with coordinate ring R and we have a subring S of R G , namely the one generated by the basic invariants, which we want to show equals R G . First we show that on an open set U ⊆ W (where an element s of S is invertible) the group action is a product action; thus the localized invariant ring R G [1/s] turns out to be S[1/s]. Then we have to find a way to cancel s: i.e., if sa ∈ S and a ∈ R G we must show that a ∈ S. This is accomplished by finding an explicit basis of the ring S and deducing the cancellation result from this. This part is the main contribution of the paper.

1.2.1 Proof of (1), (2) by SMT approach (cf. [67]) Let us denote the basic invariants by f1 , · · · , fN . We have a natural map ψ : X → AN , x → (fi (x)) In all of the three cases A, B, C above, it is not difficult to see that ψ(X)(= Spec S) gets identified with a determinantal variety D: In each case, let G denote the group in

4

1 Introduction

question, namely, G = GL(V ), Sp(V ), O(V ) in cases A, B, C respectively. Let n be the dimension of V . • In A, D is the subvariety Dn+1 (Mm,q ) of Mm,q (K) (the space of m×q matrices with entries in K) consisting of matrices of rank at most n, i.e., the matrices all of whose (n + 1)-minors vanish. • In B, D is the subvariety Dn+1 (Sk Mm ) of Sk Mm (K) (the space of skew symmetric m × m matrices with entries in K) consisting of matrices of rank at most n. • In C, D is the subvariety Dn+1 (Sym Mm ) of Sym Mm (K) (the space of symmetric m × m matrices with entries in K) consisting of matrices of rank at most n. Now f1 , · · · , fN being G-invariants, the morphism ψ goes down to a morphism ψD : Spec R G → D(= Spec S) The main idea behind the proof of the First Fundamental Theorem in [67] is to show that ψD satisfies the hypotheses of ZMT (Zariski’s Main Theorem): (i) ψD is surjective with finite fibers (ii) ψD is birational (iii) D is normal. It would then follow by ZMT that ψD is in fact an isomorphism, and we would obtain that the inclusion S → R G is in fact an equality. The verifications in [67] of (i) & (ii) turn out to be rather straight forward in view of certain geometric invariant theoretic considerations (see Chapter 10 for details). Thus to conclude that S = R G , proving normality of Spec S(= D) is the only non-trivial part. Here is where the Schubert variety connection is used in the approach in [67]. To make this more precise, in case A (respectively, B, C) above, it turns out that Mm,q (K) (respectively Sk Mm (K), Sym Mm (K)) gets identified with the “opposite cell” O− in the Grassmannian Gq,m+q (respectively the orthogonal Grassmannian SO(2m)/Pm , the symplectic Grassmannian Sp2m /Pm ); and D gets identified with O− ∩ X, for a suitable Schubert variety X in Gq,m+q (respectively in SO(2m)/Pm , Sp2m /Pm ). Hence the normality of Spec S(= D) follows, once we know the normality of Schubert varieties. Normality of Schubert varieties is a consequence of SMT. Using the SMT-basis for the homogeneous co-ordinate ring of X, we obtain a basis for D (by the process of dehomogenization). This in turn yields the Second Fundamental Theorem. Thus using Schubert varieties and their SMT, we obtain in one stroke, the proofs of the first and second fundamental theorems. At the time of the appearance of [22], SMT was developed only for “minuscule” G/P ’s. To be more precise, in his thesis written under Seshadri’s guidance, Musili [88] had extended Hodge’s results to arbitrary characteristics. Soon after, Seshadri [111] had generalized Hodge’s results to quotients by minuscule1 maximal 1 A maximal parabolic subgroup P is minuscule if the Weyl group translates of a highest

weight vector span the space of global sections on G/P of the ample generator of the Picard group of G/P . The geometry of a minuscule G/P , i.e., when P is minuscule, is very similar to that of a Grassmannian.

1.2 The subject matter in detail

5

parabolics. As is easily seen, the G/P ’s appearing in cases A, B are minuscule; and the bases for R G as given by [22] on the one hand and SMT on the other are the same. But the G/P arising in case C is not minuscule. Given this, the following problem cried out for an answer: shouldn’t there be an approach to CIT in case C similar to that of cases A, B above? Analysing carefully the work of De Concini and Procesi in case C, Lakshmibai and Seshadri [67] arrived at a conjectural SMT for Schubert varieties in quotients of classical semi-simple groups by maximal parabolic subgroups (technically speaking a little more generally but we will ignore that here for the sake of simplicity). These conjectures were later proved by them in collaboration with C. Musili in [61, 62]. Taking for granted such an SMT, case C too can be handled in a fashion entirely analogous to cases A and B. 1.2.2 SLn (K), SOn (K) actions Replacing the general linear group in case A discussed in §1.2 by the special linear group, consider D. The special linear group SL(V ) acting diagonally on V ⊕m ⊕ V ∗ ⊕q . The basic invariants in this case are the scalar products ϕij := fj (vi ) (as for the general linear group—cf. case A above) and the determinants u(I ) := det [vi1 , . . . , vin ], ξ(J ) := det [fj1 , . . . , fjn ], where 1 ≤ i1 < . . . < in ≤ m, 1 ≤ j1 < . . . < jn ≤ q— these appear only in the case when m ≥ n. Replacing in case C the orthogonal group by the special orthogonal group, consider E. The diagonal action of the special orthogonal group SO(V ) on V ⊕m . The basic invariants in this case are the bilinear products as above and the determinants u(I ) := det [vi1 , . . . , vin ], where 1 ≤ i1 < . . . < in ≤ m—these appear only if m ≥ n. De Concini and Procesi [22] treat CIT in cases A, B, and C fully. In cases D and E, they prove the first fundamental theorem, but there are no details about the second fundamental theorem. As for cases D and E, the invariant rings are not—not in any obvious way at least—rings of functions on open parts of Schubert varieties. So the approach to CIT described in §1.2.1 does not work in exactly the same way. Nevertheless Schubert varieties remain relevant. And an SMT theoretic approach to CIT in cases D and E has recently been worked out in [63, 72] respectively. In Case D, the normality of the ring S generated by the basic invariants is deduced by degenerating its spectrum to a toric variety. As a first step towards the degeneration, a basis for S is constructed. Straightening relations are then written down and the degeneration is carried out using the straightening relations. The CohenMacaulayness of S also follows immediately as a corollary of the degeneration. That the poset structure of Schubert varieties in the Grassmannian form a distributive lattice is used in the proof crucially.

6

1 Introduction

In case E, the ring S generated by basic invariants arises as the ring of functions on a branched covering of degree 2 over the symmetric determinantal variety Dn+1 (Sym Mm ). The normality of this ring is proved by showing that it is non-singular in codimension one and also Cohen-Macaulay (in view of Serre’s criterion—see [79] for example). The first property follows by noting that the branch locus is contained in the singular locus of Dn+1 (Sym Mm ) and that Dn+1 (Sym Mm ) being an open part of a Schubert variety is normal (and in particular non-singular in codimension one). The Cohen-Macaulay property however takes some work to prove. As a first step, one obtains a basis for the ring S. Then a deformation argument due to De Concini and Lakshmibai [21] is applied. They used it originally to show that Schubert varieties in quotients of classical groups by maximal parabolics are arithmetically Cohen-Macaulay for the embedding given by the ample generator of the Picard group. This finishes the discussion of the SMT approach to CIT. There is a lot more that has happened in SMT beyond what is described above. A brief history of SMT is given in §1.4 below. But the scope of this book is confined only to that part of SMT that has been discussed above. In this book we treat in detail all cases A–E via SMT.

1.3 Why this book? The main subject matter of this book, namely the connection between Schubert varieties on the one hand and CIT on the other described in §1.2 above, has not come until now within the scope of any book. The books on CIT—and there are quite a few recent ones among them, e.g., [25, 35, 53, 97, 99]—come no where near discussing this connection. In fact, except in [53] where there is a quick mention of the main papers of SMT towards the end, the books are totally silent about the connection. On the other hand, looking from the SMT side, the book [59] under preparation aims at being a comprehensive account of SMT and is authored by the main players in the creation of that theory. But here too there is very little discussion on the connection between Schubert varieties and CIT. The connection is of course mentioned in the book’s introduction which recounts in detail the history of the development of SMT. But the emphasis of the book being elsewhere, this connection is not treated in the book itself. The raison d’etre for the present book is thus clear. As seen in §1.2.1, the fact that determinantal varieties arise as open sets of Schubert varieties plays a crucial role in the connection between CIT and SMT. Although the relationship between determinantal varieties and Schubert varieties is quite classical, except for [40, 51, 89], there is not much in the literature about it. Further, to the best of our knowledge, [67] is the first work in the literature which discusses the relationship between determinantal varieties in the space of symmetric and skewsymmetric matrices on the one hand and Schubert varieties on the other. These aspects are treated in detail in the present book.

1.5 Some features of the SMT approach

7

1.4 A brief history of SMT We have discussed above the origins of SMT and how the work of De Concini and Procesi on CIT influenced the development of SMT. What follows is a brief account of the history of SMT. The book [59] under preparation aims to give a comprehensive account of SMT. Its introduction gives a detailed account of the history of SMT. But, since the present book will in all likelihood appear before [59], the following account may still be of some value. We may divide SMT into four stages. The first stage consists of the work done up to the appearance of the paper of De Concini and Procesi on CIT. As explained above, this stage comprises of the work of Hodge [41, 42], Musili [88], and Seshadri [111]. The second stage consists of the work done in roughly over a decade after the appearance of De Concini-Procesi’s paper [22] on CIT. This stage begins with the paper of Lakshmibai and Seshadri [67] which describes SMT conjecturally for Schubert varieties in quotients of classical groups by maximal parabolic subgroups. These conjectures are proved in [61, 62]. Finally, in [69], SMT is established for quotients of classical groups by parabolic subgroups (maximal or otherwise). Classical groups having been satisfactorily addressed, the third stage attempts to handle exceptional groups: [55] handles G2 , [64] handles E6 , and [70] handles the case of the affine Kac Moody group slˆ2 . Finally, as a culmination of all the work, the conjectures for a general SMT (for Schubert varieties in G/Q for an arbitrary parabolic subgroup Q of a symmetrizable Kac-Moody Group G) are formulated by Lakshmibai-Seshadri in [71]. Solving the conjectures of [71] required new ideas. This is what Littelmann has done in his work. His papers [74, 75, 77] form the fourth and final stage of SMT. An important idea of Littelmann is to view “standard tableaux” (the indexing set for the SMT basis) as certain paths—the so-called Lakshmibai-Seshadri paths. Littelmann uses the theory of quantum groups. The technique of [103] is also crucially used. Thus SMT is now complete thanks to Littelmann’s work! Our goal being the description of the connection between SMT and CIT, it suffices for us to consider SMT for quotients of classical groups by maximal parabolic subgroups. So in this book we develop SMT in the spirit of [61,62,67]. In particular, we do not discuss Littelmann’s work or quantum group theory. For details of Littelmann’s work, one could refer to the original papers [74, 75, 77], or the book [59].

1.5 Some features of the SMT approach Let us now discuss some features and advantages of the Schubert variety theoretic approach to CIT. This approach is more conceptual than that of [22], the reason being that Schubert varieties provide a powerful inductive tool facilitating the proofs. It also opens doors for generalization to other groups/situations. An important feature in the Schubert-variety-theoretic approach is the qualitative description of quadratic straightening relations—a straightening relation is the expression for a non-standard monomial as a linear sum of standard monomials. In

8

1 Introduction

this approach, the generation of the space of all monomials by standard monomials hinges on the qualitative description of certain quadratic straightening relations; further, the proof of the linear independence of standard monomials is also carried out by using these quadratic relations (and induction on the dimension of a Schubert variety). In the context of CIT, in this approach, such relations are first established on Schubert varieties, and then are specialized to rings of invariants. Even though, there are some combinatorial description of relations of any degree on a determinantal variety in [26, 101], one cannot deduce the required qualitative description of the quadratic relations from the relations found in loc.cit; the reason for this is that in [26, 101], a typical relation (on a determinantal variety) expresses a given monomial as a linear sum of monomials (not necessarily standard) which are greater than the given monomial for a suitable order (which is different from the order used in SMT) on the set of all monomials. Thus the Schubert-variety-theoretic approach seems to be indispensable from this perspective. Also, the Schubert-varietytheoretic approach seems to be the only approach which can yield the desired qualitative description of the quadratic relations - a crucial step in SMT. Similar remarks apply to the symplectic and the orthogonal group actions also. Yet another important advantage in this approach is that we obtain the proof of Cohen-Macaulayness for these rings of invariants, something one does not get in the approach of [22]. The rings of invariants in cases A, B, C described in §1.2 above being identified with open subsets of suitable Schubert varieties, the Cohen-Macaulayness for these rings of invariants follows at once from the Cohen-Macaulayness properties for Schubert varieties. The proof, via SMT, of the Cohen-Macaulayness of Schubert varieties in the Grassmannian, namely those that are related to the categorical quotients in case A, is proved in [88] (see also Chapter 4). The proof, via SMT, of the general case of Schubert varieties in G/P , where G is semi-simple and P a maximal parabolic subgroup of classical type (in particular for those related to the categorical quotients in cases A, B, C is proved by De Concini-Lakshmibai [21]. While the proof in [88] uses commutative algebra arguments, that in [21] uses “deformation technique”. Chirivi [17] has extended the latter technique to the case when G is any semi-simple algebraic group and P is any parabolic subgroup. The deformation technique consists in constructing a flat family over A1 , with the given variety as the generic fiber (corresponding to t ∈ K invertible). If the special fiber (corresponding to t = 0) is Cohen-Macaulay, then one may conclude the Cohen-Macaulayness of the given variety. Hodge algebras (cf. [20]) are typical examples where the deformation technique affords itself very well. Deformation technique is also used in [12, 33, 46]. The philosophy behind these works is that if there is a “standard monomial basis” for the co-ordinate ring of the given variety, then the deformation technique will work well in general (using the “straightening relations”). In recent times, among the several techniques of proving the Cohen-Macaulayness of algebraic varieties, particularly those that are related to Schubert varieties, two techniques have proved to be quite effective, namely, Frobenius splitting technique and deformation technique. Frobenius splitting technique is used in [104], for example, for proving the (arithmetic) Cohen-Macaulayness of Schubert varieties; fur-

1.6 The organization of the book

9

ther, in [83], the normality of Schubert varieties is proved using the Frobenius splitting technique. This technique is also used in [80, 81, 84] for proving the CohenMacaulayness of certain varieties.

1.6 The organization of the book Turning now to the organization of the book, we have tried to make it self-contained keeping in mind the needs of prospective graduate students and young researchers. After reviewing some basics in Algebraic Geometry and Algebraic Groups in Chapters 2 and 3, we first present SMT for the Grassmannian and its Schubert varieties (Musili’s thesis (cf. [88])) in Chapter 4. We then discuss the relationship between determinantal varieties and Schubert varieties in the Grassmannian in Chapter 5. Similar relationships between determinantal varieties in the space of symmetric (respectively skew-symmetric) matrices and Schubert varieties in the symplectic (respectively orthogonal) Grassmannian are established in Chapter 6 (respectively Chapter 7). The main results of SMT are stated in Chapter 8 and are proved in the Appendix. Chapter 9 is a review of GIT. In Chapter 10, using the results of Chapter 5, we describe a basis for the ring of invariants for the GLn (K)-action (cf. Example A above) thus giving a Standard Monomial Theoretic proof for DeConcini-Procesi’s results in this case. We also discuss the SLn (K)-action on the space considered in (A) above, and describe a basis for the corresponding ring of invariants in Chapter 11. Chapters 6 and 7 describe similar results for the symplectic and orthogonal group actions (cf. Examples B, C above) at the same time giving a Standard Monomial Theoretic proof for DeConcini-Procesi’s results in these cases. In Chapter 12, we discuss the SOn (K)-action on the space considered in (C) above, and describe a basis for the corresponding ring of invariants; we further deduce some related results on the moduli space M of equivalence classes of semistable rank 2 vector bundles on a smooth projective curve of genus > 2. We also describe a characteristic-free basis for the ring of invariants for the (diagonal) adjoint action of SL2 (K) on sl2 (K) ⊕ · · · ⊕ sl2 (K).    m copies

In Chapter 13, we discuss some important applications of SMT; we first present a discussion of singular loci of Schubert varieties. Next, we discuss the relationship of ladder determinantal varieties, Quiver varieties and variety of complexes to Schubert varieties, and deduce results for these varieties; in this chapter, as another application of SMT, we present the results of toric degenerations of Schubert varieties in the Grassmannian.

2 Generalities on algebraic varieties

In this chapter, we recollect some basic facts on commutative rings and algebraic varieties. For details, we refer the reader to [27, 37, 79].

2.1 Some basic definitions Definition 2.1.0.1 The Krull dimension: (or simply the dimension) of a commutative ring R (with 1) is the supremum of the lengths of chains of prime ideals of R and it is denoted by dim R. It need not be finite even for Noetherian rings. It is however finite for Noetherian local rings. If R is a Noetherian local ring with its maximal ideal m and residue field k = R/m, then the least cardinality of a set of generators of m equals dimk (m/m2 ); it is called the embedding dimension of R, denoted e(R), and is an upper bound for the dimension of R. In case R is an integral domain with its field of fractions (or quotients) K = Q(R) and R is also a finitely generated algebra over a field F , then K is a finitely generated field extension of F and dim R is precisely the transcendence degree of K over F . (This is not obvious but follows as an immediate consequence of the Noether’s Normalisation Lemma - see [86] for a statement and proof of Noether’s Normalisation Lemma.) The localisation of a ring R at a prime ideal p is denoted by Rp . The dimension of Rp is called the height of p and is denoted by htR (p). For a ring R, dimension of R is the supremum of the heights of its maximal ideals and in particular, for a local ring it is the height of its maximal ideal. Definition 2.1.0.2 A regular local ring is a Noetherian local ring R for which the Krull dimension is the same as its embedding dimension, i.e., dim(R) = e(R). We say that a Noetherian ring is regular if the localisations at all of its maximal ideals are regular local rings.

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2 Generalities on algebraic varieties

Definition 2.1.0.3 Normal domains: If R is an integral domain and K = Q(R) (its field of fractions), then we say that R is normal if it is integrally closed in K, i.e., an element of K which is a root of a monic polynomial with coefficients in R must be already in R. Normality is a local property in the sense that R is normal if and only if Rp (resp.Rm ) is normal for all prime ideals p (resp. maximal ideals m) of R. The ring of quotients S −1 R of a regular (resp. normal) ring R at any multiplicatively closed subset S is regular (resp. normal).

2.2 Algebraic varieties 2.2.1 Affine varieties Let K be an algebraically closed field of arbitrary characteristic. For a non–negative integer n ∈ Z+ , the linear space K n is called the Affine n–space over K and is denoted by An (by convention, A0 = (0), a point). Let A = K[X1 , · · · , Xn ] be the ring of polynomial functions on An . The field K(X1 , · · · , Xn ) which is the field of fractions of A is called the field of rational functions on An and is also denoted by K(An ). For a point P = (a1 , · · · , an ) ∈ An , the ai ’s are called the coordinates of P . An element f ∈ A defines a K-valued function on A by evaluation, namely, P → f (P ) = f (a1 , · · · , an ). The subring OP = OP (An ) := {f/g ∈ K(An ) | g(P ) = 0} which is a local ring is called the local ring of An at P . Recall (see [86] for example) that the ideal m =< X1 − a1 , · · · , Xn − an > is maximal in A for all (a1 , · · · , an ) ∈ K n (for every field K, algebraically closed or not). The converse is true only for algebraically closed fields K (which is known as the Hilbert’s Nullstellensatz - see [86] for further details). Thus points of An for K algebraically closed can be identified with the set Spm(A) of all maximal ideals of the ring A. Given f ∈ A, of total degree d ≥ 1, the set of zeros of f in K n , namely, V (f ) := {(a1 , · · · , an ) ∈ An | f (a1 , · · · , an ) = 0} is called the (affine) hypersurface whose equation is f and d is also called the degree of V (f ). If d = 1, V (f ) is called a hyperplane. It can be seen that a point P = (a1 , · · · , an ) ∈ V (f ) ⇐⇒ f ∈ mP where mP =< X1 − a1 , · · · , Xn − an >, i.e., V (f ) can be identified with the set of all maximal ideals of the quotient ring A / < f >. Given T ⊆ A, the set V (T ) = ∩f ∈T V (f ) of all common zeros of elements of T is called an affine algebraic (or simply an algebraic) subset of An . The family of all algebraic subsets satisfy the axioms of closed sets and the corresponding topology is called the Zariski topology on An . Coordinate Rings

√ Let J (T ) be the ideal generated by T in A and √ J (T ), the radical of J (T ): recall the definition of the radical of an ideal J , denoted J ,

2.2 Algebraic varieties

13

√ J := {f ∈ K[X1 , · · · , Xn ]|f r ∈ J for some r ≥ 1} √ We say that J is a radical ideal if J = J . Since A is Noetherian, any√ideal in A is finitely generated. Let then J (T ) =< f1 , · · · , fm > and J (T ) =< g1 , · · · , g >. Now it follows that V (T ) =



V (f ) =

f ∈T

m  j =1

V (fj ) =



V (f ) =

f ∈J (T )

 √ f ∈ J (T )

V (f ) =



V (gj ),

j =1

i.e., every algebraic subset is a finite intersection of hypersurfaces. In fact, we have   V (T ) = V (f ) = V (f1 , · · · , fm ) = V (g1 , · · · , g ) = V (f ). √ f ∈ J (T )

f ∈J (T )

Given a subset Y ⊆ An , let I (Y ) = {f ∈ An | f (y) = 0, ∀ y ∈ Y }, which is the largest ideal in A vanishing on Y , called the ideal of Y . We have the following basic facts:   V (T ) ⊇ V (S) ⇐⇒ J (T ) ⊆ J (S) V (I (Y )) = Y  I (V (T )) = J (T )

√ √ Thus J (T ) is the ideal of V (T ). If V = V (T ) and I = J (T ), the ring A/I is called the coordinate ring of V and is denoted by K[V ]. We also write V = Spec K[V ] and the points of V can be identified with the set of all maximal ideals of K[V ], i.e., with the set of all maximal ideals in A containing the ideal I of V . The Local Rings OP (V ) For P = (a1 , · · · , an ) ∈ V , the ideal mP =< X1 − a1 , · · · , Xn − an > (mod I ) is called the maximal ideal of V at P and the local ring OP (V ) := OP (A)/I = the localization of K[V ] at the maximal ideal mP is called the local ring of V at P , or also the stalk at P . An algebraic subset V whose ideal is a prime ideal p is called an affine variety (defined by p) so that its coordinate ring K[V ] = A/p is an integral domain. The field Q(K[V ]) of fractions of K[V ] is called the field of rational functions on V and it is denoted by K(V ). Definition 2.2.1.1 An affine variety V is said to be non-singular or smooth at a point P ∈ V if its local ring OP (V ) at P is regular. We say that V is non-singular or smooth if it is so at all of its points. Definition 2.2.1.2 An affine variety V is said to be normal at a point P ∈ V if its local ring OP (V ) at P is a normal domain. We say that V is normal if it is so at all of its points.

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2 Generalities on algebraic varieties

Definition 2.2.1.3 An affine variety V is said to be factorial at a point P ∈ V if its local ring OP (V ) at P is a unique factorization domain (i.e., every element in OP (V ) has a unique factorization as a product of prime elements in OP (V )). We say that V is factorial if it is so at all of its points. Definition 2.2.1.4 The topological dimension of an algebraic subset V is defined as the supremum of the lengths of chains of algebraic varieties contained in V and it is denoted by dim V . The topological dimension of V is the same as the Krull dimension of its coordinate ring K[V ]. For example, one can see that the dimension of An is n and dimension of V (f ) is n − 1 (if degree of f is positive). Projective Varieties Definition 2.2.1.5 Projective Space: Given a non-zero vector space L over K, the set of all lines (i.e., one dimensional subspaces) in L is called the projective space associated to L and is denoted by P(L). It is also called the projective n–space if L is of dimension n + 1 over K. Choosing a basis of L and writing L = K n+1 (n ≥ 0), (with respect to that basis), we write Pn = PnK for P(K n+1 ) or simply P if there is no ambiguity about K and n. Convention: Pn = a point if n = 0. A point P in Pn has n + 1 homogeneous coordinates (a0 , a1 , · · · , an ), aj ∈ K, in the sense that (1) not all aj ’s are zero and (2) both (a0 , a1 , · · · , an ) and (λa0 , λa1 , · · · , λan ) represent the same point for all λ ∈ K . Let S = K[X0 , X1 , · · · , Xn ] be the polynomial ring in n + 1 variables over K. Unlike the affine case, an element f ∈ S does not define a function on Pn . Nevertheless, we can talk about the vanishing or non-vanishing of a homogeneous polynomial f ∈ S at a point P = (a0 , a1 , · · · , an ) ∈ Pn because we have f (λ(a0 , a1 , · · · , an )) = λd f ((a0 , a1 , · · · , an )) if d is the total degree of f . This allows us to define the (projective) hypersurface V+ (f ) whose equation is a homogeneous polynomial f of total degree d, i.e., V+ (f ) = {P = (a0 , a1 , · · · , an ) ∈ Pn | f (P ) = 0}. As before, the degree of f is called the degree of V+ (f ). It is called a hyperplane if d = 1, etc. Given T ⊆ S but consisting of only homogeneous polynomials of possibly different degrees, the set V+ (T ) = ∩f ∈T V+ (f ) of all common zeros of elements of T is called a projective algebraic (or simply an algebraic) subset of Pn . The family of all algebraic subsets satisfy the axioms of closed sets and the corresponding topology is again called the Zariski topology on Pn . Now keep the natural gradation on S = m≥0 Sm (where S0 = K) and proceed exactly as before replacing “ideals” by “homogeneous ideals”, “generators” of ideals by “homogeneous generators”, etc., and define projective varieties as V+ (p) for homogeneous prime ideals p of S. The graded ring S/p is called the homogeneous coordinate ring of V+ (p). We write V+ (p) (or just V+ ) = Proj(S/p). Consider

2.2 Algebraic varieties

15

the affine variety V = V (p) = Spec S/p defined by p in K n+1 . We note that the origin (0, · · · , 0) of K n+1 is a point of the affine variety V because p is a homogeneous ideal of S. The affine variety V is called the cone over V+ , and origin is the vertex of the cone. Nonsingularity, normality, factoriality etc. are defined for a projective variety in the same way as for the affine varieties. Some of the basic facts are the following: 1. A variety which is both affine and projective is a single point. 2. Every projective variety in Pn has a finite open covering by suitable affine subvarieties of K n . 3. The affine variety V is a cone over the projective variety V+ as the base and origin as the vertex. 4. dim(V ) = 1 + dim(V+ ). 5. The linear space K n+1 is the cone over Pn . 6. The projective variety V+ is non–singular (resp. normal, factorial) at all of its points ⇔ V is non–singular (resp. normal, factorial) at all points outside the vertex. 7. V is non–singular at its vertex ⇔ V = K m is linear ⇔ V+ = Pm−1 for some m ≥ 1. 8. V is normal (resp. factorial) at its vertex ⇔ V is so at all of its points ⇔ S/p is so. Definition 2.2.1.6 Projective Normality: A projective variety V+ is said to be projectively (or arithmetically) normal (resp. factorial) if its cone V is normal (resp. factorial) at its vertex, i.e., the stalk at the vertex is a normal domain (resp. a unique factorization domain). The following example shows that projective normality is a property of the particular projective embedding of the variety (unlike the affine varieties). The projective line P1 is obviously projectively normal since its cone is the affine plane K 2 (which is non-singular). However, it can be also embedded in P3 as the quartic curve, namely, V+ = {(a 4 , a 3 b, ab3 , b4 ) ∈ P3 | (a, b) ∈ P1 }, i.e., V+ = V+ (XT − Y Z, T Y 2 − XZ 2 ), but the coordinate ring of its cone V which is K[X, Y, Z, T ]/(XT − Y Z, T Y 2 − XZ 2 ) is not normal. Using the language of Weil divisors, Cartier divisors and line bundles on V+ (as in the affine case), we define the divisor class group Cl(V+ ) and the Picard group Pic(V+ ) of V+ . In Pn = V+ (0), any hypersurface (of degree d) is a Cartier divisor. Two hypersurfaces define the same divisor class if and only if they have the same degree. The line bundle corresponding to the divisor class of a hyperplane is called the hyperplane bundle or the tautological line bundle on Pn and it is denoted by OP (1). The line bundle corresponding to the divisor class of a hypersurface of degree d is then the d th tensor power of OP (1) and it is denoted by OP (d) (= OP (1)⊗d ). The space H 0 (Pn , OP (d)) of global sections of OP (d) turns out to be Sd , the homogeneous component of S of degree d (see [37] for details).

16

2 Generalities on algebraic varieties

For a projective subvariety X = V+ (p) of Pn , the restriction of OP (1) to X gives a line bundle, called the (induced) hyperplane bundle on X (for this embedding of X in Pn ) and is denoted by OX (1). It follows that the space of sections H 0 (X, OX (d)) contains the homogeneous component (S/ p)d but need not be equal (it is well-known that equality holds for d sufficiently large). However, if X is projectively normal, then equality holds for all d. More general varieties: Let X be a locally closed (i.e., an open subset of a closed subset) in the ambient space Z (Z being An (respectively Pn ) for some n ∈ N). Then X has a natural variety structure and one refers to X as a locally closed subvariety of Z or also as a quasi-affine (respectively quasi-projective) variety. All of the above concepts extend naturally to these more general varieties. In the sequel, by an “algebraic variety”, we shall mean an affine or quasi affine or projective or quasi projective variety.

3 Generalities on algebraic groups

This chapter is a review of generalities on algebraic groups - root systems, Weyl groups, parabolic subgroups etc. For details on root systems and Weyl groups, we refer the reader to [44, 45]. For details on algebraic groups, we refer the reader to [7, 43, 47].

3.1 Abstract root systems Fix a vector space V over R with a non-degenerate positive definite inner product (·, ·). Define the reflection relative to a nonzero vector α ∈ V to be the linear transformation on V given by 2(v, α) α. sα (v) = v − (α, α) Note that sα maps α to −α and fixes the hyperplane perpendicular to α. Denote v, α =

2α 2(v, α) and α ∗ = (α, α) (α, α)

Note that we have (v, α ∗ ) = v, α. Remark 3.1.0.1. Note that α ∗ is also written as α ∨ by some authors in the literature. In the sequel we shall use both versions v, α and (v, α ∗ ) interchangeably. An abstract root system R in V is defined by the following axioms: 1. 2. 3. 4.

R is finite, spans V and does not contain 0. If α ∈ R, the only multiples of α in R are ±α. For each α ∈ R, R is stable under sα . Integrality: For α, β ∈ R, sα (β) − β = −β, αα is an integral multiple of α, i.e., β, α is an integer.

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3 Generalities on algebraic groups

The positive integer = dim(V ) is called the rank of R. A subset S = {α1 , . . . , α } of R is called a basis of the root system if S is a vector space basis of V and if every α ∈ R can be expressed as α = ci αi with all the coefficients ci being integers with the same sign. Using this property, R can be partitioned into positive roots R + and negative roots R − with respect to S. The elements of S are called simple roots. The integers αi , αj , 1 ≤ i, j ≤ , are called the Cartan integers and they determine the root system completely, up to isomorphism. The information given by the Cartan integers is summarized in a Dynkin diagram. The Dynkin diagram is a graph with vertices indexed by the simple roots and the number of edges between αi and αj for i = j is αi , αj αj , αi  with an arrow pointing to the smaller of the two roots if they have different lengths. All irreducible root systems are completely classified by their Dynkin diagrams. xi = 0} Example. Let V be the hyperplane in R3 consisting of {(x1 , x2 , x3 ) ∈ R3 | equipped with the usual inner product. Let {e1 , e2 , e3 } denote the standard basis for R3 . Let R = {±(e1 − e2 ), ±(e2 − e3 ), ±(e1 − e3 )}. Then R is a root system (of rank 2), known as of “Type A2 ” in V . The set S := {α1 , α2 } = {e1 − e2 , e2 − e3 } is a basis of the root system R. The Cartan integer αi , αj  is 2 or −1 according as i = j or i = j . The Weyl group W of the root system R is the subgroup of GL(V ) generated by the sα for α ∈ R. It follows from the axioms of root systems that W is also finite; further, W is a Coxeter group, and is (minimally) generated by the simple reflections sα1 , . . . , sα , denoted s1 , . . . , s corresponding to the simple roots. For w ∈ W , the length of w, denoted (w), is defined to be the minimum of the length of an expression for w as a product of simple reflections; if (w) = p, then an expression w = sa1 sa2 . . . sap is called a reduced expression for w and the (ordered) sequence sa1 , sa2 , . . . , sap or simply a1 , a2 , . . . , ap is called a reduced word for w. Note that for a given w, there can be many reduced expressions. For instance, in the above example, we have W = the symmetric group S3 with simple reflections s1 := the transposition (1, 2) and s2 := the transposition (2, 3). The two reduced expressions s1 s2 s1 , s2 s1 s2 give the same permutation w = (321). Here and throughout the text we will denote a permutation w as (w1 w2 . . . wn ) (here, 1 is mapped to w1 , 2 to w2 , etc). The following facts about Weyl groups are often used. FACTS: • There is a unique element in W of longest length, denoted by w0 . • The set of all reflections in W , is obtained by conjugating all simple reflections. • For w ∈ W , (w) equals #{β ∈ R + | w −1 (β) < 0}. For details, see [44, 45]. Any vector λ ∈ V such that λ, α ∈ Z for every α ∈ R is called an abstract weight. By the integrality axiom, roots are weights. The set of all weights form a lattice Λ, called the weight lattice which contains the lattice generated by R as an abelian subgroup of finite index. The lattice generated by R is usually called the root lattice. If S = {α1 , . . . , α } is a basis of R, then Λ has a corresponding basis consisting of the fundamental weights: 1 , . . . ,  , defined by the condition that i , αj  = δij

3.2 Root systems of algebraic groups

19

(Kronecker delta) for all i, j ∈ {1, · · · , }. A weight λ = ci i is said to be dominant if all ci are non-negative integers, and is said to be regular if ci = 0, for all i. In the sequel, Λ+ will denote the set of all dominant weights in Λ. The Weyl chambers in V are the connected components in the complement of the union of hyperplanes orthogonal to the roots. These chambers are in one-toone correspondence with the bases of the root system. The chamber containing the dominant weights with respect to S is also called the dominant chamber.

3.2 Root systems of algebraic groups 3.2.1 Linear algebraic groups An affine algebraic subvariety G of K n which is also a group such that its group operations are “morphisms” (i.e., polynomial maps) in the category of affine varieties is called an affine or linear algebraic group or an affine group variety over K 2 ([7, 43]). The group GLn (K)(⊂ Mn (K) = K n ) of all non–singular n × n matrices over K, called the general linear group, is a universal example of a linear alge2 braic group over K; this is closed in K n +1 with defining ideal being the principal ideal generated by (Z · det(Xij ) − 1) in K[{Xij }1≤i,j ≤n , Z]. This is indeed universal, thanks to a theorem of Chevalley, which says that any affine algebraic group over K is a closed subgroup of GLn (K) for some n (see [7] for a proof of Chevalley’s result). Another equally universal example is the special linear group SLn (K) = 2 {A ∈ K n | det(A) = 1} whose defining ideal is the principal ideal generated by (det(Xij ) − 1). There are other classical examples like the symplectic and orthogonal groups, and also other examples, the so-called exceptional groups. Recall (cf. [7, 43]: A torus is an algebraic group isomorphic to a product of copies of K ∗ . Maximal tori exist in a given algebraic group (by reasons of dimensions); further, all maximal tori are conjugates and hence have the same dimension. Let l be the dimension of a maximal torus; l is defined as the rank of G. A Borel subgroup in an algebraic group G is a maximal connected solvable subgroup; further, all Borel subgroups in G are conjugates (and hence have the same dimension). The radical of G, denoted R(G), is the connected component through the identity element of the intersection of all the Borel subgroups in G. An algebraic group G is semisimple, if R(G) is trivial. The group SLn (K) is an example of a semisimple algebraic group. For the rest of this chapter, we shall suppose G to be a connected, semisimple algebraic group (defined over the algebraically closed base field K). Let us fix a maximal torus T in G. Let X(T ) := Homalg. gp. (T , K ∗ ) be the character group of T (written additively). Let V be any finite dimensional T -module. Then the complete reducibility of T implies (in fact, equivalent to the fact) that V is the direct sum of weight spaces, i.e., V = ⊕χ∈X(T ) Vχ , where Vχ = {v ∈ V : tv = χ(t)v, ∀t ∈ T }. We say χ ∈ X(T ) is a weight in V if Vχ = 0. Let Φ(T , V ) be the set of non-zero weights in V .

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3 Generalities on algebraic groups

Denote the Lie algebra of G by g or Lie G. For any g ∈ G, consider Inn(g) : G → G given by x → gxg −1 . This induces the adjoint action on the Lie algebra Ad(g) : Lie G → Lie G. For V = Lie G as a T -module via the adjoint action, the corresponding Φ(T , Lie G) is called the root system of G relative to T ; it is an abstract root system. One considers Φ(T , Lie G) to be a root system contained in the vector space X(T ) ⊗Z R of rank l (= dimT ). We shall denote this root system by R. The Weyl group W associated to Φ(T , Lie G) is isomorphic to N (T )/T where N(T ) is the normalizer of T in G. Let B be a Borel subgroup of G such that B ⊃ T . The subset Φ(T , Lie B) of Φ(T , Lie G) is called the set of positive roots of R relative to B, denoted R + . The indecomposable elements in R + , i.e., the set of all α such that α cannot be expressed as a positive sum of other elements in R + , is a set of simple roots for the root system R which we shall denote by S. Thus, choosing B is equivalent to choosing a set of simple roots S in R. The fundamental weights {i : 1 ≤ i ≤ l} are defined as in Section 3.1. We have the inclusions root lattice ⊆ X(T ) ⊆ weight lattice. The elements in X(T ) which lie in the dominant chamber with respect to S are called dominant characters. Also, if G is simply connected, then X(T ) equals the weight lattice Λ. Example. Let G = SLn (K) be the special linear group of rank n − 1, T the subgroup of diagonal matrices and B the subgroup of upper triangular matrices. Then T (resp. B) is a maximal torus (resp. a Borel subgroup). Further, N (T ) is the subgroup consisting of matrices with exactly one nonzero entry in each row and column, and W = N(T )/T is the symmetric group Sn . We have Lie G = {n × n matrices with trace 0}. We further have, LieG = h ⊕ n+ ⊕ n− , where h = { diagonal matrices in LieG}, n+ = ⊕1≤i<j ≤n (Eij ), n− = ⊕1≤j
3.2 Root systems of algebraic groups

21

Root Subgroups The Lie algebra of G, denoted by g, contains a one-dimensional subalgebra gα for each root α ∈ R. Each gα is a T -weight space of weight α; gα is called a root space. Furthermore, if h denotes the Lie algebra of T , then g decomposes as a direct sum of T -weight spaces

gα . g=h⊕ α∈R

For each β ∈ R, there exists a unique connected T -stable subgroup Uβ of G (i.e., Uβ is normalized by T ) with gβ as its Lie algebra. The subgroup Uβ is called a root subgroup. Let Ga denote the additive group K. For β ∈ R, there exists an isomorphism θβ : Ga ∼ = Uβ such that for all t ∈ T , x ∈ Ga , tθβ (x)t −1 = θβ (β(t)x). Let Bu denote the unipotent part of B, i.e., the subgroup of B consisting of all the unipotent elements in B. Then we have an identification of varieties  Uβ , Bu ∼ = β∈R +

the product being taken in any fixed order. Further, Bu is normalized by T , and B is the semi-direct product of Bu and T . Example. In the example of Section 3.2, (Ei,j ), the one dimensional span of Ei,j for 1 ≤ i, j ≤ n, i = j , are all the root spaces. The root subgroup Ui,j , 1 ≤ i, j ≤ n, i = j is the subgroup of G consisting of matrices with 1’s along the diagonal and the only possible non-zero entry occurs at the (i, j )-th place. Bu is the subgroup of G consisting of unipotent upper triangular matrices. 3.2.2 Parabolic subgroups A parabolic subgroup of G is any closed subgroup Q ⊂ G such that G/Q is a projective variety (or equivalently, a complete variety, since G/Q is quasi projective (see [7, 43])). Fact: A subgroup Q of G is parabolic if and only if it contains a Borel subgroup. Let Q be a parabolic subgroup containing B. Let R(Q) be the radical of Q, namely, the connected component through the identity element of the intersection of all the Borel subgroups of Q. Let Ru (Q) be the unipotent radical of Q (the sub+ be the subset of R + defined by group of unipotent elements of R(Q)), and let RQ + − + + R + \ RQ = {β ∈ R + | Uβ ⊂ Ru (Q)}. Let RQ = −RQ (= {−β, β ∈ RQ }), + − RQ = RQ ∪ RQ , and SQ = S ∩ RQ . Then RQ is a subroot system of R called the + − root system associated to Q, with SQ as a set of simple roots and RQ (resp. RQ ) as the set of positive (resp. negative) roots of RQ relative to SQ . Conversely, given a subset J of S, there exists a parabolic subgroup Q of G such + that Q contains B. Namely, Q is generated by B and {U−α | α ∈ R and α = β∈J aβ β}.

22

3 Generalities on algebraic groups

Thus the set of parabolic subgroups containing B is in bijection with the power set of S. Note that for Q = B (resp. G), SQ is the empty set (resp. the whole set S). The subgroup of Q generated by T and {Uα | α ∈ RQ } is called the Levi subgroup associated to SQ , and is denoted LQ . We have that Q is the semidirect product of Ru (Q) and LQ (see [7, 43] for details). The set of maximal parabolic subgroups containing B is in one-to-one correspondence with S. Namely given α ∈ S, the parabolic subgroup Q where SQ = S \ {α} is a maximal parabolic subgroup, and conversely. In the sequel, we shall also denote the maximal parabolic subgroup Q, where SQ = S \ {αi } by Pi . The Weyl Group of a Parabolic Subgroup Given a parabolic subgroup Q, let WQ be the subgroup of W generated by {sα | α ∈ SQ }. WQ is called the Weyl group of Q. Note that WQ ∼ = NQ (T )/T , where NQ (T ) is the normalizer of T in Q. min of minimal representatives of W/W : In each Definition 3.2.2.1 The set WQ Q min be coset wWQ ∈ W/WQ , there exists a unique element of minimal length. Let WQ the set of minimal length representatives of W/WQ . We have min = {w ∈ W | l(ww  ) = l(w) + l(w  ), for all w  ∈ WQ }. WQ

In other words, each element w ∈ W can be written uniquely as w = u · v min , v ∈ W and l(w) = l(u) + l(v). The set W min may be also be where u ∈ WQ Q Q characterized as min = {w ∈ W | w(α) > 0, for all α ∈ SQ }. WQ min is also (Here by a root β being > 0 we mean β ∈ R + .) In the literature, WQ denoted by W Q ; in the sequel, we shall use both of the notations. max of maximal representatives of W/W : In each Definition 3.2.2.2 The set WQ Q max be the set of coset wWQ there exists a unique element of maximal length. Let WQ maximal length representatives of W/WQ : max = {w ∈ W | w(α) < 0 for all α ∈ SQ }. WQ

Further, if we denote by wQ the unique element of maximal length in WQ , then we have max min = {wwQ | w ∈ WQ }. WQ

3.3 Schubert varieties The projective variety G/Q is called a generalised flag variety. This terminology stems from the fact that for the case of G = GLn (K), and Q = B, the (Borel) sub group of upper triangular matrices, it is indeed a flag variety; namely, let F = {all flags F : V0 ⊂ V1 ⊂ · · · ⊂ Vn = K n , dim Vi = i}

3.3 Schubert varieties

23

We have a transitive action of GLn (K) on F, and B is the isotropy at the standard flag F0 (with Vi = the span of {e1 , · · · , ei }, where {e1 , . . . , en } denotes the standard basis of K n ). Thus we obtain a bijection between GLn (K)/B and F. Hence F acquires a projective variety structure, and F with this variety structure is called the flag variety. Returning to the case of a general G/Q, we have that for the induced action of T on Q, there are only finitely many fixed points {ew := wWQ | w ∈ W/WQ }. For w ∈ W/WQ , the B-orbit CQ (w) := Bew in G/Q is a locally closed subset of G/Q, called the Bruhat or Schubert cell. The Zariski closure of CQ (w) with the canonical reduced structure is the Schubert variety associated to w, and is denoted XQ (w). min (or also by W max ). Note that Thus Schubert varieties in G/Q are indexed by WQ Q if Q = B, then WQ = {id}, and the Schubert varieties in G/B are indexed by the elements of W . Remark 3.3.0.1. We use the notation X(w) for the Schubert variety in G/Q indexed by w ∈ W when Q = B or when the parabolic subgroup is understood from context. Dimension of XB (w): Let w ∈ W . The isotropy subgroup in G at the (T -fixed) point ew (in G/B) is wBw −1 ; hence, the isotropy subgroup in Bu at ew is generated by the root subgroups {Uβ , β ∈ R + | Uβ ⊂ wBw −1 }, i.e., {Uβ , β ∈ R + | w −1 (β) > 0}. Hence we get an identification  CB (w) ∼ Uβ = {β∈R + | w −1 (β)<0}

Now (w) equals #{β ∈ R + | w −1 (β) < 0} (cf. §3.1, Facts). Thus CB (w) is isomorphic to the affine space K (w) . Hence we obtain dim XB (w) (= dim CB (w)) = (w) Note that for w = id, X(id) = C(id) = {a point}, the unique point cell. Dimension of XQ (w): Let w ∈ W/WQ . Denoting the unique representative for w in min (resp. W max ) by w min (resp. w max ), we have that under the canonical projection WQ Q Q Q min ) maps birationally onto X (w), and X (w max ) = πQ : G/B → G/Q, XB (wQ Q B Q −1 πQ (XQ (w)). Hence we obtain

min min dim XQ (w) = dim XB (wQ ) = (wQ )

Big cell and opposite big cell: We have, G/B = X(w0 ), w0 being the element in W of largest length. The cell CB (w0 ) is the unique cell of maximal dimension (= (w0 ) = # R + ); it is affine, open and dense in G/B, called the big cell of G/B. It is denoted as O. Let B − be the Borel subgroup opposite to B (note that B − = w0 Bw0−1 = w0 Bw0 , since w0−1 = w0 ). The B − -orbit B − eid is again affine, open and dense in G/B, and is called the opposite big cell of G/B. It is an affine neighborhood of eid , and is denoted as O− . More generally, for τ ∈ W , τ B − τ −1 · eτ

24

3 Generalities on algebraic groups

is affine, open and dense in G/B. It is an affine neighborhood of eτ , and is denoted as Oτ− . Opposite cell in XB (w): Let w ∈ W . Then YB (w) := XB (w) ∩ O− is called the opposite cell in XB (w) (though for a general w, it need not be a cell). It is a closed sub variety of the affine space O− (∼ = AN , N being the dimension of G/B equal to + # R ), and hence is affine. In particular, it is an open neighborhood of eid in XB (w). More generally, for τ, w ∈ W, τ ≤ w, XB (w) ∩ Oτ− is an open neighborhood of eτ in XB (w). Remark 3.3.0.2. For a general G/Q, Q being any parabolic subgroup, the above concepts - big cell, opposite big cell, opposite cell in a Schubert variety etc., are defined in the obvious way. Bruhat decomposition: We have ˙

G/Q =

Bew Q(mod Q),

w∈W Q

XQ (y) =

˙

Bew Q(mod Q), y ∈ W Q

{w∈W Q , ew ∈XQ (y)}

The Bruhat-Chevalley order: There is a partial order on W Q , known as the BruhatChevalley order (or just Bruhat order), induced by the partial order on the set of Schubert varieties given by inclusion, namely, for w1 , w2 ∈ W Q , w1 ≥ w2 ⇐⇒ XQ (w1 ) ⊇ XQ (w2 ). Taking Q = B, we obtain a partial order on W (the Weyl group of G). This partial order plays an important role in the theory of Schubert varieties. Chevalley has shown (cf. Chevalley [15]) that this order can be stated equivalently in terms of the Weyl group, namely w1 ≥ w2 ∈ W if every (equivalently, any) reduced expression for w1 contains a sub-expression which is a reduced expression for w2 . For example, s1 s2 s3 s2 ≥ s1 s3 s2 . Another description for Bruhat-Chevalley order is given by the following: For w ∈ W , let X(w (d) ) denote the projection of X(w) under G/B → G/Pd , 1 ≤ d ≤ l; then for u, v ∈ W, u ≤ v if and only if X(u(d) ) ⊆ X(v (d) ) for all 1 ≤ d ≤ l. 3.3.1 Weyl and Demazure modules Let gC = Lie GC , and U (gC ) be the universal enveloping algebra of gC . Let U + (gC ) be the subalgebra of U (gC ) generated by {Xα , α ∈ S}, and UZ+ (gC ) be the Kostant Z-form of U + (gC )) (recall (cf. [52]) that UZ+ (gC ) is the Z-subalgebra of U + (gC ) Xn

generated by { n!α , α ∈ R + , n ∈ N}). Let λ be a dominant weight, and V (λ) be the irreducible G-module over C. Fix a highest-weight vector uλ (i.e., annulled by Xβ for all β ∈ R + ) in V (λ) of weight λ (for the T -action); we have that the weight λ in V (λ) has multiplicity one. For w ∈ W , fix a representative nw for w in NT (G) (the normalizer of T in G), and set uw,λ = nw · uλ , known as an extremal weight vector; it is a weight vector in V (λ)

3.3 Schubert varieties

25

(for the T -action) of weight w(λ), and is unique up to scalars. Having fixed λ, we shall denote uλ (resp. uw,λ ) by just u (resp. uw ). Set Vw,Z (λ) = UZ+ (gC )uw . For any field K, let Vw,λ = Vw,Z (λ) ⊗ K, w ∈ W . Then VK (λ) := Vw0 ,λ (= Vw0 ,Z (λ) ⊗ K) is the Weyl module with highest weight λ, and for w ∈ W, Vw,λ is the Demazure module corresponding to w (and λ). Note, the Bruhat-Chevalley order also appears in the context of Demazure modules as follows: Given a dominant weight λ, let Pλ be the stabilizer of K · u, where u as above, is a highest-weight vector u in VK (λ); note that the Weyl group WPλ of Pλ is the subgroup of W generated by ; {sα , α ∈ S, sα (λ) = λ}. The extremal weight vectors in VK (λ) are indexed by WPmin λ further, for w1 , w2 ∈ WPmin , we have, w ≤ w ⇔ V ⊆ V , equivalently, u 1 2 w1 ,λ w2 ,λ w1 λ is in Vw2 ,λ . Fix a dominant weight λ. Then VK (λ)∗ is again a Weyl module; it is the Weyl module with highest weight i(λ), i being the Weyl involution i := −w0 . Note that i(λ) = −w0 (λ) is the negative of the lowest weight w0 (λ) in VK (λ). More generally, the weights in VK (λ)∗ are the negatives of the weights in VK (λ). For w ∈ W , as above, let uw be an extremal weight vector (unique up to scalars) in VK (λ). Let pw be an extremal weight vector (again unique up to scalars) in VK (λ)∗ of weight −w(λ). There exists a canonical G-invariant pairing (, ) on VK (λ)×VK (λ)∗ ; further, we have, (uw , pw ) = δw,w 3.3.2 Line bundles on G/Q For the study of G/B, there is no loss in generality in assuming that G is simply connected; in particular, as remarked in §3.2, we have that the character group X(T ) coincides with the weight lattice Λ. Henceforth, we shall suppose that G is simply connected. Let X = G/B, and consider the canonical projection π : G → X. Then π : G → X is a principal B-bundle with B as both the structure group and fiber. Let Gm denote the multiplicative group K ∗ and let λ ∈ X(T ). Then λ defines a character λB : B → Gm obtained by composing the natural map B → T with λ : T → Gm (recall that B is the semi-direct product of T and Bu (= the unipotent part of B)). We shall denote λB also by just λ. Now λ : B → Gm gives rise to an action of B on K(= Ga ), namely b · k = λ(b)k, b ∈ B, k ∈ K. Set E = G × K/ ∼, where ∼ is the equivalence relation defined by (gb, b · k) ∼ (g, k), g ∈ G, b ∈ B, k ∈ K. Then E is the total space of a line bundle, say L(λ), over X. Thus we obtain a map L : X(T ) → Pic(G/B),

λ → L(λ),

where Pic(G/B) is the Picard group of G/B (namely, the group of isomorphism classes of line bundles on G/B). We recall (cf. Chevalley [16]) that the above map is in fact an isomorphism of groups since G is simply connected. On the other hand, consider the prime divisors X(w0 si ), 1 ≤ i ≤ l on G/B. Let Li = OG/B (X(w0 si )) be the line bundle defined by X(w0 si ), 1 ≤ i ≤ l. The Picard group Pic(G/B) is a free abelian group generated by the Li ’s, and under the isomorphism L : X(T ) ∼ = Pic(G/B), we have L(ωi ) = Li , 1 ≤ i ≤ l (see

26

3 Generalities on algebraic groups

⊗λ,αi  Chevalley [16]). Thus for λ = li=1 λ, αi ωi , we have L(λ) = ⊗li=1 Li . We shall write L(λ) ≥ 0 if λ ≥ 0, i.e., if λ is dominant. As above, let E denote the total space of the line bundle L(λ), over X(= G/B). Let σ : E → X be the canonical map σ (g, c) = gB. Let Mλ = {f ∈ k[G] | f (gb) = λ(b)f (g), g ∈ G, b ∈ B}. Then Mλ can be identified with the space of sections H 0 (G/B, L(λ)) := {s : X → E | σ ◦ s = idX } as follows. Let f ∈ Mλ . To f , we associate a section s : X → E by setting s(gB) = (g, f (g)). To see that s is well defined, consider g  = gb, b ∈ B. Then (g  , f (g  )) = (gb, f (gb)) = (gb, λ(b)f (g)) = (gb, b · f (g)) ∼ (g, f (g)). From this, it follows that s is well defined. Conversely, given s ∈ H 0 (G/B, L(λ)), consider gB ∈ G/B. Let s(gB) = (g  , f (g  )), where g  = gb for some b ∈ B (note that g  B = gB, since σ ◦ s = idX ). Now the point (g  , f (g  )) may also be represented by (g, λ(b)−1 f (gb)) (since (g  , f (g  )) = (gb, f (gb)) ∼ (g, λ(b)−1 f (gb)). Thus given g ∈ G, there exists a unique representative of the form (g, f (g)) for s(gB). This defines a function f : G → K. Further, this f has the property that for b ∈ B, f (g) = λ(b)−1 f (gb), i.e., f (gb) = λ(b)f (g), b ∈ B, g ∈ G. Thus we obtain an identification Mλ = H 0 (G/B, L(λ)). It can be easily checked that the above identification preserves the respective G-module structures. The structure of the line bundle L(λ) can often be described in terms of the weight λ. For example, Chevalley has proven the following facts (see [16] and also [47]): 1. H 0 (G/B, L(λ)) = 0 if and only if λ is dominant. 2. L(λ) is ample ⇔ L(λ) is very ample ⇔ λ is dominant and regular. (A line bundle L on an algebraic variety X is very ample if there exists an immersion i : X → PN such that i ∗ (OPN (1)) = L. A line bundle L on X is ample if Lm is very ample for some positive integer m ≥ 1. See [37] for details.) Summary Let λ ∈ Λ+ , i.e., λ is a dominant weight. We have (see Jantzen [47] for example), H 0 (G/B, L(λ)) ∼ = VK (λ)∗ where VK (λ)∗ , as above, is the dual Weyl module corresponding to λ. Now using the fact that the multiplicity of the weight i(λ) in VK (λ)∗ equals one, we obtain that H 0 (G/B, L(λ)) is G-indecomposable. The indecomposability of H 0 (G/B, L(λ)) together with the complete reducibility of G in characteristic 0 implies that if charK = 0, then H 0 (G/B, L(λ)) is irreducible. Thus we obtain geometric realizations of irreducible G-modules in characteristic 0.

3.3 Schubert varieties

27

We further have the following facts (see [47] for details): 1. There exists a bijection between Λ+ and the set of isomorphism classes of finite dimensional irreducible G-modules (in any characteristic). 2. H 0 (G/B, L(λ)) = {f ∈ K[G] | f (gb) = λ(b)f (g), g ∈ G, b ∈ B}. 3. H 0 (G/B, L(λ)) = 0 if and only if λ ∈ Λ+ . 4. L(λ) is ample ⇐⇒ L(λ) is very ample ⇐⇒ λ is dominant and regular. 5. Let λ be dominant. There exists a unique B-stable line in H 0 (G/B, L(λ)), and the G-submodule generated by this line is the irreducible G-module with highest weight i(λ), i(= −w0 ) being the Weyl involution. 6. Let λ be dominant. Then H 0 (G/B, L(λ)) is an indecomposable G-module. 7. Let λ be dominant. Then the weights in H 0 (G/B, L(λ)) are the negatives of the weights in the Weyl module VK (λ); in particular, the extremal weights in (the Weyl module) H 0 (G/B, L(λ)) are given by {−w(λ), w ∈ W }. In the sequel, the extremal weight vector in H 0 (G/B, L(λ)) of weight −w(λ) (which is unique up to scalars) will be denoted by pw . 8. Let λ be dominant. We have a natural identification ∗ ,w∈W H 0 (X(w), Lλ ) ∼ = Vw,λ

9. Let λ =  be a fundamental weight with P as the maximal parabolic subgroup. Then L( ) is the ample generator of Pic G/P (∼ = Z). 10. In characteristic 0, H 0 (G/B, Lλ ) is G-irreducible with highest weight i(λ) and conversely. Thus in characteristic 0, {H 0 (G/B, Lλ ), λ ∈ Λ+ } gives (up to isomorphism) all finite-dimensional irreducible G-modules. 3.3.3 Equations defining a Schubert variety Let G be semi-simple, and Pi where i ≤ l (l being the rank G) be a maximal parabolic subgroup with i as the associated fundamental weight. Let Li be the ample generator of Pic G/Pi (∼ = Z). Now Li induces a projective embedding G/Pi → P(VK (i )); in fact, for τ ∈ W Pi , the point τ Pi in G/Pi gets identified with the point in P(VK (i )) representing the one-dimensional span of an extremal weight vector in VK (i ) of weight τ (i ). Let w ∈ W , and wi be the projection of w under W → W/WPi . We have the following: Fact 1: ([10, Theorem 3.5.2], [47, II.14.15]) The restriction map H 0 (G/Pi , Lm i )→ 0 H (X(wi ), Lm ), m ≥ 0 is surjective. i Fact 2: ([105], [10, Theorem 3.5.2]) Fix a basis Di for the kernel of the surjective restriction map from H 0 (G/Pi , Li ) to H 0 (X(wi ), Li ). Then the ideal sheaf of X(wi ) in G/Pi is generated by Di . Denoting R=



H 0 (G/Pi , Lm i )

m∈Z+

R(wi ) =



m∈Z+

H 0 (X(wi ), Lm i )

28

3 Generalities on algebraic groups

Facts 1 & 2 imply that the restriction map R → R(wi ) is surjective, and that the kernel is generated (as an ideal) by Di . We further have, Fact 3: ([50]) The ideal sheaf of X(w) in G/B is generated by {Di , 1 ≤ i ≤ l}. Denoting l

A= H 0 (G/B, Lai i ) a

A(w) =

a

i=1

H 0 (X(w),

l

Lai i ),

i=1

where a = (a1 , . . . , al ) ∈ Zl+ , Fact 3 implies that the restriction map A → A(w) is surjective, and its kernel is generated as an ideal by {Di , 1 ≤ i ≤ l}, i.e., by elements of total degree 1. Remark 3.3.3.1. In the literature, one describes this by saying “the ideal sheaf of a Schubert variety in G/B is generated in degree one”.

4 Grassmannian Variety

In this chapter, we study the Schubert varieties in the Grassmannian. After introducing the Grassmannian and its Schubert varieties in §1 and §2, we present the Standard monomial theory for Schubert varieties ([41, 42, 88]) in the Grassmannian in §3 and §4. We then present a proof of the “vanishing theorems” for Schubert varieties in §5, and deduce normality and Cohen-Macaulayness for Schubert varieties in §6.

4.1 The Plücker embedding Let us fix the integers 1 ≤ d < n and let V = K n . The Grassmannian Gd,n is the set of d-dimensional subspaces U ⊂ V . Let U be an element of Gd,n , and a1 , . . . , ad a basis of U , where ⎛ ⎞ a1j ⎜a2j ⎟ ⎟ aj = ⎜ ⎝ . . . ⎠ , with aij ∈ k, for 1 ≤ i ≤ n, 1 ≤ j ≤ d. anj (here, each vector aj is written as a column vector with respect to the standard basis of K n ). Thus the basis {a1 , . . . , ad } gives rise to an n × d matrix A = (aij ) of rank d, whose columns are the vectors a1 , . . . , ad . Let a1 , . . . , ad be another basis giving rise to an n × d matrix A . Then there exists a change of basis matrix C ∈ GLd (K), such that A = AC. Conversely, given two n × d matrices A and A of rank d, such that A = AC, for some matrix C ∈ GLd (K), the columns of both A and A are d linearly independent vectors in V , generating one and the same d-dimensional subspace of V . Identifying an n × d matrix with a point in the affine space And , we see that Gd,n can be viewed as (And \ Z)/ ∼, where Z is the set of n × d matrices of rank less than d, and the equivalence relation ∼ is defined by A ∼ A

if there exists a C ∈ GLd (K) such that A = AC.

(Note in particular that Pn−1 = G1,n .)

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4 Grassmannian

0 0 /GL (K), Thus denoting Mn×d = And \ Z, Gd,n is simply the orbit space Mn×d d 0 for the action of GLd (K) on Mn×d by multiplication on the right.

4.1.1 The partially ordered set Id,n Define the set Id,n := {i = (i1 , . . . , id )|1 ≤ i1 < · · · < id ≤ n}, Define a partial order ≥ on Id,n by defining i ≥ j ⇔ it ≥ jt , ∀t. 4.1.2 Plücker embedding and Plücker coordinates   Let N = #Id,n ; then N = dn , and the coordinates of the affine space ∧d V = K N will be indexed by the set Id,n . For a non-zero v ∈ K N , we shall denote by [v] the point determined by v in PN −1 . Let X=V · · ⊕ V = K nd .  ⊕ · dtimes

The exterior product map ∧d : X → ∧d V sends the element (a1 , . . . , ad ) to the point a1 ∧ · · · ∧ ad in ∧d V . Identifying ∧d V with K N , the i-th coordinate of a1 ∧ · · · ∧ ad is given by the determinant of the d-minor of the n×d matrix A = (aij ) (where the columns of the matrix A = (aij ) are the vectors a1 , . . . , ad ), with row indices i1 , . . . , id . Note that a1 ∧· · ·∧ad = 0 if and only if the matrix A belongs to the set Z of matrices of rank less than d. Moreover, it is easily seen that given (a1 , . . . , ad ), (a1 , . . . , ad ) in X, such that the associated n × d matrices A, A , are equivalent (i.e., there exists a matrix C ∈ GLd (K) such that A = AC), then a1 ∧ · · · ∧ ad = det(C)a1 ∧ · · · ∧ ad . This shows that the map ∧d induces a well defined map p : Gd,n → P(∧d V ) = PN −1 called the Plücker map. For i ∈ Id,n , we shall denote the i-th component of p by pi , or by pi1 ,...,id , where i = (i1 , . . . , id ); the pi ’s, with i ∈ Id,n , are called the Plücker coordinates. For U in Gd,n , as above, let An,d be the matrix representing U ; then pi1 ,...,id (U ) = det(Ai1 ,...,id ), where Ai1 ,...,id denotes the d × d sub matrix of A consisting of the rows with indices i1 , . . . , id . Given a point U in Gd,n represented by the n × d matrix A, and numbers 1 ≤ i1 , . . . , id ≤ n (not necessarily distinct, nor in increasing order), Ai1 ,...,id will denote the matrix whose rows are the rows of A with indices i1 , . . . , id , taken in the order in which they appear in A. Also, we define pi1 ,...,id (U ) = det(Ai1 ,...,id ). It is easy to see

4.1 The Plücker embedding

31

that pi1 ,...,id = 0 if i1 , . . . , id are not distinct, and pi1 ,...,id = (−1)l(σ ) piσ (1) ,...,iσ (d) if i1 , . . . , id are distinct and σ is the permutation of the set {1, . . . , d}, such that σ (1) < · · · < σ (d). For each i ∈ Id,n consider the point ei of Gd,n represented by the n × d matrix whose entries are all 0, except the ones in the ij -th row and j -th column, which are equal to 1, 1 ≤ j ≤ d. Clearly, for i, j ∈ Id,n ,  1, if i = j ; pi (ej ) = 0, otherwise. Theorem 4.1.2.1 The Plücker map is injective. Proof. Let U, U  be two points in Gd,n such that p(U ) = p(U  ). Then there exists l = (l1 , . . . , ld ) ∈ Id,n such that pl (U ) = pl (U  ) = 0, and we may assume that pl (U ) = pl (U  ) = 1. If U and U  are represented by the n × d matrices A, respectively A , we have det(Al1 ,...,ld ) = det(Al1 ,...,ld ) = 1. Replacing A and A with

  −1 AC, respectively A C  , where C = A−1 l1 ,...,ld , C = A l1 ,...,ld , we can assume that Al1 ,...,ld = Al1 ,...,ld = Idd . For 1 ≤ i ≤ n and 1 ≤ j ≤ d, we have

aij = det Al1 ,...,lj −1 ,i,lj +1 ,...,ld = pl1 ,...,lj −1 ,i,lj +1 ,...,ld (U ) = pl1 ,...,lj −1 ,i,lj +1 ,...,ld (U  ) = det Al1 ,...,lj −1 ,i,lj +1 ,...,ld = aij . Thus A = A , hence U = U  , which shows that p is injective. In view of Theorem 4.1.2.1, we identify Gd,n with Im p and consider Gd,n ⊂ PN −1 set theoretically. Next we will show that Gd,n is closed in PN −1 . 4.1.3 Plücker quadratic relations Theorem 4.1.3.1 The Grassmannian Gd,n ⊂ PN −1 consists of the zeroes in PN −1 of the following quadratic polynomials: d+1 

(−1)λ pi1 ,...,id−1 ,jλ pj1 ,...,jλ ,...,jd+1

(*)

λ=1

where i1 , . . . , id−1 and j1 , . . . , jd+1 are any numbers between 1 and n. (A relation obtained by equating an expression given by (*) to 0 is usually referred to as a Plücker relation). Proof. Fix 1 ≤ i1 , . . . , id−1 , j1 , . . . , jd+1 ≤ n. Let U ∈ Gd,n be represented by the n × d matrix A. Then (*) evaluated at U equals d+1  (−1)λ det(Ai1 ,...,id−1 ,jλ ) det(Aj1 ,...,jλ ,...,jd+1 ). λ=1

32

4 Grassmannian

Expanding the first determinants along their last rows, the above expression equals d d+1    μ (−1)λ (−1)d+μ ajλ ,μ det(Ai1 ,...,id−1 ) det(Aj1 ,...,jλ ,...,jd+1 ), λ=1

μ=1

 μ

where Ai1 ,...,id−1 is obtained from Ai1 ,...,id−1 by deleting the μ-th column. The last expression can be written in the form d 

 μ

(−1)d+μ det(Ai1 ,...,id−1 )

μ=1

or

d+1  (−1)λ ajλ ,μ det(Aj1 ,...,jλ ,...,jd+1 ), λ=1

d 

 μ

(−1)d+μ det(Ai1 ,...,id−1 )(−1) det(μ Aj1 ,...,jλ ,...,jd+1 )

μ=1

where μ Aj1 ,...,jλ ,...,jd+1

is obtained from the matrix Aj1 ,...,jλ ,...,jd+1 by adding the μ-th column of A as its first column. Note that this matrix has two identical columns, and therefore its determinant is 0. This implies that the last expression is 0 and hence we obtain that the point U in Gd,n is a zero of the polynomials given by (*). Conversely, let q = (qi )i∈Id,n be a zero of the polynomials given by (*). Let ql1 ,...,ld = 1, for some (l1 , . . . , ld ) ∈ Id,n . For 1 ≤ i ≤ n, 1 ≤ j ≤ d let aij = ql1 ,...,lj −1 ,i,lj +1 ,...,ld . Let A = (aij ). Note that Al1 ,...,ld = Idd . Clearly, rankA = d (since det Al1 ,...,ld = 1). For j ∈ Id,n , let us denote pj (A) = det Aj1 ,...,jd . We shall now show that for j ∈ Id,n , we have pj (A) = qj , from which it will follow that the point U in Gd,n represented by A satisfies pj (U ) = qj , j ∈ Id,n , i.e., q ∈ Gd,n . We prove this by decreasing induction on #{l ∩ j }, the starting point of induction result in this being #{l ∩ j } = d − 1 (i.e. l and j differ in just one place), and the case follows from our definition of A. In the Plücker relation ql qj + ±ql  qj  = 0, l  differs from l in just one place (namely the d-th place), while #{j  ∩ l} > #{j ∩ l} (since some entry in {j \ l} has been replaced by ld ). Thus we obtain ql  = pl  (A), qj  = pj  (A). Also, from the first part, the d-minors of A satisfy the Plücker relations. Hence we obtain pl (A)qj (A) = ql qj , which implies pj (A) = qj (since pl (A) = ql = 1). 4.1.4 More general quadratic relations The Plücker coordinates satisfy a more general type of quadratic relations described as follows: Let i, j ∈ Id,n . Suppose that i, j are not comparable, say, i ≥ j . This implies that there exists a t, t ≤ d such that ir ≥ jr , 1 ≤ r ≤ t − 1, and it < jt . Let [i, j ] denote the set of permutations σ of the set {i1 , · · · , it , jt , · · · , jd } such that i1σ < · · · < itσ ; jtσ < · · · < jdσ . Then the Plücker coordinates pi , pj satisfy the

4.1 The Plücker embedding

following relation: Qi,j :



sign(σ ) pi σ pj σ = 0,

33

(**)

σ

where i σ = (i1σ , · · · , itσ , it+1 , · · · , id ) ↑, j σ = (j1 , · · · , jt−1 , jtσ , · · · , jdσ ) ↑, and σ runs over [i, j ] (here, for a d-tuple (l1 , · · · , ld ), li ∈ Z, (l1 , · · · , ld ) ↑ denotes the d-tuple obtained from (l1 , · · · , ld ) by arranging the entries in the ascending order). The proof of this is very similar to the above proof. For σ = id, i σ is either = 0 (namely, if there is a repetition in σ {i1 , · · · , itσ , it+1 , · · · , id }) or i σ > i (since in {i1 , · · · , it , jt , · · · , jd } any j > any i, and ir ≥ jr , 1 ≤ r ≤ t − 1); similarly, for σ = id, j σ is either = 0 (namely, if there is a repetition in {j1 , · · · , jt−1 , jtσ , · · · , jdσ }) or j σ < j . Hence (**) may be reformulated as  pi pj = ±pα pβ , (***) where α, β run over a certain subset of Id,n such that α ≥ i and β < j .  4.1.5 The cone G d,n over Gd,n Let A = K[Gd,n ], the homogeneous coordinate ring of Gd,n for the Plücker embed ding. Let G d,n = Spec A, the cone over Gd,n . Let X = V ⊕ · · · ⊕ V (d copies). With respect to the standard basis {e1 , . . . , en } of V (= K n ), X may be identified with Mn×d , the space of n × d matrices with entries in K. With this identification, we have K[X] = K[xij , 1 ≤ i ≤ n, 1 ≤ j ≤ d]. Consider the canonical morphism π : X → ∧d V , π(u1 , . . . , ud ) = u1 ∧ . . . ∧ ud . The morphism π corresponds to the algebra homomorphism π ∗ : K[xi , i ∈ Id,n ] → K[xij ], π ∗ (xi ) = pi . The  image of π is precisely G d,n , the cone over Gd,n (note that π(X) is precisely the set of decomposable vectors in ∧d V ). Thus K[Gd,n ] gets identified with the subring of K[xij ] generated by {pi | i ∈ Id,n }. 4.1.6 Identification of G/Pd with Gd,n Let G = SLn (K). Let Pd be the maximal parabolic subgroup with S \ {αd } as the associated set of simple roots. Then      ∗ ∗  Pd = A ∈ G  A = , 0(n−d)×d ∗ WPd = Sd × Sn−d . For the natural action of G on P(∧d V ), we have, the isotropy at [e1 ∧ · · · ed ] is Pd while the orbit through [e1 ∧ · · · ed ] is Gd,n . Thus we obtain a surjective morphism π : G → Gd,n , g → g · a, where a = [e1 ∧ · · · ed ]. Further, the differential dπe : Lie G → T (Gd,n )a (= the tangent space to Gd,n at a) is easily seen to be surjective. Hence we obtain an identification fd : G/Pd ∼ = Gd,n (cf. Borel [7],

34

4 Grassmannian

Proposition 6.7). With this identification, we have H 0 (G/Pd , Lωd ) = (∧d V )∗ (here ωd is the fundamental weight associated to Pd and Lωd is the associated line bundle on G/Pd ; note that Lωd is the ample generator of Pic G/Pd (∼ = Z)).

4.2 Schubert varieties of Gd,n For 1 ≤ t ≤ n, let Vt be the subspace of V spanned by {e1 , . . . , et }. For each i ∈ Id,n , the Schubert variety associated to i is defined to be Xi = {U ∈ Gd,n | dim(U ∩ Vit ) ≥ t, 1 ≤ t ≤ d}. For the action of G on P(∧d V ) the T -fixed points are precisely the points corresponding to the T -eigenvectors in ∧d V . Now

Kei , as T -modules, ∧d V = i∈Id,n

where for i = (i1, · · · , id ), ei = ei1 ∧ · · · ∧ eid . Thus the T -fixed points in P(∧d V ) are precisely ei , i ∈ Id,n , and these points, obviously, belong to Gd,n . Further,the Schubert variety Xi associated   to i is simply the Zariski closure of the B-orbit B ei through the T -fixed point ei (with the canonical reduced structure), B being the subgroup of upper triangular matrices in G. 4.2.1 Bruhat decomposition Let i ∈ Id,n . Let Ci = B[ei ] be the Schubert cell associated to i. The Ci ’s provide a cell decomposition of Gd,n . Let X = V ⊕ · · · ⊕ V be as above. Let π : X → ∧d V , (u1 , . . . , ud ) → u1 ∧ . . . ∧ ud , and p : ∧d V \ {0} → P(∧d V ), u1 ∧ . . . ∧ ud → [u1 ∧ . . . ∧ ud ]. Let vi denote the point (ei1 , . . . , eid ) ∈ X. As above, identifying X with Mn×d , vi gets identified with the n × d matrix whose entries are all zero except the ones in the ij -th row and j -th column, 1 ≤ j ≤ d, which are equal to 1. We have B · vi = {A ∈ Mn×d | xij = 0, i > ij , and t xit t = 0}. Denoting B · vi by Di , we have Di = {A ∈ Mn×d | xij = 0, i > ij }. Further, π(B · vi ) = p −1 (Ci ), !i . From this, we obtain π(Di ) = X Theorem 4.2.1.1 Xi ⊆ Xj if and only if i ≤ j . ˙ i≤j Bei (Bruhat Decomposition). In particular, we have, Xj = ∪ Thus, under the set-theoretic bijection between the set of Schubert varieties and the set Id,n , the partial order on the set of Schubert varieties given by inclusion induces the partial order ≥ on Id,n . Note that Id,n can be identified with Sn /Sd × Sn−d (here, Sn is the permutation group on n symbols). In the sequel, we shall identify Id,n with the set of “minimal representatives” of Sn /Sd × Sn−d in Sn ; to be very precise, a d-tuple i ∈ Id,n will be identified with the element (i1 , . . . , id , j1 , . . . , jn−d ) ∈ Sn , where {j1 , . . . , jn−d } is the complement of {i1 , . . . , id } in {1, . . . , n} arranged in increasing order.

4.2 Schubert varieties of Gd,n

35

4.2.2 Dimension of Xi Let Ei = Bu · vi , where Bu is the subgroup of unipotent upper triangular matrices in G(= SLn (K)). Let Ud be the subgroup of unipotent upper triangular matrices in GLd (K). Then Ei is Ud -stable and the canonical surjective map π0 : Ei → Ci , Avi → A·[ei ], A ∈ Bu , is in fact the orbit map (note that Ci = B ·[ei ] = Bu ·[ei ], as [ei ] is a T -fixed point). For M ∈ Ei , it is easily checked that the isotropy subgroup in Ud at M is {id}. Thus all the Ud -orbits in Ei are isomorphic to Ud . Thus π0 : Ei → Ci is a principal fibration with fiber Ud . Hence we obtain     1 1 dim Ci = dim Ei − d(d −1) = it −d − d(d −1) = it − t= (it −t). 2 2 Thus we obtain Proposition 4.2.2.1 dim Xi =

d

d

d

d

t=1

t=1

t=1

t=1



(it − t).

1≤t≤d

Remark 4.2.2.2. We have Gd,n = Xi , where i = (n + 1 − d, n + 2 − d, . . . , n). In particular, dim Gd,n = d(n − d). Remark 4.2.2.3. Each i ∈ Id,n determines a λ = (id − d, . . . , i2 − 2, i1 − 1) and dim Xi is simply |λ|(= λi , λi = id+1−i − (d + 1 − i)). 4.2.3 Further results on Schubert varieties Lemma 4.2.3.1 Let i, j ∈ Id,n . Then  pj X = 0 ⇐⇒ i ≥ j . i

Proof. From §4.1.2, we have pj |Xi = 0 ⇐⇒ ej ∈ Xi . In view of Bruhat decomposition, we have ej ∈ Xi ⇐⇒ j ≤ i. The result follows from this. Lemma 4.2.3.2 Let X1 , X2 be two Schubert varieties in Gd,n . Then X1 ∩ X2 is irreducible, i.e. X1 ∩ X2 is a Schubert variety (set-theoretically). Proof. Let X1 = X(τ1 ), X2 = X(τ2 ), where τ1 = (a1 , . . . , ad ), τ2 = (b1 , . . . , bd ). Let X1 ∩ X2 = ∪X(wi ), where wi < τ1 , wi < τ2 . Let cj = min{aj , bj }, 1 ≤ j ≤ d, and τ = (c1 , . . . , cd ). Then, clearly τ ∈ Id,n , and τ < τi , i = 1, 2. We have wi ≤ τ , and hence X1 ∩ X2 = X(τ ). Lemma 4.2.3.3 Let X1 , X2 be two distinct Schubert divisors in a Schubert variety X(τ ). Then X1 ∩ X2 is irreducible of codimension 2 in X. Proof. Let X1 = X(τ1 ), X2 = X(τ2 ). Let τ = (c1 , . . . , cd ), τ1 = (a1 , . . . , ad ), τ2 = (b1 , . . . , bd ). Since X(τi ) has codimension 1 in X(τ ), the entries in τi are the same as the entries in τ , but for one place, where the entry is one less than the corresponding entry in τ , i = 1, 2. Let then j such that at = ct , t = j , aj = cj − 1, and k such that bt = ct , t = k, bk = ck − 1. Let w = (f1 , . . . , fd ), where ft = ct , t = j, k, fj = cj − 1, fk = ck − 1. Then X1 ∩ X2 = X(w), and dimX(w) = dimX(τ ) − 2 (clearly, in view of Proposition 4.2.2.1).

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4.3 Standard monomial theory for Schubert varieties in Gd,n 4.3.1 Standard monomials Let R0 be the homogeneous coordinate ring of Gd,n for the Plücker embedding, and for τ ∈ Id,n , let R(τ ) be the homogeneous coordinate ring of the Schubert variety X(τ ) (we shall denote XPd (τ ) by just X(τ )). In this section, we present a Standard monomial theory for X(τ ) (cf. [41, 42, 88]), and as a consequence obtain a basis for R(τ )m , m ∈ Z+ , prove the vanishing theorems, arithmetic normality and arithmetic Cohen-Macaulayness for X(τ ). Standard monomial theory consists in constructing explicit bases for R(τ ), τ ∈ Id,n . Definition 4.3.1.1 A monomial f = pτ1 · · · pτm is said to be standard if τ1 ≥ · · · ≥ τm .

(*)

Such a monomial is said to be standard on X(τ ), if in addition to condition (*), we have τ ≥ τ1 . Remark 4.3.1.2. Note that in the presence  of condition (*), the standardness of f on X(τ ) is equivalent to the condition that f X(τ ) = 0. Thus given a standard monomial  f = pτ1 · · · pτm , f X(τ ) is either 0 or remains standard on X(τ ). 4.3.2 Linear independence of standard monomials Theorem 4.3.2.1 The standard monomials on X(τ ) of degree m are linearly independent in R(τ ). Proof. We proceed by induction on dim X(τ ). If dim X(τ ) = 0, pid is the only standard monomial on X(τ ), and the result is obvious. Let dim X(τ ) > 0. Let r 

ci Fi , ci ∈ k,

(∗)

i=1

be a linear relation of standard monomials Fi of degree m. Let Fi = pwi1 . . . pwim . We divide the proof into the following two cases: Case 1: wi1 = τ, ∀i. First observe that X(τ ) being an integral scheme, pτ is a non-zero divisor in R(τ ); we cancel pτ and obtain a linear relation among monomials of degree m − 1. The required result follows by induction on m (note that when m = 1, the result follows by linear independence of {pi , i ∈ Id,n }). Case 2: wi1 < τ for some i. For simplicity, assume that w11 < τ , and w11 is a minimal element of {wj 1 | wj 1 < τ }. Let us denote w11 by φ. Then for i ≥ 2, Fi |X(φ) is either 0, or is standard on X(φ). Hence restricting (∗) to X(φ), we obtain a nontrivial standard sum on X(φ) being zero, which is not possible (by induction hypothesis). Hence we conclude that wi1 = τ for all i, 1 ≤ i ≤ m. Canceling pτ , we obtain a linear relation among standard monomials on X(τ ) of degree m − 1. Using induction on m, the required result follows.

4.3 Standard monomial theory for Schubert varieties in Gd,n

37

4.3.3 Generation by standard monomials Theorem 4.3.3.1 Let F = pw1 . . . pwm be any monomial in the Plücker coordinates of degree m. Then F is a linear combination of standard monomials of degree m. Proof. For F = pw1 . . . pwm , define NF = l(w1 )N m−1 + l(w2 )N m−2 + . . . l(wm ), where N # 0, say N > d(n − d)(= dim Gd,n ) and l(w) = dimX(w). If F is standard, there is nothing to prove. Let t be the first violation of standardness, i.e., pw1 . . . pwt−1 is standard, but pw1 . . . pwt is not. Hence wt−1 ≥ wt , and using the quadratic relation (cf. the relation (***) in §4.1.4)  pwt−1 pwt = ±pα pβ , (*) α,β

F can be expressed as F = Fi , with NFi > NF (since α > wt−1 for all α on the right hand side of (∗)). Now the required result is obtained by decreasing induction on NF (the starting point of induction, i.e. the case when NF is the largest, corresponds to standard monomial F = pθm , where θ = (n+1−d, n+2−d, · · · , n), in which case F is clearly standard; note that X(θ ) = Gd,n ). Combining Theorems 4.3.2.1 and 4.3.3.1, we obtain Theorem 4.3.3.2 Standard monomials on X(τ ) of degree m give a basis for R(τ )m . As a consequence of Theorem 4.3.3.2, we have a qualitative description of a typical quadratic relation on a Schubert variety X(w) as given by the following Proposition 4.3.3.3 Let w, τ, φ ∈ Id,n , w > τ, φ. Further let τ, φ be non-comparable (so that pτ pφ is a non-standard degree 2 monomial on X(w)). Let  p τ pφ = cα,β pα pβ , cα,β ∈ k ∗ (*) α,β

be the expression for pτ pφ as a sum of standard monomials on X(w). Then for every α, β on the right hand side we have, α > both τ and φ, and β < both τ and φ. Such a relation as in (*) is called a straightening relation. Proof. Among all the α’s choose a minimal one, call it α0 . Restricting (*) to X(α0 ), we have, the restriction of the right hand side is a non-zero sum of standard monomials on X(α0 ) (note that the restriction of pα pβ to X(α0 ) is non-zero if and only if α = α0 - by the minimality of α0 , and there is at least one term, namely pα0 pβ whose restriction to X(α0 ) is non-zero). Hence in view of linear independence of standard monomials on X(α0 ), we obtain that the restriction of the left hand side to X(α0 ) is non-zero. From this we conclude α0 ≥ both τ and φ; in fact, α0 > both τ and φ (for, if α0 equals one of them, say, α0 = τ , then this would imply (by weight considerations) that β = φ which in turn would imply τ and φ are comparable (since

38

4 Grassmannian

α and β are), a contradiction. Any α is ≥ some minimal α0 . Hence α > both τ and φ. Now replacing τ and φ by w0 τ and w0 φ, we conclude from above w0 β > both w0 τ and w0 φ, i.e., β < both τ and φ. 4.3.4 Equations defining Schubert varieties in the Grassmannian Let τ ∈ Id,n . Let πτ be the map R0 → R(τ ) (the restriction map). Let ker πτ = Jτ . Let Zτ = {all standard monomials F | F starts with pφ for some φ ≤ τ }. We shall now give a set of generators for Jτ in terms of Plücker coordinates. Lemma 4.3.4.1 Let Iτ = (pw , τ ≥ w) (ideal in R). Then Zτ is a basis for Iτ . Proof. Let F ∈ Iτ . Then writing F as a linear combination of standard monomials   F = a i Fi + bj Gj , where in the first sum each Fi starts with pτi1 , with τ ≥ τi1 , and in the second sum each Gj starts with φj 1 , with τ ≥ φj 1 . This implies (cf. Lemma 4.2.3.1) that ai Fi ∈ Iτ , and hence we obtain  bj Gj ∈ Iτ . This now implies that considered as an element of R(τ ), bj Gj is equal to 0 (note that Iτ ⊂ Jτ ). Now the linear independence of standard monomials on X(τ ) implies that bj = 0 for all j . The required result now follows. Proposition 4.3.4.2 Let τ ∈ Id,n . Then R(τ ) = R0 /Iτ . Proof. We have, R(τ ) = R0 /Jτ (where Jτ is as above). We shall now show that the inclusion Iτ ⊂ Jτ is in fact an equality. Let F ∈ R0 . Writing F as a linear combination of standard monomials   bj Gj , F = a i Fi + where in the first sum each Fi starts with pτi1 , with τ ≥ τi1 , and in the second sum each Gj starts with φj 1 , with τ ≥ φj 1 , we have, ai Fi ∈ Iτ , and hence we obtain F ∈ J τ ⇐⇒ bj Gj ∈ Jτ (since ai Fi ∈ Iτ , and Iτ ⊂ Jτ ) ⇐⇒ πτ (F )(= bj Gj ) is zero ⇐⇒ bj Gj (= a sum of standard monomials on X(τ )) is zero on X(τ ) ⇐⇒ bj = 0 for all j (in view of the linear independence of standard monomials on X(τ )) ⇐⇒ F = ai Fi ⇐⇒ F ∈ Iτ . Hence we obtain Jτ = Iτ .

4.4 Standard monomial theory for a union of Schubert varieties

39

Equations defining Schubert varieties: Let τ ∈ Id,n . We have (cf. Lemma 4.2.3.1), the kernel of (R0 )1 → (R(τ ))1 has a basis given by {pw , τ ≥ w}. Combining this fact with Proposition 4.3.4.2 and Lemma 4.3.4.1 we obtain that the ideal Jτ (= the kernel of the restriction map R0 → R(τ )) is generated by {pw | τ ≥ w, w ∈ Id,n }. Hence Jτ is generated by the kernel of (R0 )1 → (R(τ ))1 . Thus we obtain that X(τ ) is scheme-theoretically (even at the cone level) the intersection of Gd,n with all hyperplanes in P(∧d k n ) containing X(τ ). Further, as a closed subvariety of Gd,n , X(τ ) is defined (scheme-theoretically) by the vanishing of pw , τ ≥ w, w ∈ Id,n .

4.4 Standard monomial theory for a union of Schubert varieties In this section, we prove results similar to Theorems 4.3.2.1, 4.3.3.1, 4.3.3.2 for a union of Schubert varieties. Let Xi be Schubert varieties in Gd,n . Let X = ∪Xi . Definition 4.4.0.1 A monomial F in the Plücker coordinates is standard on the union X = ∪Xi if it is standard on some Xi . 4.4.1 Linear independence of standard monomials on X = ∪Xi Theorem 4.4.1.1 Monomials standard on X = ∪Xi are linearly independent. Proof. If possible, let

r 

ai Fi = 0

(*)

i=1

be a nontrivial relation among standard monomials on X. Let Fi = pτi1 . . . pτiri . For simplicity, denote τ11 by just τ . Then restricting (∗) to X(τ ), we obtain a nontrivial relation among standard monomials on X(τ ), which is a contradiction (note that for any i, Fi |X(τ ) is either 0 or remains standard on X(τ ); further, F1 |X(τ ) is non-zero). 4.4.2 Standard Monomial Basis Theorem 4.4.2.1 Let X = ∪ri=1 X(τi ), and S the homogeneous coordinate ring of X. Then the standard monomials on X give a basis for S. Proof. For τ ∈ Id,n , let Iτ be as in Lemma 4.3.4.1. Let us denote It = Iτt , 1 ≤ t ≤ r. We have R(τt ) = R0 /It (cf. Proposition 4.3.4.2). Let S = R0 /I . Then I = ∩It (note that being the intersection of reduced ideals, I is also reduced, and hence the set theoretic equality X = ∪Xi is also scheme theoretic). Let F ∈ I . Further, let us write F as a linear combination of standard monomials  F = c k Fk .

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4 Grassmannian

Then in view of Lemma 4.3.4.1, we obtain that each Fk starts with pτk1 , where τk1 ≤ τi , 1 ≤ i ≤ r. A typical element in R0 /I may be written as π(f ), for some f ∈ R0 , where π is the canonical projection R0 → R0 /I . Let us write f as a sum of standard monomials   bl Hl , f = aj Gj + where each Gj (resp. Hl ) starts with pτ such that τ ≤ τi , 1 ≤ i ≤ r (resp. τ ≤ τi for some i). We have π(f ) = bl Hl (since aj Gj ∈ I ). Thus we obtain that S (as a vector space) is generated by monomials standard on X. This together with the linear independence of standard monomials on X implies the required result. 4.4.3 Consequences Theorem 4.4.3.1 Let X1 , X2 be two Schubert varieties in Gd,n . Then (1) X1 ∪ X2 is reduced. (2) X1 ∩ X2 is reduced. Proof. (1). For a closed subscheme Y in Gd,n , let I (Y ) denote the ideal defining Y in Gd,n . We have I (X1 ∪ X2 ) = I (X1 ) ∩ I (X2 ), and hence I (X1 ∪ X2 ) is reduced (since I (X1 ), I (X2 ) are reduced). (2) Let X1 = X(w1 ), X2 = X(w2 ). Let A be the homogeneous coordinate ring of X1 ∩ X2 . Let A = R0 /I . Then I = I1 + I2 . Let X1 ∩ X2 = ∪ri=1 X(τi ) (set theoretically). Let F ∈ I . Then in the expression for F as a linear combination of standard monomials  F = aj Fj , √ then by each Fj starts with pτj 1 , where τj 1 ≤ either w1 or w2 . If B = R0 / I , Theorem 4.4.2.1, under π : R0 → B, ker π consists of all f such that f = ck fk , fk being standard monomials such that each fk starts with√pφk1 , where φk1 ≤ τi , for all i, 1 ≤ i ≤ r. Hence φk1 ≤ either w1 or w2 . Hence I = I , and the required result follows from this. Theorem 4.4.3.2 (Pieri’s formula) X(τ ) ∩ {pτ = 0} = ∪ri=1 X(τi ),

scheme theoretically,

where the varieties X(τi ), 1 ≤ i ≤ r are all the Schubert divisors in X(τ ). Proof. Let X = ∪ri=1 X(τi ), and let B be the homogeneous coordinate ring of X. Let B = R(τ )/I . Clearly, (pτ ) ⊆ I, (pτ ) being the principal ideal in R(τ ) generated by pτ . Let f ∈ I . Writing f as   cj Hj , f = bi Gi + where each Gi is a standard monomial in R(τ ) starting with pτ and each Hj is a , we have, bi Gi ∈ I . standard monomial in R(τ ) starting with pθj 1 , where θj 1 < τ cj Hj being a sum of This now implies cj Hj is zero on ∪ri=1 X(τi ). But now r X(τ ), we have by Theorem 4.4.1.1, c = 0, for all j . standard monomials on ∪ i j i=1 Thus we obtain f = bi Gi , and hence f ∈ (pτ ). This implies I = (pτ ). Hence we obtain B = R(τ )/(pτ ), and the result follows from this.

4.5 Vanishing theorems

41

4.5 Vanishing theorems Cohomology of sheaves Let X be an algebraic scheme, Ab(X) the category of sheaves of abelian groups on X. Let Ab denote the category of abelian groups. Let Γ (X, ·) be the global section functor. It is left exact, i.e., if 0 → F → G → H → 0 is a short exact sequence of sheaves (i.e., 0 → Fx → Gx → Hx → 0 is exact for every x ∈ X), then 0 → Γ (X, F) → Γ (X, G) → Γ (X, H) is exact. The cohomology functors H i (X, ·) are defined to be the right derived functors of Γ (X, ·). For any sheaf F, the groups H i (X, F) are defined as the cohomology groups of F. Thus H i (X, F) are just the cohomology groups of the complex obtained by applying Γ (X, ·) to an injective resolution of F (see [37] for details). In the discussion below, we will be using the following well-known fact from cohomology theory (see [112] for example): Proposition 4.5.0.1 Let 0 → F → G → H → 0 be a short exact sequence of sheaves on an algebraic scheme X. This gives rise to a cohomology long exact sequence · · · → H i−1 (X, H) → H i (X, F) → H i (X, G) → H i (X, H) → · · · Let now X be a union of Schubert varieties. Let S(X, m) be the set of standard monomials on X of degree m, and s(X, m) the cardinality of S(X, m). If X = X(τ ) for some τ , S(X, m) and s(X, m) will also be denoted by just S(τ, m), respectively s(τ, m). Lemma 4.5.0.2 (1) Let Y = Y1 ∪ Y2 . Then s(Y, m) = s(Y1 , m) + s(Y2 , m) − s(Y1 ∩ Y2 , m). (2) s(τ, m) = s(τ, m − 1) + s(Hτ , m), where Hτ is the union of all the Schubert divisors in Xτ . (1) and (2) are easy consequences of the results of the previous section. Let L = L(ωd ). We shall denote the restriction of L to a subvariety X of Gd,n also by just L. Proposition 4.5.0.3 Let r be an integer ≤ d(n − d) (= dim Gd,n ). Suppose that all Schubert varieties X in Gd,n of dimension at most r have the following two properties: (1) H i (X, Lm ) = 0, for i ≥ 1, m ≥ 0. (2) The set S(X, m) is a basis for H 0 (X, Lm ), m ≥ 0. Then all unions and intersections of Schubert varieties of dimension at most r satisfy (1) and (2). Proof. We will prove the result by induction on r. Let Sr denote the set of Schubert varieties X in Gd,n of dimension at most r. Let Y = ∪tj =1 Xj , Xj ∈ Sr . Let Y1 = ∪t−1 j =1 Xj , and Y2 = Xt . Consider the exact sequence

0 → OY → OY1 ⊕ OY2 → OY1 ∩Y2 → 0,

42

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where OY → OY1 ⊕ OY2 is the map f → (f |Y1 , f |Y2 ) and OY1 ⊕ OY2 → OY1 ∩Y2 is the map (f, g) → (f − g)|Y1 ∩Y2 . Tensoring with Lm , we obtain the long exact sequence → H i−1 (Y1 ∩Y2 , Lm ) → H i (Y, Lm ) → H i (Y1 , Lm )⊕H i (Y2 , Lm ) → H i (Y1 ∩Y2 , Lm ) →

Now Y1 ∩ Y2 is reduced (cf. Theorem 4.4.3.1) and Y1 ∩ Y2 ∈ Sr−1 . Hence, by the induction hypothesis (1) and (2) hold for Y1 ∩ Y2 . In particular, if m ≥ 0, then (2) implies that the map H 0 (Y1 , Lm ) ⊕ H 0 (Y2 , Lm ) → H 0 (Y1 ∩ Y2 , Lm ) is surjective. Hence we obtain that the sequence 0 → H 0 (Y, Lm ) → H 0 (Y1 , Lm ) ⊕ H 0 (Y2 , Lm ) → H 0 (Y1 ∩ Y2 , Lm ) → 0 is exact. This implies H 0 (Y1 ∩ Y2 , Lm ) → H 1 (Y, Lm ) is the zero map; we have, H 1 (Y, Lm ) → H 1 (Y1 , Lm ) ⊕ H 1 (Y2 , Lm ) is also the zero map (since by induction H 1 (Y1 , Lm ) = 0 = H 1 (Y2 , Lm )). Hence we obtain H 1 (Y, Lm ) = 0, m ≥ 0, and for i ≥ 2, the assertion that H i (Y, Lm ) = 0, m ≥ 0 follows from the long exact cohomology sequence above (and induction hypothesis). This proves the assertion (1) for Y . To prove assertion (2) for Y , we observe h0 (Y, Lm ) = h0 (Y1 , Lm ) + h0 (Y2 , Lm ) − h0 (Y1 ∩ Y2 , Lm ) = s(Y1 , m) + s(Y2 , m) − s(Y1 ∩ Y2 , m). Hence Lemma 4.5.0.2 and induction hypothesis imply that h0 (Y, Lm ) = s(Y, Lm ). This together with linear independence of standard monomials on Y proves assertion (2) for Y . Proposition 4.5.0.4 Let X be a Schubert variety in Gd,n . Let Y = ∪tj =1 Xj be a union of Schubert divisors in X. Let L be as in Proposition 4.5.0.3. Suppose H i (Xj , Lm ) = 0, for m < 0, 0 ≤i ≤ dim Y − 1(= dim Xj − 1). Then H i (Y, Lm ) = 0, for m < 0, 0 ≤ i ≤ dimY − 1. Proof. We will prove the result by induction on t and dim Y (=dim X − 1). As in the proof of Proposition 4.5.0.3, we set Y1 = ∪t−1 j =1 Xj , Y2 = Xt , and consider the long exact cohomology sequence → H i−1 (Y1 ∩Y2 , Lm ) → H i (Y, Lm ) → H i (Y1 , Lm )⊕H i (Y2 , Lm ) → H i (Y1 ∩Y2 , Lm ) →

We have, Y1 ∩ Y2 = ∪t−1 j =1 Xj ∩ Xt . By Lemma 4.2.3.3, Xj ∩ Xt , 1 ≤ j ≤ t − 1 is irreducible of codimension 1 in Xj (and Xt ). Hence by induction on dim Y , we have, H i (Y1 ∩ Y2 , Lm ) = 0, for m < 0, 0 ≤ i ≤ dim Y1 − 2(= dim Y − 2); further, by induction on t, H i (Y1 , Lm ) = 0, m < 0, 0 ≤ i < dimY1 , and by hypothesis, H i (Y2 , Lm ) = 0, m < 0, 0 ≤ i < dimY1 . The required result now follows.

4.5 Vanishing theorems

43

Theorem 4.5.0.5 (Vanishing Theorems) Let L = Lωd . Note that L is the restriction to Gd,n of the tautological bundle on P(Λd V ). Let X be a Schubert variety in Gd,n . Denoting the restriction of L to X also by just L, we have (a) H i (X, Lm ) = 0 for i ≥ 1, m ≥ 0. (b) H i (X, Lm ) = 0 for 0 ≤ i < dim X, m < 0. (c) The set S(X, m) is a basis for H 0 (X, Lm ). Proof. We prove the result by induction on m and dim X. If dim X = 0, X is just a point, and the result is obvious. Assume now that dim X"≥ 1. Let X = X(τ ). Let X1 , . . . , Xs be all the Schubert divisors in X, and Y = si=1 Xi . Then by Pieri’s formula (cf. Theorem 4.4.3.2), we have, Y = X(τ ) ∩ {pτ = 0} (scheme-theoretically). Hence the sequence 0 → OX (−1) → OX → OY → 0 is exact (note that OX (−1) = L−1 , and that the map L−1 → OX is given by pτ which is a global section of L). Tensoring it with Lm , and taking cohomology, we obtain the long exact cohomology sequence · · · → H i−1 (Y, Lm ) → H i (X, Lm−1 ) → H i (X, Lm ) → H i (Y, Lm ) → · · · . Let m ≥ 0, i ≥ 2. Then the induction hypothesis on dim X implies (in view of Proposition 4.5.0.3) that H i (Y, Lm ) = 0, i ≥ 1. Hence we obtain that the sequence 0 → H i (X, Lm−1 ) → H i (X, Lm ), i ≥ 2, is exact. If i = 1, again the induction hypothesis implies the surjectivity of H 0 (X, Lm ) → H 0 (Y, Lm ). This in turn implies that the map H 0 (Y, Lm ) → H 1 (X, Lm−1 ) is the zero map, and hence we obtain that the sequence 0 → H 1 (X, Lm−1 ) → H 1 (X, Lm ) is exact. Thus we obtain that 0 → H i (X, Lm−1 ) → H i (X, Lm ), m ≥ 0, i ≥ 1 is exact. But H i (X, Lm ) = 0, m # 0, i ≥ 1 (cf. [109]). Hence we obtain H i (X, Lm ) = 0 for i ≥ 1, m ≥ 0,

(1)

h0 (X, Lm ) = h0 (X, Lm−1 ) + h0 (Y, Lm )

(2)

and where = In particular, assertion (a) follows from (1). For m ≥ 1, the induction hypothesis on m implies that h0 (X, Lm−1 ) = s(X, m − 1) (note that when m = 1, H 0 (X, Lm−1 ) = H 0 (X, OX ) = k, and hence h0 (X, Lm−1 ) = s(X, m − 1)). On the other hand, the induction hypothesis on dim X implies (in view of Proposition 4.5.0.3) that h0 (Y, Lm ) = s(Y, m). Hence we obtain h0 (X, Lm )

dim H 0 (X, Lm ).

h0 (X, Lm ) = s(X, m − 1) + s(Y, m).

(3)

Now (3) together with part 2 of Lemma 4.5.0.2 implies that h0 (X, Lm ) = s(X, m). Hence (c) follows in view of the linear independence of standard monomials on X(τ ) (cf. Theorem 4.3.2.1).

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To prove (b), consider the long exact cohomology sequence · · · → H i−1 (Y, Lm ) → H i (X, Lm−1 ) → H i (X, Lm ) → · · · We have, H i (Y, Lm ) = 0, m ∈ Z, 1 ≤ i ≤ dim Y − 1 (by induction hypothesis and Propositions 4.5.0.3 and 4.5.0.4). Hence we obtain the exact sequence 0 → H i (X, Lm−1 ) → H i (X, Lm ), 2 ≤ i < dim X. But H i (X, Lm ) = 0, m # 0 (cf. [109]). Hence we obtain, H i (X, Lm ) = 0, 2 ≤ i < dim X, m ∈ Z (in particular for m < 0). It remains to prove (b) for i = 0, 1. Let then m < 0. Induction hypothesis together with Proposition 4.5.0.4 implies that H 0 (Y, Lm ) = 0. Hence we obtain the exact sequence 0 → H 1 (X, Lm−1 ) → H 1 (X, Lm ), m < 0. Thus we obtain inclusions H 1 (X, Lm−1 ) ⊂ H 1 (X, Lm ) ⊂ · · · ⊂ H 1 (X, L−1 ). But now the isomorphism H 0 (X, OX ) ∼ = H 0 (Y, OY ) together with the fact that 1 1 −1 H (X, OX ) = 0 implies that H (X, L ) = 0. This now implies (in view of the above inclusions) that H 1 (X, Lm ) = 0, m < 0. Now H 0 (Y, Lm ) = 0 implies H 0 (X, Lm−1 ) ∼ = H 0 (X, Lm ), m < 0.

(4)

Again the isomorphism H 0 (X, OX ) ∼ = H 0 (Y, OY ) implies that H 0 (X, L−1 ) = 0, 0 m and it follows that H (X, L ) = 0, m < 0 (note that in view of (4), we have, H 0 (X, Lm ) ∼ = H 0 (X, L−1 ) = 0). Corollary 4.5.0.6 We have # 0 1. R = m∈Z Lm ), where X = Gd,n . # + H (X, 0 2. R(τ ) = m∈Z+ H (X(τ ), Lm ), for all τ ∈ Id,n . Proof. The assertions follow immediately from Theorems 4.3.3.2 and 4.5.0.5.

4.6 Arithmetic Cohen-Macaulayness, normality and factoriality In this section, we present a proof of the arithmetic Cohen-Macaulayness and arithmetic normality of Schubert varieties in the Grassmannian. We also give a characterization of factorial Schubert varieties. We follow the treatment as in [88, 89]. Let X → PN be a closed sub scheme which we shall suppose to be equidimensional (i.e., all irreducible components of X are of the same dimension) of dimension d. Let I be the homogeneous ideal defining X and A = K[x0 , . . . , xN ]/I the homogeneous coordinate ring of X. Let L denote the tautological line bundle on PN , as  and A(0) denote well as its restriction to X. Let (0) denote the vertex of the cone X,  at the vertex of X. the local ring OX,(0) 

4.6 Arithmetic Cohen-Macaulayness, normality and factoriality

45

 is Cohen-Macaulay if and only if X  is Lemma 4.6.0.1 Let X be as above. Then X Cohen-Macaulay at its vertex. Proof. Proof of the implication ⇒ is clear. Let then the vertex (0) be a Cohen Then X  is Cohen-Macaulay in a neighbourhood U of (0). Macaulay point of X.  \ {(0)} → X meets U . But It is clear that each fiber of the natural projection π : X  \ {(0)}, and hence X  is Cohen-Macaulay these fibers are simply the Gm -orbits in X at each point of π −1 (x) for all x ∈ X. Theorem 4.6.0.2 The Schubert variety X(τ ) in Gd,n is arithmetically Cohen) is Cohen-Macaulay. Macaulay, i.e., the cone X(τ Proof. Let us denote X = X(τ ), d = dimX. Then in view of the above Lemma,  is Cohen-Macaulay at its vertex, i.e., to show that A(0) is it suffices to show that X Cohen-Macaulay. In view of Serre-Grothendieck Criterion, we are reduced to show i (X, i  O ) denotes the local coho OX that H(0)  ) = 0, for all i ≤ d (here, H(0) (X,  X mology; also, recall (see [79] for example) Serre-Grothendieck Criterion that A(0) is i (X,  OX Cohen-Macaulay if and only if H(0)  ) = 0, for all i ≤ d). To prove this, let us  denote U = X \ {(0)}. Then we have a long exact sequence i  i  i (X, OX OX · · · → H(0)  ) → H (X,  ) → H (U, OU ) → · · ·

 OX  is affine), and that the restriction We have that H i (X,  ) = 0, i ≥ 1 (since X 0 0  map H (X, OX  ) → H (U, OU ) is injective (since U is a dense open subset of X). Hence we obtain the following: 0 (X,  OX 1. H(0) ) = 0 1  O ) → 0 is exact 0  OX 2. 0 → H 0 (X,  ) → H (U, OU ) → H(0) (X,  X  OX 3. H i−1 (U, OU ) ∼ = H i (X,  ), i ≥ 2 (0)

On the other hand, the natural map U → X realizes U as the scheme over X defined by the sheaf of OX -algebras ⊕r∈Z Lr . Hence we have isomorphisms (∗)

H i (U, OU ) = ⊕r∈Z H i (X, Lr ), i ≥ 0

We have (4)

i  (X, OX H(0)  ) = 0, i = 0, 1

(for i = 0, this is just (1) above; for i = 1, this follows from (2) and Corol O  )(= R(τ ) = A) = lary 4.5.0.6 (note that by Corollary 4.5.0.6, we have H 0 (X, X ⊕r∈Z+ H 0 (X, Lr )(= ⊕r∈Z H 0 (X, Lr ), since H 0 (X, Lr ) = 0 for r < 0 (cf. Theorem 4.5.0.5 (b))). Let then 2 ≤ i ≤ d. In view of (*) and Theorem 4.5.0.5 (a),(b), we have H i−1 (U, OU ) = 0, 2 ≤ i ≤ d This together with (3) implies (5)

i  (X, OX H(0)  ) = 0, 2 ≤ i ≤ d

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i (X,  O  ) = 0, i ≤ d. This completes the Thus we obtain (from (4) and (5)) that H(0) X proof of the Theorem.

As a consequence, we have the following Theorem 4.6.0.3 The Schubert variety X(τ ) in Gd,n is arithmetically normal, i.e. ) is normal. the cone X(τ Proof. First observe that the above Theorem together with Chevalley’s result that Schubert varieties are regular in codimension 1 (cf. Theorem A.12.1.10 of Appendix) implies that X(τ ) is normal (in view of Serre Criterion$ for normality—a ring A is normal if and only if A has S2 and R1 ). Let Y = P( d V ). The canonical homomorphism φm : H 0 (Y, Lm ) → H 0 (X(τ ), Lm ) m ). We have (see [37] for examis surjective for m # 0. Let Sm = H 0 (X(τ ), L# ple) that image of (φm ) equals R(τ )m , and S = m∈Z+ Sm is the integral closure of R(τ ) in its quotient field. Now by Corollary 4.5.0.6, we have R(τ )m = Sm for all m, and hence we obtain that R(τ ) is normal (note that if a projective variety X → Pn is normal, then it is arithmetically normal if and only if the restriction map H 0 (Pn , Lm ) → H 0 (X, Lm ) is surjective for all m, L being the tautological line bundle on Pn ). Therefore X(τ ) is arithmetically normal.

4.6.1 Factorial Schubert varieties Recall that an algebraic variety X is said to be factorial at a point P ∈ X, if the stalk OX,P is factorial, i.e., OX,P is a unique factorization domain; the variety X is said to be factorial if it is factorial at all P ∈ X. If in addition X is projective, then X is said to be projectively (or also arithmetically) factorial if the cone Xˆ is factorial at its vertex. Theorem 4.6.1.1 ([89]) The following are equivalent for a Schubert variety X. 1. 2. 3. 4. 5. 6.

X is projectively factorial (for the Plücker embedding). X is factorial. X is factorial at the point cell eid . X has a unique Schubert divisor. X is non–singular. X is itself a suitable Grassmannian.

5 Determinantal varieties

In this chapter, we give a self-contained exposition of the classical determinantal varieties from the view-point of their relationship to Schubert varieties. We follow the same treatment as in [67].

5.1 Recollection of facts Let us fix the integers 1 ≤ d < n and let V = K n . The Grassmannian Gd,n , and the set Id,n be as in Chapter 4. For i ∈ Id,n , let Xi be the associated Schubert variety in Gd,n . Recall the following facts from Chapter 4: Fact 1. Xi ⊆ Xj if and only if i ≤ j . ˙ i≤j Bei . Fact 2. Bruhat Decomposition: Xj = ∪ Fact 3. dim Xi = 1≤t≤d (it −t). Fact 4. Let i, j ∈ Id,n . Then pj X = 0 ⇐⇒ i ≥ j . i Fact 5. Under the set-theoretic bijection between the set of Schubert varieties and the set Id,n , the partial order on the set of Schubert varieties given by inclusion induces the partial order ≥ on Id,n . Let R be the homogeneous co-ordinate ring of Gd,n for the Plücker embedding, and for τ ∈ Id,n , let R(τ ) be the homogeneous co-ordinate ring of the Schubert variety X(τ ). We have from Chapter 4 Theorem 5.1.0.1 Standard monomials on X(τ ) of degree m give a basis for R(τ )m . As seen in Chapter 4, Theorem 5.1.0.1 implies the following qualitative description of a typical quadratic relation on a Schubert variety X(w): Proposition 5.1.0.2 Let w, τ, φ ∈ Id,n , w > τ, φ. Further let τ, φ be non-comparable (so that pτ pφ is a non-standard degree 2 monomial on X(w)). Let  cα,β pα pβ , cα,β ∈ K ∗ (*) p τ pφ = α,β

48

5 Determinantal varieties

be the expression for pτ pφ as a sum of standard monomials on X(w). Then for every α, β on the R.H.S. we have, α ≥ both τ and φ, β ≤ both τ and φ. Remark 5.1.0.3. Such a relation as in (*) is called a straightening relation. 5.1.1 Equations defining Schubert varieties in the Grassmannian Let τ ∈ Id,n . Let πτ be the map R → R(τ ) (the restriction map, R being the homogeneous co-ordinate ring of Gd,n for the Plücker embedding). Let ker πτ = Iτ . The following Proposition (cf. Chapter 4) gives a set of ideal generators for Iτ in terms of Plücker co-ordinates. Proposition 5.1.1.1 Iτ is generated by {pw , τ ≥ w} ( as an ideal in R). Further, Zτ := {all standard monomials F | F starts with pφ for some φ ≤ τ } is a basis for Iτ . 5.1.2 Evaluation of Plücker coordinates on the opposite big cell in Gd,n Let Pd be the parabolic subgroup of G consisting of all matrices of the form   ∗∗ 0∗ where the 0-matrix is of size n − d × d. Then we have an identification ϕd : G/Pd ∼ = Gd,n . Denote by O − the subgroup of G generated by {Uα | α ∈ R − \ RP− }. Then O − consists of the elements of G of the form   0d×(n−d) Id A(n−d)×d In−d where Id (resp. In−d )is the d × d (resp. n − d × n − d) identity matrix. Further, the restriction of the canonical morphism θd : G → G/Pd to O − is an open immersion, and θd (O − ) = B − eid , where eid is the coset Pd of G/Pd ; also, ϕd (B − eid ) is the opposite cell in Gd,n . Thus the opposite cell in Gd,n gets identified with O − , and in the sequel we shall denote B − eid by just O − . Note that O − ∼ = Ad(n−d) ). = Mn−d, d (∼ d Consider the morphism ξd : G → P(∧ V ), where ξd = fd ◦ θd , fd being the Plücker embedding (cf. chapter 4), and θd (as above) the natural projection G → G/Pd . Then pj (ξd (g)) is simply the minor of g consisting of the first d columns and the rows with indices j1 , . . . , jd . Let z ∈ O − , and let z correspond to A ∈ Mn−d, d . The Plücker coordinate pj evaluated at z is simply a certain minor of A, which may be explicitly described as follows. Let j = (j1 , . . . , jd ), and let jr be the largest entry ≤ d. Let {k1 , . . . , kd−r } be the complement of {j1 , . . . , jr } in {1, . . . , d}. Then this minor of A is given by column indices k1 , . . . kd−r , and row indices jr+1 , . . . , jd (here the rows of A are indexed as d +1, . . . , n). Conversely, given a minor of A, say, with column indices b1 , . . . , bs , and row indices id−s+1 , . . . , id , it is the evaluation of the Plücker coordinate pi at A, where i = (i1 , . . . , id ) may be described as follows:

5.2 Determinantal varieties

49

{i1 , . . . , id−s } is the complement of {b1 , . . . , bs } in {1, . . . , d}, and id−s+1 , . . . , id are simply the row indices (again, the rows of A are indexed as d + 1, . . . , n). Convention. If j = (1, . . . , d), then pj evaluated at A is 1. We shall consider the element 1 (in K[Mn−d, d ]) as the minor of A with row indices (and column indices) given by the empty set. 5.1.3 Ideal of the opposite cell in X(w) For a Schubert variety X(w) in Gd,n , let us denote B − eid ∩ X(w) by Y (w). We consider Y (w) as a closed subvariety of O − . In view of §5.1.1.1, we obtain that the ideal defining Y (w) in O − is generated by {pi | i ∈ Id,n , w ≥ i}.

5.2 Determinantal varieties Let Z = Mm,n (K), the space of all m × n matrices with entries in K. We shall identify Z with Amn . We have K[Z] = K[xi,j , 1 ≤ i ≤ m, 1 ≤ j ≤ n]. 5.2.1 The variety Dt Let X = (xij ), 1 ≤ i ≤ m, 1 ≤ j ≤ n be a m × n matrix of indeterminates. Let A ⊂ {1, · · · , m}, B ⊂ {1, · · · , n}, #A = #B = s, where s ≤ min {m, n}. We shall denote by [A|B] the s-minor of X with row indices given by A, and column indices given by B. For t, 1 ≤ t ≤ min {m, n}, let It (X) be the ideal in K[xi,j ] generated by {[A|B], A ⊂ {1, · · · , m}, B ⊂ {1, · · · , n}, #A = #B = t}. Let Dt (K) (or just Dt ) be the determinantal variety, a closed subvariety of Z, with It (X) as the defining ideal. 5.2.2 Identification of Dt with Yφ Let G = SLn (K). Let r, d be such that r + d = n. Let X be a r × d matrix of indeterminates. As above, let us identify the opposite cell O − in G/Pd (∼ = Gd,n ) as   Id . O− = X We have a bijection between {Plücker co-ordinates pi , i = {1, 2, · · · , d}} and {minors of X} (note that if i = {1, 2, · · · , d}, then pi = the constant function 1). For example, take r = 3 = d. We have,   I3 − . O3 = X3×3

50

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We have, p(1,2,4) = [{1}|{3}], p(2,4,6) = [{1, 3}|{1, 3}]. In the sequel, given a Plücker coordinate pi (on Od− ), we shall denote the associated minor (of Xr,d ) by Δi . The set Zt : Let Zt = {i ∈ Id,n | Δi is a t minor of X}. The partial order on the Plücker co-ordinates induces a partial order on Zt . Lemma 5.2.2.1 Let τ be the d-tuple, τ = (1, 2, · · · , d − t, d + 1, d + 2, · · · , d + t). Then τ ∈ Zt . Further, τ is the unique smallest element in Zt . Proof. Clearly τ ∈ Zt . Let Δ be a t-minor of X, and pi the associated Plücker coordinate. Let i = (i1 , · · · , id ). We have, for 1 ≤ k ≤ d − t, ik ≤ d, and for d − t + 1 ≤ k ≤ d, ik > d. Clearly τ is the smallest such d-tuple. Remark 5.2.2.2. With τ as in Lemma 5.2.2.1, note that the associated minor Δτ of X has row and column indices given by {d + 1, d + 2, · · · , d + t}, {d + 1 − t, d + 2 − t, · · · , d} respectively, i.e., the right most top corner t minor of X, the rows of X being indexed as d + 1, . . . , n. Remark 5.2.2.3. With τ as in Lemma 5.2.2.1, note that τ is the smallest d-tuple (j1 , · · · , jd ) such that jd−t+1 ≥ d + 1. Lemma 5.2.2.4 Let τ be as in Lemma 5.2.2.1. Let Nt = {i ∈ Id,n | pτ |Xi = 0}. Let φ be the d-tuple, φ = (t, t + 1, · · · , d, n + 2 − t, n + 3 − t, · · · , n) (note that φ consists of the two blocks [t, d], [n + 2 − t, n] of consecutive integers - here, for i < j , [i, j ] denotes the set {i, i + 1, · · · , j }). Then φ is the unique largest element in Nt . Proof. Let i ∈ Nt , say, i = (i1 , · · · , id ). We have, pτ |Xi = 0 if and only if i ≥ τ , i.e., if and only if id−t+1 ≥ d + 1 (cf. Remark 5.2.2.3), i.e., id−t+1 ≤ d. Now it is easily checked that φ is the largest d-tuple (j1 , · · · , jd ) such that jd−t+1 ≤ d. Remark 5.2.2.5. Let φ be as in Lemma 5.2.2.4. As observed in the proof of Lemma 5.2.2.4, we have φ is the largest d-tuple (j1 , · · · , jd ) such that jd−t+1 ≤ d. Corollary 5.2.2.6 Let i ∈ Zt . Then pi |Xφ = 0, φ being as in Lemma 5.2.2.4. Proof. We have (cf. Lemma 5.2.2.4), pτ |Xφ = 0, τ being as in Lemma 5.2.2.1 and hence φ ≥ τ which in turn implies (in view of Lemma 5.2.2.1), φ ≥ i. Hence we obtain pi |Xφ = 0. Theorem 5.2.2.7 Let φ be as in Lemma 5.2.2.4, and let Yφ = O− ∩Xφ (the opposite cell in Xφ ). Then Dt ∼ = Yφ .

5.2 Determinantal varieties

51

Proof. Let Iφ be the ideal defining Yφ in O− . We have (cf. §5.1.3), Iφ is generated by Mt := {pi , i ≤ φ}. Also, I (Dt ) (the ideal defining Dt in O− ) is generated by {pi , i ∈ Zt }. Let i ∈ Zt , say, i = (i1 , · · · , id ). We have (cf. Lemma 5.2.2.4) i ≤ φ, and hence pi ∈ Mt . Let now pi ∈ Mt , say, i = (i1 , · · · , id ). This implies that id−t+1 ≥ d + 1 (cf. Remark 5.2.2.5), and hence it corresponds to a s-minor in X, where s ≥ t. From this it follows that pi ∈ the ideal generated by {pi , i ∈ Zt }, i.e., pi belongs to I (Dt ). Thus we have shown Iφ = I (Dt ) and the result follows from this. Corollary 5.2.2.8 Dt is normal, Cohen-Macaulay of dimension (t − 1)(n − (t − 1)), where note that n = r + d. Proof. The normality and Cohen-Macaulayness of Schubert varieties (cf. Chapter 4) imply corresponding properties for Dt (in view of Theorem 5.2.2.7). Regarding dim Dt , we have dim Dt = dimXφ = (t − 1)(n − (t − 1)) (recall from Chapter 4 that if w = (a1 , · · · , ad ) ∈ Id,n , then dim X(w) = i=d i=1 (ai − i)). 5.2.3 The bijection θ Let n = r + d. Given i ∈ Id,n , let m be such that im ≤ d, im+1 > d. Set Ai = {n + 1 − id , n + 1 − id−1 , · · · , n + 1 − im+1 }, Bi = the complement of {i1 , i2 · · · , im } in {1, 2, · · · , d}. Define θ : Id,n \ {(12 · · · d)} → {all minors of X} by setting θ (i) = [Ai |Bi ]. Clearly, θ is a bijection. 5.2.4 The partial order  Define a partial order on the set of all minors of X as follows: Let [A|B], [A |B  ] be two minors of size s, s  respectively. Let A = {a1 , · · · , as }, B = {b1 , · · · , bs }, A = {a1 , · · · , as }, B = {b1 , · · · , bs }. We define [A|B]  [A |B  ] if s ≤ s  , aj ≥ aj , bj ≥ bj , 1 ≤ j ≤ s. Note that the bijection θ reverses the respective partial orders, i.e., given i, i  ∈ Id,n , we have, i ≤ i  ⇐⇒ θ (i)  θ (i  ). Using this partial order, we define standard monomials in [A, B]’s: Definition 5.2.4.1 A monomial [A1 , B1 ] · · · [As , Bs ], s ∈ N is standard if [A1 , B1 ]  · · ·  [As , Bs ]. In view of Theorem 5.1.0.1 and Theorem 5.2.2.7, we obtain Theorem 5.2.4.2 Standard monomials in [A, B]’s with # A ≤ t − 1 form a basis for K[Dt ], the algebra of regular functions on Dt .

52

5 Determinantal varieties

Proof. Theorem 5.2.2.7 identifies Dt as the principal open set X(φ)pid ; hence we get an identification K[Dt ] ∼ = K[X(φ)](pid ) where K[X(φ)](pid ) denotes the homogeneous localization; note that pid is a lowest weight vector in H 0 (Gd,n , L(ωd )). The fact that standard monomials {pτ1 · · · pτm , m ∈ N, φ ≥ τ1 } generate K[X(φ)] implies that {

p(τ1 ) p(τm ) ··· , m ∈ N, φ ≥ τ1 } pid pid

generates K[Dt ]. Note that the condition that φ ≥ τ corresponds to the condition that the minor corresponding to τ has size ≤ t − 1 (note that under the order-reversing bijection θ , φ corresponds to the pair ((1, · · · , t − 1), (1, · · · , t − 1))). The linear independence of standard monomials {pτ1 · · · pτm , m ∈ N, φ ≥ τ1 } clearly implies the p(τm ) 1) linear independence of { p(τ pid · · · pid , m ∈ N, φ ≥ τ1 }. Further, clearly pτ1 · · · pτm is standard if and only if

p(τ1 ) pid

m) · · · p(τ pid is. The result now follows from this.

In particular, taking t = 1 + min{m, n}, we recover the result due to DoubiletRota-Stein ([26], Theorem 2): Theorem 5.2.4.3 Standard monomials in [A, B]’s form a basis for K[Z](∼ = K[xij , 1 ≤ i ≤ m, 1 ≤ j ≤ n]). 5.2.5 Cogeneration of an ideal As above, let X be a matrix of variables of size r × d, and let ≥ be a partial order on the set of all minors of X. Definition 5.2.5.1 Given a minor M of X, the ideal IM of K[xij , 1 ≤ i ≤ r, 1 ≤ j ≤ d] generated by {M  | M  a minor of X such that M  ≥ M} is called the ideal cogenerated by M (for the given partial order). Proposition 5.2.5.2 Given a minor M of X, let τM be the element of Id,n such that θ (τM ) = M. Then IM = I (YτM ), where YτM = Od− ∩ XPd (τM ) and I (YτM ) is the ideal defining YτM as a closed subvariety of Od− . Proof. We have (cf. Theorem 5.2.2.7), I (YτM ) is generated by {pi | i ∈ Id,n , | τM ≥ i}. The required result follows from this in view of the bijection θ . Corollary 5.2.5.3 {IM , M a minor of X} is precisely the set of ideals of the opposite cells in Schubert varieties in Gd,n . Proof. The result follows in view of Proposition 5.2.5.2 and the bijection θ (cf. §5.2.3). Corollary 5.2.5.4 Let M be the (t −1)-minor of X consisting of the first (t −1)-rows of X and the first (t − 1)-columns of X. Then It (X) (= the ideal of Dt ) equals IM . Proof. We have (cf. Theorem 5.2.2.7), It (X) = I (Yφ ), φ being as in Theorem 5.2.2.7. Now, under the bijection θ , we have θ (φ) = [{1, 2, · · · , t − 1} | {1, 2, · · · , t − 1}]. The result now follows from Proposition 5.2.5.2.

5.2 Determinantal varieties

53

5.2.6 The monomial order ≺ and Gröbner bases We introduce a total order on the variables as follows: xm1 > xm2 > · · · > xmn > xm−1 1 > xm−1 2 > · · · > xm−1 n > . . . > x11 > x12 > · · · > x1n . This induces a total order, namely the lexicographic order, on the set of monomials in K[X] = K[x11 , . . . , xmn ], denoted by ≺. The largest monomial (with respect to ≺) present in a polynomial f ∈ K[X] is called the initial term of f , and is denoted by in(f ). Note that the initial term (with respect to ≺) of a minor of X is equal to the product of its elements on the skew diagonal. Given an ideal I ⊂ K[X], denote by in(I ) the ideal generated by the initial terms of the elements in I . A set G ⊂ I is called a Gröbner basis of I (with respect to the monomial order ≺), if in(I ) is generated by the initial terms of the elements in G. Note that a Gröbner basis of I generates I as an ideal. We recall the following (cf. [38]) Theorem 5.2.6.1 Let M = [i1 , . . . , ir |j1 , . . . , jr ] be a minor of X, and I the ideal of K[X] cogenerated by M. For 1 ≤ s ≤ r + 1, let Gs be the set of all s-minors [i1 , . . . , is |j1 , . . . , js ] satisfying the conditions  ≤ ir−1 , . . . , i2 ≤ ir−s+2 , is ≤ ir , is−1  js−1

if s ≤ r, then i1 >

≥ js−1 , . . . , j2 ir−s+1 or js < js .

≥

j2 , j1

(1)

≥ j1 (2)

Then the set G = ∪r+1 i=1 Gi is a Gröbner basis for the ideal I with respect to the monomial order ≺. Theorem 5.2.6.2 Let t, 1 ≤ t ≤ min {m, n}. Let Ft = {M  | M  is a minor of X of size t}. Then Ft is a Gröbner basis for It (X). Proof. We have Dt is generated by Ft . Let M be the (t − 1)-minor [A|B] of X, where A = {m + 2 − t, m + 3 − t, · · · , m}, B = {1, 2, · · · , t − 1}. Let us write A = {i1 , · · · , ir }, B = {j1 , · · · , jr }. We have r = t − 1, il = m + l − r, jl = l, 1 ≤ l ≤ r. Let M  ∈ G, G being as in Theorem 5.2.6.1, say, M  = [i1 , . . . , is |j1 , . . . , js ] be a minor of X of size s, s ≤ r + 1 (= t).The inequalities regarding i’s and i  ’s in condition (1) of Theorem 5.2.6.1 are redundant, since il = m + l − r, 1 ≤ l ≤ r. Similarly, the inequalities regarding j ’s and j  ’s in condition (1) of Theorem 5.2.6.1 are again redundant (since jl = l, 1 ≤ l ≤ r; also, condition (2) reduces to the condition that if s ≤ r(= t − 1), then i1 > ir−s+1 (since js = s). Therefore, for the above choice of M, the conditions (1) and (2) of Theorem 5.2.6.1 are equivalent to if s ≤ r (= t − 1), then i1 > ir−s+1 (= it−s ). Let s ≤ t − 1. The condition that i1 > it−s (= m + 1 − s) implies is > m + 1, which is not possible. Hence Gs = ∅, if s ≤ t − 1; thus we obtain G = Gt . Now for

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s = t, as discussed above, condition (1) Theorem 5.2.6.1 is satisfied by any t-minor M  while condition (2) is vacuous. Hence we obtain Gt is the set of all t-minors, i.e., G = Ft . The result now follows from Theorem 5.2.6.1. We conclude this chapter with the following Theorem 5.2.6.3 The singular locus of Dt is Dt−1 . This theorem will be proved in Chapter 13 (cf. Theorem 13.3.0.3) as an application of SMT.

6 Symplectic Grassmannian

In this chapter we give an exposition of the determinantal varieties in Sym Mn (the space of symmetric n×n matrices) from the view-point of their relationship to Schubert varieties in the symplectic (or also Lagrangian) Grassmannian varieties. After reviewing some basic algebraic group-theoretic results on symplectic groups, we establish the identification of Sym Mn with the “opposite cell” in a suitable symplectic Grassmannian, which then yields an identification of a determinantal variety in Sym Mn with the “opposite cell” in a suitable Schubert variety in the symplectic Grassmannian (cf. [67]). Using the standard monomial basis for the homogeneous co-ordinate ring of a Schubert variety in the symplectic Grassmannian, we obtain a standard monomial basis for the ring of regular functions on a determinantal variety in Sym Mn . This basis in fact coincides with that of DeConciniProcesi (cf. [22]). Thus Schubert-variety-theoretic approach gives a different proof of DeConcini-Procesi’s basis for these varieties. Let V = K 2n together with a non-degenerate, skew-symmetric bilinear form (·, ·). Let H = SL(V ) and G = Sp(V ) = {A ∈ SL(V ) | A leaves the form (·, ·) invariant }. Taking the matrix of the form (with respect to the standard basis {e1 , ..., e2n } of V ) to be   0 J E= −J 0 where J is the anti-diagonal (1, . . . , 1) of size n × n, we may realize Sp(V ) as the fixed point set of a certain involution σ on SL(V ), namely G = H σ , where t −1 E −1 . Thus σ : H −→ H is given by σ (A) = E(A) t G = Sp(2n) = {A ∈ SL(2n) | AEA = E} t −1 = {A ∈ SL(2n) | E −1 (A) E = A} t −1 −1 = {A ∈ SL(2n) | E(A) E = A}

= Hσ. (Note that E −1 = −E). Denoting by TH (resp. BH ) the maximal torus in H consisting of diagonal matrices (resp. the Borel subgroup in H consisting of upper

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6 Symplectic Grassmannian

triangular matrices) we see easily that TH , BH are stable under σ . We set TG = σ . Then it can be seen easily that T is a maximal torus in G and B THσ , BG = BH G G is a Borel subgroup in G; in particular, TG consists of diagonal matrices of the form diag (t1 , · · · , tn , tn−1 , · · · , t1−1 ). We note the following specific facts for this group. See [67] for details.

6.1 Some basic facts on Sp(V ) Let NG (resp. NH ) denote the normalizer in G (resp. H ) of TG (resp. TH ). We have, NG ⊂ NH ; further, NH is stable under σ , and we have NG = NHσ NG /TG → NH /TH Thus we obtain WG → WH where WG , WH denote the Weyl groups of G, H respectively (with respect to TG , TH respectively). Further, σ induces an involution on WH : w = (a1 , · · · , a2n ) ∈ WH , σ (w) = (c1 , · · · , c2n ), ci = 2n + 1 − a2n+1−i and WG = WHσ Thus we obtain WG = {(a1 · · · a2n ) ∈ S2n | ai = 2n + 1 − a2n+1−i , 1 ≤ i ≤ 2n}. (here, S2n is the symmetric group on 2n letters). Thus w = (a1 · · · a2n ) ∈ WG is known once (a1 · · · an ) is known. We shall denote an element (a1 · · · a2n ) in WG by just (a1 · · · an ). For example, (4231) ∈ S4 represents (42) ∈ WG , G = Sp(4). Root system of type Cn : σ induces an involution on X(TH ), the character group of TH : χ ∈ X(TH ), σ (χ)(D) = χ(σ (D)), D ∈ TH Let i , 1 ≤ i ≤ 2n be the character in X(TH ), i (D) = di , the i-th entry in D(∈ TH ). We have σ (i ) = −2n+1−i Now it is easily seen that under the canonical surjective map ϕ : X(TH ) → X(TG ) we have ϕ(i ) = −ϕ(2n+1−i ), 1 ≤ i ≤ 2n Let RH := {i − j , 1 ≤ i, j ≤ 2n}, the root system of G (relative to TH ), and + := {i − j , 1 ≤ i < j ≤ 2n}, the set of positive roots (relative to BH ). We have RH the following:

6.1 Some basic facts on Sp(V )

1. 2. 3. 4.

57

+ σ leaves RH (resp. RH ) stable. For α, β ∈ RH , ϕ(α) = ϕ(β) ⇔ α = σ (β). ϕ is equivariant for the canonical action of WG on X(TH ), X(TG ). σ = {±( −  RH i 2n+1−i ), 1 ≤ i ≤ n}.

+ Let RG (resp. RG ) the set of roots of G with respect to TG (resp. the set of positive roots with respect to BG ). Using the above facts and the explicit nature of the adjoint representation of G on Lie G, we deduce that + + = ϕ(RH ) RG = ϕ(RH ), RG + + ) gets identified with the orbit space of RH (resp. RH ) In particular, RG (resp. RG modulo the action of σ . Thus we obtain the following identification:

RG = {±(i ± j ), 1 ≤ i < j ≤ n} ∪ {±2i , i = 1, ..., n} + = {(i ± j ), 1 ≤ i < j ≤ n} ∪ {2i , i = 1, ..., n}. RG + The set SG of simple roots in RG is given by

SG := {αi = i − i+1 , 1 ≤ i ≤ n − 1} ∪ {αn = 2n }. Let us denote the simple reflections in WG by {si , 1 ≤ i ≤ n}, namely, si = reflection with respect to i − i+1 , 1 ≤ i ≤ n − 1, and sn = reflection with respect to 2n . Then we have (cf. [8])  ri r2n−i , if 1 ≤ i ≤ n − 1 si = if i = n rn , where ri denotes the transposition (i, i + 1) in S2n , 1 ≤ i ≤ 2n − 1. Continuing the example above, we have, (42) = s1 s2 s1 . Chevalley basis. For 1 ≤ i ≤ 2n, set i  = 2n + 1 − i. The involution σ : SL(2n) → SL(2n), A → E(t A)−1 E −1 , induces an involution σ : sl(2n) → sl(2n), A → −E(t A)E −1 (= E(t A)E, since E −1 = −E). In particular, we have, for 1 ≤ i, j ≤ 2n  −Ej  i  , if i, j are both ≤ n or both > n σ (Eij ) = if one of {i, j } is ≤ n and the other > n Ej  i  , where Eij is the elementary matrix with 1 at the (i, j )-th place and 0 elsewhere. Further Lie Sp(2n) = {A ∈ sl(2n) | E(t A)E = A}. The Chevalley basis {Hαi : αi ∈ SG } ∪ {Xα : α ∈ RG } for Lie Sp(2n) may be given as follows:

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6 Symplectic Grassmannian

Hi −i+1 = Eii − Ei+1,i+1 + E(i+1) ,(i+1) − Ei  i  H2n = Enn − En n Xj −k = Ej k − Ek  j  Xj +k = Ej k  + Ekj  X2m = Emm X−(j −k ) = Ekj − Ej  k  X−(j +k ) = Ek  j + Ej  k X−2m = Em m . Length Formula: For w ∈ WG , let us denote l(w, WH ) (resp. l(w, WG )), the length of w as an element of WH (resp. WG ). For w = (a1 , · · · , a2n ) ∈ WH , denote mw := #{i ≤ n | ai > n} Proposition 6.1.0.1 For w = (a1 , · · · , a2n ) ∈ WG , we have l(w, WG ) =

1 (l(w, WH ) + mw ) 2

Proof. Set + + SH (w) = {β ∈ RH | w(β) < 0}, SG (w) = {β ∈ RG | w(β) < 0}

We have l(w, WH ) = #SH (w), l(w, WG ) = #SG (w) The canonical map ϕ : X(TH ) → X(TG ) induces a surjective map RH → RG . Further, we have, for α ∈ RH , α > 0 ⇔ ϕ(α) > 0 Hence ϕ induces a surjective map SH (w) → SG (w) Further, σ leaves SH (w) stable, and SH (w)σ = {α | α = i − 2n+1−i , 1 ≤ i ≤ n, w(α) < 0 in RH } Now if α = i − 2n+1−i , then w(α) = ai − a2n+1−i . Hence w(α) < 0 in RH ⇔ a2n+1−i < ai , i ≤ n But now a2n+1−i = 2n + 1 − ai (since w ∈ WG ). Hence w(α) < 0 in RH ⇔ n < ai , 1 ≤ i ≤ n From this, we obtain #SH (w)σ = m(w) Also, we have, #SH (w) = 2#SG (w) − #SH (w)σ The required result now follows from this.

6.1 Some basic facts on Sp(V )

59

6.1.1 Schubert varieties in G/BG For w ∈ WH , let CH (w) be the Schubert cell BH wBH (mod BH ) in H /BH ; if w ∈ WG , we shall denote by CG (w) the Schubert cell BG wBG (mod BG ) in G/BG . We observe that σ induces a natural involution on H /BH . Proposition 6.1.1.1 Let w ∈ WG . The Schubert cell CH (w) is stable under σ , and CH (w)σ = CG (w), w ∈ WG . u denote the unipotent part of B . Clearly, B u is Proof. The first part is clear. Let BH H H u at the point σ -stable (since BH is σ -stable). Let B1 be the isotropy subgroup of BH wH (∈ CH (w)); then B1 is stable under σ (since σ (w) = w). We have  u = Uα (∗) BH + α∈RH

(here, Uα denotes the root subgroup of H associated to α). We see easily  B1 = Uα + {α∈RH , w −1 (α)>0}

Let B2 =



Uα

+ {α∈RH , w −1 (α)<0}

u , B are σ -stable implies that B is σ -stable. Then (*) together with the facts that BH 1 2 Let x ∈ CH (w); then x has a presentation x = bw for a unique b ∈ B2 . Hence x ∈ CH (w)σ ⇔ σ (b) = b ⇔ b ∈ BG ; and therefore x ∈ CH (w)σ ⇔ x ∈ CG (w).

Maximal parabolics. For 1 ≤ d ≤ n, we let Pd be the maximal parabolic subgroup of G with SG \ {αd } as the associated set of simple roots. Then it can be seen easily that WGPd , the set of minimal representatives of WG /WPd can be identified with the set of all (a1 · · · ad ) satisfying (1) and (2) below: (1) 1 ≤ a1 < a2 < · · · < ad ≤ 2n (2) for 1 ≤ i ≤ 2n, if i ∈ {a1 , ..., ad } then 2n + 1 − i ∈ {a1 , ..., ad } Bruhat–Chevalley order. For w1 = (a1 · · · a2n ) ∈ WG and w2 = (b1 · · · b2n ), w1 , w2 we have w2 ≥ w1 ⇔ {b1 , ..., bd } ↑ ≥ {a1 , ..., ad } ↑ for each 1 ≤ d ≤ n (cf. [102]). Here {a1 , . . . , ad }↑, {b1 , . . . , bd }↑ are the corresponding d-tuples arranged in ascending order. Hence for w ∈ WG , denoting by w (d) the element in WGPd which represents the coset wWPd , we have for w1 , w2 ∈ WG and 1 ≤ d ≤ n, (d)

(d)

w2 ≥ w1 , 1 ≤ d ≤ n ⇐⇒ {b1 , . . . , bd } ↑ ≥ {a1 , . . . , ad } ↑ .

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6 Symplectic Grassmannian (d)

(d)

Further, w2 ≥ w1 ⇐⇒ w2 ≥ w1 , 1 ≤ d ≤ n. But now, the latter condition is equivalent to w2 ≥ w1 in WH . Thus we obtain that the partial order on WG is induced by the partial order on WH (cf. [102]). In particular, for w1 = (a1 · · · ad ), w2 = (b1 · · · bd ), w1 , w2 ∈ WGPd , we have w2 ≥ w1 ⇔ {b1 , . . . , bd } ↑ ≥ {a1 , . . . , ad } ↑. In the sequel, we shall denote an element (a1 · · · an ) in WGPd by just (a1 · · · ad ). The fact that the partial order on WG is induced by the partial order on WH together with Proposition 6.1.1.1 yields the following: Proposition 6.1.1.2 Let w ∈ WG ; let XG (w) (resp. XH (w)) be the associated Schubert variety in G/BG (resp. H /BH ). Under the canonical inclusion G/BG → H /BH , we have XG (w) = XH (w) ∩ G/BG . Further, the intersection is schemetheoretic. Proof. Denote Z := XH (w) ∩ G/BG . We have XG (w) ⊆ XH (w) ∩ G/BG (clearly), and is in fact an irreducible component of Z (in view of Proposition 6.1.1.1). Let Z  be another irreducible component of Z. Then Z  = XG (w  ), for some w  ∈ WG (Z being closed and BG -stable in G/BG , is a union of Schubert varieties in G/BG ). Now the inclusion XG (w  ) ⊂ Z implies in particular that CG (w  )(= CH (w  )σ ) is contained in Z. Hence we obtain that CH (w  )σ ⊆ XH (w) ∩ CH (w  ); this implies in particular that CH (w  ) ⊆ XH (w) (since XH (w) ∩ CH (w  ) is non-empty, and XH is BH -stable), and hence we obtain that w  ≤ w in WH , and hence also in WG (in view of the fact that the partial order on WG is induced by the partial order on WH ). Thus we get that Z  (= XG (w  )) ⊂ XG (w). The first part of the Lemma follows from this. The second part follows from the fact (cf. Chapter 13, Theorem 13.2.1.10) that for any τ ∈ WG , τ ≤ w, we have that the tangent space to XH (w) at the T -fixed point τ BH is σ -stable, and its subspace of σ -fixed points is the tangent space to XG (w) at the T -fixed point τ BG .

6.2 The variety G/Pn Let Qn denote the maximal parabolic subgroup of H associated to the simple root n − n+1 . As above, let Pn denote the maximal parabolic subgroup of G associated to the simple root 2n . It is easily seen that Qn is σ -stable so that σ induces an involution on H /Qn . Let Z be the subgroup of H consisting of matrices of the form   I dn 0 , Y ∈ Mn Y I dn (here, Mn denotes the n × n matrices with entries in K). The canonical morphism H → H /Qn induces a morphism ψH : Z → H /Qn We have (cf. Chapter 5) Fact: ψH is an open immersion, and ψH (Z) gets identified with the opposite big cell − in H /Qn . OH

6.2 The variety G/Pn

61

− − σ Lemma 6.2.0.1 OH is σ - stable, and we have an identification Sym Mn ∼ ) . = (OH   I dn 0 ∈ Z, we have, Proof. For z = Y I dn   I dn 0 σ (z) = J t Y J I dn

(where recall that J is the anti-diagonal (1, · · · , 1) of size n × n). Hence, Z is σ -stable, and the first part of the Lemma follows (in view of Fact above). We have, Z σ = {z ∈ Z | J t Y J = Y } Letting Y = J X, we have J t Y J = Y ⇔t X = X The second part of Lemma follows from this. Consider H /Qn as the Grassmannian of n-dimensional subspaces of V := K 2n . It is well-known that the set of maximal totally isotropic subspaces (for the skewsymmetric form (, ) on V ) is a closed subvariety of H /Qn isomorphic to G/Pn , the symplectic (or also the Lagrangian) Grassmannian. Lemma 6.2.0.2 ψH induces an inclusion ψH : Z σ → G/Pn .   I dn 0 considered as a point of H /Qn (= the Grassmannian Proof. Now z = Y I dn of n dimensional subspacesof K2n ) corresponds to the n-dimensional subspace I dn spanned by the columns of . Let us denote this n-dimensional subspace by Y Uz . We have,     0 J  I dn t t =0 J Y J = Y ⇔ I dn Y Y −J 0 Hence we obtain that z ∈ Z σ ⇔ Uz is a maximal totally isotropic subspace. The required result follows from this.  Let usdenote the restriction of the morphism ψH : Z → H /Qn , Y → I dn 0 − to Sym Mn by ψG . Let OG the opposite big cell in G/Pn . J Y I dn − 6.2.1 Identification of Sym Mn with OG

Let Ln denote the tautological line bundle on H /Qn (= Gn,2n ) for the Plücker embedding. Then H 0 (H /Qn , Ln ) is the dual Weyl H -module with highest weight ωn . Proposition 6.2.1.1 Let I d = (1, · · · , n)(∈ In,2n ); denote f := pI d , the Plücker co-ordinate corresponding to I d. We have

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1. f is a lowest weight vector in H 0 (H /Qn , Ln ). − 2. OH is the principal open set (H /Qn )f . Proof. (1). We have H 0 (H /Qn , Ln ) = (Λn (K 2n ))∗ Further, for i = (i1 , · · · , in ) ∈ In,2n , pi = ei∗1 ∧ · · · ∧ ei∗n (here {et∗ , 1 ≤ t ≤ 2n} denotes the basis of V ∗ dual to the standard basis {et , 1 ≤ t ≤ 2n} of V = K 2n ). Hence it follows that pi is a TH -weight vector of weight −(i1 + · · · + in ). Now (1) follows from this. (2) follows from the fact that the opposite big cell is precisely the set of points in H /Qn where a lowest weight vector (in H 0 (H /Qn , Ln )) does not vanish. 6.2.2 Canonical dual pair Let Y ∈ Mn . Given two s-tuples A := (a1 , · · · , as ), B := (b1 , · · · , bs ), in Is,n , s ≤ n, let us denote by p(A, B)(Y ), the s-minor of Y with row and column indices given by A, B respectively. Consider the identification (∗)

Mn ∼ =



  I dn , Y ∈ Mn JY

where J is as above. Let i := (i1 , · · · , in ) ∈ In,2n , and let pi be the associated Plücker co-ordinate on Gn,2n (= H /Qn ). Let fi denote the restriction of pi to Mn (under the identification (*)). Then fi = p(i(A), i(B)), where i(A), i(B) are given as follows: Let r be such that ir ≤ n, ir+1 > n; let s = n − r. Then i(A) is the s-tuple given by (2n + 1 − in , · · · , 2n + 1 − ir+1 ), while i(B) is the s-tuple given by the complement of (i1 , · · · , ir ) in (1, · · · , n). We refer to (i(A), i(B)) as the canonical dual pair associated to i.

6.2.3 The bijection θ Define a partial order ≥ on the set of minors of Y ∈ Mn as follows: Given A, B, A , B  where A, B ∈ Is,n , and A , B  ∈ Is  ,n , p(A, B) > p(A , B  ) if s ≤ s  , and at ≥ at , bt ≥ bt , 1 ≤ t ≤ s. Then the map i → (i(A), i(B)) defines an order-reversing bijection θ between In,2n and minors of Y . In the above bijection, the n-tuple (1, · · · , n) will correspond to the constant function 1 (corresponding to the minor with the set of row (resp. column) indices being the empty set).

6.2 The variety G/Pn

63

6.2.4 The dual Weyl G-module with highest weight ωn Let Ln be the restriction of Ln to G/Pn . For i ∈ In,2n , let pi be the restriction of pi to G/Pn ; note that pi ∈ H 0 (G/Pn , Ln ). Let p(i(A), i(B)) be the restric− tion of pi to Mn , Mn being identified with the opposite cell OH as above (cf. (*));  let p (i(A), i(B)) be the restriction of p(i(A), i(B)) to Sym Mn . Let f be a lowest weight vector in H 0 (H /Qn , Ln ), and f  the restriction of f to G/Pn . Proposition 6.2.4.1 H 0 (G/Pn , Ln ) is the dual Weyl G-module with highest weight ωn . Proof. We first observe that pi , i ∈ In,2n is non-zero; this follows from the fact that p  (i(A), i(B)) is non-zero on Sym Mn (Sym Mn = Z σ being identified with an open subset of G/Pn ). Now f being a lowest weight vector in H 0 (H /Qn , Ln ) − − -stable, where BH is (cf. Proposition 6.2.1.1), the one-dimensional span Kf is BH − the Borel subgroup of H opposite to BH . Also, one sees easily that BH is σ -stable, − σ − − ) (= G ∩ BH ) is precisely BG , the Borel subgroup of G opposite to BG . and (BH − implies that f  is a lowest This fact together with the fact that Kf is stabilized by BG 0  weight vector in H (G/Pn , Ln ); further, weight of f  (= weight of f ) = −(1 + · · · + n ) = −ωn Hence we obtain that a highest weight vector in H 0 (G/Pn , Ln ) has weight w0 (G)(−ωn ) (note that the Weyl involution i = −w0 (G) is I d on X(TG )—here, w0 (G) is the element of largest length in WG ). Hence we obtain that Ln is the ample generator of Pic G/Pn so that H 0 (G/Pn , Ln ) is the fundamental representation of G with highest weight ωn . The result now follows from this. As an immediate consequence we obtain − Theorem 6.2.4.2 We have an identification Z σ ∼ . = OG − , we have that Z ∼ Proof. Under the identification ψ : Z ∼ = (H /Qn )f (cf. = OH Proposition 6.2.1.1,(2)); thus Z is precisely the set of points in H /Qn where the lowest weight vector f (∈ H 0 (H /Qn , Ln )) does not vanish. Hence we obtain that

(G/Pn )f  = Z ∩ G/Pn  Id Let z ∈ Z, say z = . Then z ∈ G/Pn if and only if the n-dimensional subspace JY   Id spanned by the columns of is totally isotropic, i.e, if and only if JY     0 J  I dn I dn t Y =0 Y −J 0 

i.e, if and only if J tYJ = Y

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6 Symplectic Grassmannian

Hence, letting Y = J X, we obtain that − OG (= (G/Pn )f  ) = Z ∩ G/Pn = Sym Mn = Z σ

This implies the required result. − − σ = (OH ) = Sym Mn . Corollary 6.2.4.3 OG (Thus the opposite cell in H /Qn induces the opposite cell in G/Pn .) − , and Proof. This follows from the above Theorem, the identification Z ∼ = OH Lemma 6.2.0.1.

Corollary 6.2.4.4 dim G/Pn = 12 n(n + 1). 6.2.5 Identification of Dt (Sym Mn ) with YPn (ϕ) Fix an integer t where 1 ≤ t ≤ n. Let Dt (Sym Mn ) denote the subscheme of Sym Mn defined by the vanishing of t minors in Sym Mn . In this subsection, we shall describe an identification of Dt (Sym Mn ) with the opposite cell YPn (ϕ) in a suitable Schubert variety XPn (ϕ). As a consequence of Proposition 6.1.1.2, we have the following: Proposition 6.2.5.1 Let w ∈ W Pn . Let XPn (w) (resp. XQn (w)) be the associated Schubert variety in G/Pn (resp. H /Qn ). Then under the canonical inclusion G/Pn ⊂ H /Qn , we have, XPn (w) = XQn (w) ∩ G/Pn (scheme-theoretically). Fix a t as above. Let ϕ ∈ W Pn be defined by ϕ = (t, t + 1, · · · , n, 2n + 2 − t, 2n + 3 − t, · · · , 2n) (note that ϕ consists of two blocks [t, n], [2n + 2 − t, 2n] of consecutive integers— − here, for i < j, [i, j ] denotes the set {i, i + 1, · · · , j }). Let YPn (ϕ) := XPn (ϕ) ∩ OG , the opposite cell in XG (ϕ). ∼ O − induces an isomorphism Theorem 6.2.5.2 The isomorphism Sym Mn = G

Dt (Sym Mn ) ∼ = YPn (ϕ)

Proof. We have scheme-theoretic identifications Dt (Sym Mn ) ∼ = Dt (Mn ) ∩ Sym Mn Sym Mn ∼ = O− G

− YPn (ϕ) ∼ = YQn (ϕ) ∩ OG

On the other hand, we have an isomorphism (cf. Chapter 5) Dt (Mn ) ∼ = YQn (ϕ) The required result now follows.

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65

Corollary 6.2.5.3 Dt (Sym Mn ) is normal, Cohen-Macaulay of dimension n(t − 1) − 12 (t − 1)(t − 2). Proof. The normality and Cohen-Macaulayness follow since Schubert varieties in G/Pn are normal and Cohen-Macaulay (cf. §A.12 of Appendix). The assertion on the dimension follows in view of Proposition 6.1.0.1 (note that for w ∈ W P , we have, dim XP (w) = l(w)). 6.2.6 Admissible pairs and canonical pairs Let (τ, ϕ) be an admissible pair in W Pn (see Appendix A.2.4 for the definition of admissible pairs). We have ([56]) that (τ, ϕ) is an admissible pair with XPn (ϕ) being a divisor in XPn (τ ) if and only if there is a pair i, (i + 1) present in ϕ (as an element of In,2n - also recall that for 1 ≤ j ≤ 2n, j  = 2n + 1 − j ), for some i ≤ n, and τ is obtained from ϕ by replacing i, (i + 1) by (i + 1), i  . Let us refer to this as a elementary double move. Hence, (τ, ϕ) is an admissible pair if and only if either τ = ϕ(in which case it is a trivial admissible pair) or τ is obtained from ϕ by a sequence of elementary double moves. From this we obtain the following Lemma 6.2.6.1 Let τ, ϕ ∈ W Pn . Then (τ, ϕ) is a non-trivial admissible pair if and only if the following hold: (i) τ > ϕ (as elements of In,2n ) (ii) mτ > 0 (iii) mτ = mϕ where recall that for θ := (a1 , · · · , a2n ) ∈ In,2n , mθ = #{i ≤ n | ai > n}. 6.2.7 Canonical pairs We now define an injection of In,2n (= W Qn ) into W Pn × W Pn . We first associate to i ∈ In,2n , a pair (i(α), i(β)) to be called the canonical pair as follows: This is defined similar to the canonical dual pair (i(A), i(B)) (cf. §6.2.1). In §6.2.1, we used the identification    I dn , Y ∈ Mn (K) (∗) Mn (K) ∼ = JY where J is the anti-diagonal (1, · · · , 1) of size n × n. If i ∈ In,2n , then the restriction to Mn of the Plücker co-ordinate pi (under the identification (*)) evaluated at Y corresponds to the minor of Y with row and column indices given by i(A), i(B) respectively. To define the canonical pair for an i ∈ In,2n we use the identification:    Y (∗∗) Mn (K) ∼ , Y ∈ Mn (K) = J For i ∈ In,2n , the pair (i(α), i(β)) is defined to be the set of row and column indices of the minor of Y which is the evaluation at Y of the restriction to Mn (K) of the Plücker

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6 Symplectic Grassmannian

co-ordinate pi (under the identification (**)). Let i = (i1 , · · · , in ). Then i(α), i(β) have the following description: Let r be such that ir ≤ n, ir+1 > n. For 1 ≤ j ≤ 2n, let j  = 2n + 1 − j . Then i(α) is the r-tuple (i1 , · · · , ir ), while i(β) is the r-tuple which is the complement  , i  , · · · , i  . Note that the canonical pair (i(α), i(β)), and the in (1, · · · , n) of ir+1 n r+2 canonical dual pair (i(A), i(B)) are related as follows: i(A) = complement of i(β) in (1, · · · , n), i(B) = complement of i(α) in (1, · · · , n) Further we have the map i → (i(α), i(β)) is an order-preserving bijection between In,2n and minors of Y ; while the map i → (i(A), i(B)) is an order-reversing bijection between In,2n and minors of Y (the partial order among the minors of Y being as in §6.2.3). In the above bijections, the n-tuple (1, · · · , n) will correspond to the constant function 1 (corresponding to the minor with the set of row (resp. column) indices being the empty set). 6.2.8 The inclusion η : In,2n → W Pn × W Pn First observe that for τ := (a1 , · · · , ar , b1 , · · · , bs ) ∈ W Pn , where a1 < · · · < ar ≤ n < b1 < · · · < bs , we have, {b1 , · · · , bs } is just the complement in {n + 1, n+2, · · · , 2n} of {a1 , · · · , ar } (arranged in ascending order). Hence, τ is completely determined by (a1 , · · · , ar ). Denoting I :=

∪ Ir,n

1≤r≤n

the map τ → (a1 , · · · , ar ) defines a bijection bij

υ : W Pn → I Set A := {(I, J ) ∈ ∪ Ir,n × Ir,n , | I ≥ J } r≤n

Then the bijection υ induces a bijection: bij

ρ : {admissible pairs in W Pn } → A (cf. Lemma 6.2.6.1; note that for I = J , we obtain W Pn (being identified with trivial admissible pairs)). We shall refer to the elements of A as admissible elements in I × I. Using the bijection υ, we define η to be the injection η : In,2n → W Pn × W Pn , i → (i(α), i(β))

6.2 The variety G/Pn

67

Identifying W Pn with the diagonal set in W Pn × W Pn , we have that η(i) ∈ W Pn if and only if i(α) = i(β). Further, if i(α) = i(β), then η(i) is an admissible pair if and only if the following hold: (i) i(α) ≥ i(β)) (ii) # i(α)(= # i(β)) < n. We shall refer to an i ∈ In,2n such that η(i) ∈ A as an admissible element of In,2n . Let us denote An,2n := {admissible elements in In,2n } Then we have a bijection bij

An,2n → {admissible pairs in W Pn } The comparison order ≥ among admissible pairs in W Pn (namely, (τ, ϕ) ≥ (τ  , ϕ  ) ⇔ ϕ ≥ τ  ) induces a comparison order on An,2n : i ≥ j, if i(β) ≥ j(α) (here, an s-tuple (a1 , · · · , as ) ≥ an t-tuple (b1 , · · · , bt ), if s ≤ t and ar ≥ br , 1 ≤ r ≤ s). 6.2.9 A standard monomial basis for Dt (Sym Mn ) Fix 1 ≤ t ≤ n. For A := (a1 , · · · , as ), B := (b1 , · · · , bs ), in Is,n , s ≤ n, let p  (A, B) denote the restriction of p(A, B)(∈ K[Mn ]) to Sym Mn ; if i is the element of In,2n corresponding to (A, B) under the bijection θ (cf. §6.2.3), we shall denote p  (A, B) also by p  (i). Definition 6.2.9.1 A monomial p  (i1 ) · · · p  (ir ), it ∈ An,2n is standard if i1 ≥ · · · ≥ ir , i.e., if i1 (α) ≥ i1 (β) ≥ i2 (α) ≥ · · · ≥ ir (β). Note that for any t ≤ r, we have, it (α) ≥ it (β) (since it ∈ An,2n ). Using the identification Mn ∼ =

   Y , Y ∈ Mn J

and the isomorphism (cf. Theorem 6.2.5.2) Dt (Sym Mn ) ∼ = YG (ϕ) we obtain Theorem 6.2.9.2 K[Dt (Sym Mn )] has a basis consisting of standard monomials of the form p  (A1 , B1 ) · · · p  (Ar , Br ), A1 ≥ B1 ≥ A2 ≥ · · · ≥ Br , r ∈ N, A1 ≤ (t, t + 1, · · · , n).

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Proof. We have following facts: • K[XPn (ϕ)] has a basis consisting of standard monomials on XPn (ϕ) of the form p(τ1 , δ1 ) · · · p(τr , δr ), r ∈ N, ϕ ≥ τ1 ≥ δ1 ≥ τ2 ≥ · · · ≥ δr , (τi , δi ) being admissible pairs (cf. Chapter 8, Theorem 8.2.0.7). • i → (i(α), i(β)) is an order-preserving bijection between In,2n and minors of (a generic matrix) Y ∈ Mn . Further, under this bijection, the admissible elements of In,2n are mapped bijectively onto admissible pairs in W Pn . bij

• Under the bijection υ : W Pn → I, we have, υ(ϕ) = ([t, n], [2n + 2 − t, 2n]) (cf. Theorem 6.2.5.2). • The canonical pair corresponding to ϕ(= ([t, n], [(t − 1) , 1 ])) is ((t, t + 1, · · · , n), (t, t + 1, · · · , n)). These facts together with Theorem 6.2.5.2 identify K[Dt (Sym Mn )] with K[XG (ϕ)](pid ) (the homogeneous localization of K[XPn (ϕ)] at pid ). Now, the fact that pid ≤ pτ,δ (for all admissible pairs (τ, δ)) implies that p(τ1 , δ1 ) · · · p(τr , δr ), r ∈ N is standard if and only if p(τp1id,δ1 ) · · · p(τprid,δr ) is. Also, the fact that the standard monomials {p(τ1 , δ1 ) · · · p(τr , δr ), ϕ ≥ τ1 , r ∈ N} generate K[XPn (ϕ)] implies that the standard monomials {

p(τ1 , δ1 ) p(τr , δr ) ··· , ϕ ≥ τ1 , r ∈ N} pid pid

generate K[Dt (Sym Mn )](= K[XG (ϕ)](pid ) ); further, the linear independence of the standard monomials {p(τ1 , δ1 ) · · · p(τr , δr ), ϕ ≥ τ1 , r ∈ N} clearly implies the linear independence of the standard monomials { p(τp1id,δ1 ) · · · p(τprid,δr ) , ϕ ≥ τ1 , r ∈ N}. The result now follows from this. 6.2.10 De Concini-Procesi’s basis for Dt (Sym Mn ) In arriving at Theorem 6.2.9.2, we have used the order-preserving bijection between In,2n and minors of Y ∈ Mn (K) given by the canonical pairs of elements of In,2n . If instead we use the order-reversing bijection between In,2n and minors of Y ∈ Mn (K) given by the canonical dual pairs of elements of In,2n , then we obtain De ConciniProcesi’s basis for Dt (Sym Mn ) (cf. [22]) described below. First, we note (or recall) the following: (i) The canonical dual pair of i := ([t, n], [(t − 1) , 1 ])(= ϕ) is given by i(A) = (1, · · · , t − 1), i(B) = (1, · · · , t − 1) (ii) Under the identification Mn ∼ = we have



  I dn , Y ∈ Mn JY

pi (Y ) = p(i(A), i(B))(Y )

6.2 The variety G/Pn

69

(iii) Defining η to be the injection η : In,2n → W Pn × W Pn , i → (i(A), i(B)) we have that η (i) ∈ W Pn if and only if i(A) = i(B). Further, if i(A) = i(B), then η (i) is an admissible pair if and only if the following hold: (1) i(A) ≤ i(B)) (2) # i(α)(= # i(β)) < n. We shall refer to such i’s as dual admissible elements of In,2n , and denote An,2n := {dual admissible elements in In,2n } (iv) The comparison order ≥ among admissible pairs in W Pn induces a comparison order on An,2n : i ≤ j, if i(B) ≤ j(A) Definition 6.2.10.1 A monomial p  (i1 ) · · · p  (ir ), it ∈ An,2n , is standard if i1 ≤ · · · ≤ ir , i.e., if i1 (A) ≤ i1 (B) ≤ i2 (A) ≤ · · · ≤ ir (B). Note that for any t ≤ r, we have, it (A) ≤ it (B) (since it ∈ An,2n ). Theorem 6.2.10.2 K[Dt (Sym Mn )] has a basis consisting of standard monomials of the form p  (A1 , B1 ) · · · p  (Ar , Br ), A1 ≤ B1 ≤ A2 ≤ · · · ≤ Br , r ∈ N, A1 ≥ (1, · · · , t − 1). (Note that the condition that (1, · · · , t − 1) ≤ A ≤ B amounts to the condition that #A(= #B) ≤ t − 1, i.e., p  (A, B)’s are minors of size ≤ t − 1.) Proof. Proof is similar to that of Theorem 6.2.9.2. As in the proof of Theorem 6.2.9.2, we have the following facts: • K[XPn (ϕ)] has a basis consisting of standard monomials on XPn (ϕ) of the form p(τ1 , δ1 ) · · · p(τr , δr ), r ∈ N, ϕ ≥ τ1 ≥ δ1 ≥ τ2 ≥ · · · ≥ δr , (τi , δi ) being admissible pairs. • i → (i(A), i(B)) is an order-reversing bijection between In,2n and minors of (a generic matrix) Y ∈ Mn . Further, under this bijection, the dual admissible elements of In,2n are mapped bijectively onto admissible pairs in W Pn . bij

• Under the bijection υ : W Pn → I, we have, υ(ϕ) = [t, n](= (t, t +1, · · · , n)) (cf. Theorem 6.2.5.2). • The canonical dual pair corresponding to ϕ(= ([t, n], [(t − 1) , 1 ])) is ((1, · · · , t − 1), (1, · · · , t − 1)). These facts together with Theorem 6.2.5.2 imply the required result. Taking t = n + 1, we have Dt (Sym Mn ) equals Sym Mn , and we obtain Theorem 6.2.10.3 K[Sym Mn ] has a basis consisting of standard monomials of the form p  (A1 , B1 ) · · · p  (Ar , Br ), A1 ≤ B1 ≤ A2 ≤ · · · ≤ Br , r ∈ N.

7 Orthogonal Grassmannian

In this chapter we give an exposition of determinantal varieties in Sk Mn (the space of skew symmetric n × n matrices) from the view-point of their relationship to Schubert varieties in the orthogonal Grassmannian varieties (cf. [67]). After reviewing some basic algebraic group-theoretic results on even special orthogonal groups, we establish the identification of Sk Mn with the “opposite cell” in a suitable orthogonal Grassmannian, which then yields an identification of a determinantal variety in Sk Mn with the “opposite cell” in a suitable Schubert variety in the orthogonal Grassmannian. Using the standard monomial basis for the homogeneous co-ordinate ring of a Schubert variety in the orthogonal Grassmannian, we obtain a standard monomial basis for the ring of regular functions on a determinantal variety in Sk Mn . This basis in fact coincides with that of DeConcini-Procesi (cf. [22]). Thus the Schubertvariety-theoretic approach gives a different proof of DeConcini-Procesi’s basis for these varieties.

7.1 The even orthogonal group SO(2n) Let K be an algebraically closed field of characteristic not equal to 2. Let V = K 2n together with a non-degenerate symmetric bilinear form (·, ·). Taking the matrix of the form (·, ·) (with respect to the standard basis {e1 , . . . , e2n } of V ) to be E, the anti-diagonal (1, . . . , 1) of size 2n × 2n. We may realize G = SO(V ) as the fixed point set SL(V )σ , where σ : SL(V ) → SL(V ) is given by σ (A) = E(t A)−1 E. Set H = SL(V ). Denoting by TH (resp. BH ) the maximal torus in H consisting of diagonal matrices (resp. the Borel subgroup in H consisting of upper triangular matrices) we see σ . Then it follows easily that TH , BH are stable under σ . We set TG = THσ , BG = BH that TG is a maximal torus in G and BG is a Borel subgroup in G. Some basic facts on SO(V ): Let NG (resp. NH ) denote the normalizer in G (resp. H ) of TG (resp. TH ). We have, NG ⊂ NH ; further, NH is stable under σ , and we have

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7 Orthogonal Grassmannian

NG = NHσ NG /TG → NH /TH Thus we obtain WG → WH where WG , WH denote the Weyl groups of G, H respectively (with respect to TG , TH respectively). Further, σ induces an involution on WH : w = (a1 , · · · , a2n ) ∈ WH , σ (w) = (c1 , · · · , c2n ), ci = 2n + 1 − a2n+1−i and WG = {w ∈ WHσ | w is an even permutation} Thus we obtain WG = {(a1 · · · a2n ) ∈ S2n | ai = 2n + 1 − a2n+1−i , 1 ≤ i ≤ 2n, and mw is even}. where mw = #{i ≤ n | ai > n}. Thus w = (a1 · · · a2n ) ∈ WG is known once (a1 · · · an ) is known. Root system of type Dn : σ induces an involution on X(TH ), the character group of TH : χ ∈ X(TH ), σ (χ)(D) = χ(σ (D)), D ∈ TH Let i , 1 ≤ i ≤ 2n be the character in X(TH ), i (D) = di , the i-th entry in D(∈ TH ). We have σ (i ) = −2n+1−i Now it is easily seen that the under the canonical surjective map ϕ : X(TH ) → X(TG ) we have ϕ(i ) = −ϕ(2n+1−i ), 1 ≤ i ≤ 2n Let RH := {i − j , 1 ≤ i, j ≤ 2n}, the root system of G (relative to TH ), and + := {i − j , 1 ≤ i < j ≤ 2n}, the set of positive roots (relative to BH ). We have RH the following: 1. 2. 3. 4.

+ ) stable. σ leaves RH (resp. RH For α, β ∈ RH , ϕ(α) = ϕ(β) ⇔ α = σ (β) ϕ is equivariant for the canonical action of WG on X(TH ), X(TG ). σ = {±( −  RH i 2n+1−i ), 1 ≤ i ≤ n}

+ ) the set of roots of G with respect to TG (resp. the set of Let RG (resp. RG positive roots with respect to BG ). Using the above facts and the explicit nature of the adjoint representation of G on Lie G, we deduce that + + + σ σ ), RG = ϕ(RH \ (RH ) ) RG = ϕ(RH \ RH

7.1 The even orthogonal group SO(2n)

73

+ + In particular, RG (resp. RG ) gets identified with the orbit space of RH (resp. RH ) modulo the action of σ minus the fixed point set under σ . Thus we obtain the following identification:

RG = {±(i ± j ), 1 ≤ i < j ≤ n} + RG = {(i ± j ), 1 ≤ i < j ≤ n}. + are given by The simple roots in RG

{αi = i − i+1 , 1 ≤ i ≤ n − 1} ∪ {αn = n−1 + n }. Let us denote the simple reflections in WG by {si , 1 ≤ i ≤ n}, namely, si = reflection with respect to i −i+1 , 1 ≤ i ≤ n−1, and sn = reflection with respect to n−1 +n . Then we have (cf. [8])  ri r2n−i , if 1 ≤ i ≤ n − 1 si = rn−1 rn rn−1 rn rn+1 rn , if i = n where ri denotes the transposition (i, i + 1) in S2n , 1 ≤ i ≤ 2n − 1. Chevalley basis. Recall, for 1 ≤ k ≤ 2n, we set k  = 2n + 1 − k. The involution σ : SL(2n) → SL(2n), A → E(t A)−1 E, induces an involution σ : sl(2n) → sl(2n), A → −E(t A)E. In particular, we have, for 1 ≤ i, j ≤ 2n, σ (Eij ) = −Ej  i  , where Eij is the elementary matrix with 1 at the (i, j )-th place and 0 elsewhere. Further, Lie SO(2n) = {A ∈ sl(2n) | E(t A)E = −A}. The Chevalley basis for Lie SO(2n) may be given as follows: Hi −i+1 = Eii − Ei+1,i+1 + E(i+1) ,(i+1) − Ei  i  Hn−1 +n = En−1,n−1 + En,n − En ,n − E(n−1) ,(n−1) Xj −k = Ej k − Ek  j  Xj +k = Ej k  − Ekj  X−(j −k ) = Ekj − Ej  k  X−(j +k ) = Ek  j − Ej  k . Length Formula: For w ∈ WG , let us denote l(w, WH ) (resp. l(w, WG )), the length of w as an element of WH (resp. WG ). For w = (a1 , · · · , a2n ) ∈ WH , denote mw := #{i ≤ n | ai > n} Proposition 7.1.0.1 For w = (a1 , · · · , a2n ) ∈ WG , we have l(w, WG ) = WH ) − mw ).

1 2 (l(w,

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Proof. Set + + SH (w) = {β ∈ RH | w(β) < 0}, SG (w) = {β ∈ RG | w(β) < 0}

We have l(w, WH ) = #SH (w), l(w, WG ) = #SG (w) σ → The canonical map ϕ : X(TH ) → X(TG ) induces a surjective map ϕ : RH \ RH RG . Further, we have,

for α ∈ RH , α > 0 ⇔ ϕ(α) > 0 Hence ϕ induces a surjective map SH (w) \ SH (w)σ −→ SG (w) (note that σ leaves SH (w) stable), and SH (w)σ = {α | α = i − 2n+1−i , 1 ≤ i ≤ n, w(α) < 0 in RH } If α = i − 2n+1−i , then w(α) = ai − a2n+1−i . Hence w(α) < 0 in RH ⇔ a2n+1−i < ai , i ≤ n But now a2n+1−i = 2n + 1 − ai (since w ∈ WG ). Hence w(α) < 0 in RH ⇔ n < ai , 1 ≤ i ≤ n From this, we obtain #SH (w)σ = m(w) Also, we have, #SH (w) = 2#SG (w) + #SH (w)σ The required result now follows from this. 7.1.1 Schubert varieties in G/BG For w ∈ WH , let CH (w) be the Schubert cell BH wBH (mod BH ) in H /BH ; if w ∈ WG , we shall denote by CG (w) the Schubert cell BG wBG (mod BG ) in G/BG . We observe that σ induces a natural involution on H /BH . Proposition 7.1.1.1 Let w ∈ WG . The Schubert cell CH (w) is stable under σ , and CH (w)σ = CG (w), w ∈ WG . u denote the unipotent part of B . Clearly, B u is Proof. The first part is clear. Let BH H H u at the point σ -stable (since BH is σ -stable). Let B1 be the isotropy subgroup of BH

7.1 The even orthogonal group SO(2n)

75

wH (∈ CH (w)); then B1 is stable under σ (since σ (w) = w). We have  u (∗) BH = Uα + α∈RH

(here, Uα denotes the root subgroup of H associated to α). We see easily  B1 = Uα + {α∈RH , w −1 (α)>0}

Let B2 =



Uα

+ {α∈RH , w −1 (α)<0}

u , B are σ -stable implies that B is σ -stable. Then (*) together with the facts that BH 1 2 Let x ∈ CH (w); then x has a presentation x = bw for a unique b ∈ B2 . Hence x ∈ CH (w)σ ⇔ σ (b) = b ⇔ b ∈ BG ; and therefore x ∈ CH (w)σ ⇔ x ∈ CG (w).

Maximal parabolics. For 1 ≤ d ≤ n, we let Pd be the maximal parabolic subgroup of G with S \ {αd } as the associated set of simple roots. Then it can be seen easily that WGPd , d = n − 1, can be identified with ⎧  ⎫  (1)1 ≤ a1 < a2 < · · · < ad ≤ 2n ⎪ ⎪  ⎨ ⎬  (a1 · · · ad )  (2)for 1 ≤ i ≤ 2n, if i ∈ {a1 , ..., ad } .  ⎪ ⎪ ⎩ ⎭  then 2n + 1 − i ∈ / {a1 , ..., ad } For d = n − 1, if w ∈ WGPd , then w ≡ wui (mod WPn−1 ), 0 ≤ i ≤ n, i = n − 1, where

⎧ ⎪ ⎨sαn , ui = id, ⎪ ⎩ sαi sαi+1 · · · sαn−2 sαn ,

if i = n if i=0 if 1 ≤ i ≤ n − 2.

Note that the set {wui , 0 ≤ i ≤ n, i = n − 1} is totally ordered under the Bruhat order; note also that given w ∈ W , there are n different n − 1-tuples representing the coset wWPd , namely, the tuples given respectively by the first n − 1 entries in wui , 0 ≤ i ≤ n, i = n − 1. For w ∈ W , denoting by w (d) the element of P (n−1) (n−1) WGd representing wWPd , we can have w1 = w2 , with {a1 , . . . , an−1 } ↑ and {b1 , . . . , bn−1 } ↑ being different (here, {i1 , . . . , is } ↑ denotes the entries i1 , . . . , is arranged in ascending order). For w ∈ W , say w = (a1 · · · a2n ), we see easily that w (d) = {a1 , . . . , ad } ↑, 1 ≤ d ≤ n, d = n − 1 and w (n−1) = the least (under ≥) in the totally ordered set Y

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7 Orthogonal Grassmannian

where (i)

(i)

Y = {(y1 , . . . , yn−1 ) ↑ 0 ≤ i ≤ n, i = n − 1}. (i)

(i)

y1 , . . . , yn−1 being the first (n−1) entries in wui , 0 ≤ i ≤ n, i = n−1. (Here, the partial order ≥ is the usual partial order, namely, (i1 , . . . , in−1 ) ≥ (j1 , . . . , jn−1 ), if it ≥ jt , 1 ≤ t ≤ n − 1, where (i1 , . . . , in−1 ), (j1 , . . . , jn−1 ) are two increasing sequences of (n − 1)-tuples.) Bruhat–Chevalley order. For 1 ≤ i ≤ 2n, let i  = 2n + 1 − i, and |i| = min {i, i  }. We shall denote the Bruhat–Chevalley order on WG by . Given w1 = (a1 · · · a2n ), w2 = (b1 · · · b2n ), w1 , w2 ∈ WG , we have w2  w1 if and only if the following two conditions hold [102]. 1. For 1 ≤ d ≤ n, we have {b1 , . . . , bd } ↑ ≥ {a1 , . . . , ad } ↑, for all d. 2. Let (c1 , . . . , cd ) (resp. (e1 , . . . , ed )) be the increasing sequence {a1 , . . . , ad } ↑ (resp. {b1 , . . . , bd } ↑). Suppose for some r, 1 ≤ r ≤ d, and some i, 0 ≤ i ≤ d − r, {|ci+1 |, . . . , |ci+r |} = {|ei+1 |, . . . , |ei+r |} = {n + 1 − r, . . . , n} as sets (order doesn’t matter). Then #{j : i + 1 ≤ j ≤ i + r and cj > n}, and #{j : i + 1 ≤ j ≤ i + r and ej > n} should both be even or both be odd. Remark 7.1.1.2. Thus the Bruhat order  on WG is not induced from the Bruhat order on WH . Following the terminology in [102], we shall refer to {|ci+1 |, . . . , |ci+r |} and {|ei+1 |, . . . , |ei+r |} as analogous parts if they satisfy the hypothesis in Condition 2 above, and if {c1 , . . . , cd } and {e1 , . . . , ed } have analogous parts of the same parity, we say they are D-compatible. Remark 7.1.1.3. (a) Let (c1 , . . . , cd ), (e1 , . . . , ed ) ∈ WGPd , where (c1 , . . . , cd )  {e1 , . . . , ed ). Suppose (c1 , . . . , cd ), (e1 , . . . , ed ) have analogous parts. Then it is easily seen that the condition (2) is equivalent to the condition that #{j, 1 ≤ j ≤ d | cj > n} and #{j, 1 ≤ j ≤ d and ej > n} are both even or both odd. As a consequence, we have the following Proposition 7.1.1.4 The Bruhat-Chevalley order  on W Pn is induced by the BruhatChevalley order  on W Qn ,Pn (resp. Qn ) being the maximal parabolic subgroup of G (resp. H = SL2n (K)) associated to the simple root n−1 + n (resp. n − n+1 ). Proof. Let w1 , w2 ∈ WGPn , and let w1 ≥ w2 (as elements of WH Qn ). Let w1 = (c1 , . . . , cn ), w2 = (e1 , . . . , en ) Then we have (c1 , . . . , cn ) ≥ (e1 , . . . , en ) (since w1 ≥ w2 in WH Qn ). Condition (2) above is trivially satisfied (cf. Remark 7.1.1.3). The result now follows.

7.2 The variety G/Pn

77

7.2 The variety G/Pn As above, let Qn denote the maximal parabolic subgroup of H associated to the simple root n −n+1 , and Pn denote the maximal parabolic subgroup of G associated to the simple root n−1 + n . It is easily seen that Qn is σ -stable so that σ induces an involution on H /Qn . Let Z be the subgroup of H consisting of matrices of the form   I dn 0 , Y ∈ Mn (K) Y I dn (here, Mn (K) denotes the n×n matrices with entries in K). The canonical morphism H → H /Qn induces a morphism ψH : Z → H /Qn Recall (cf. Chapter 5) Fact: ψH is an open immersion, and ψH (Z) gets identified with the opposite big cell − in H /Qn . OH − − σ is σ - stable, and we have an identification Sk Mn ∼ ) , Lemma 7.2.0.1 OH = (OH where Sk Mn is the space of skew symmetric n × n matrices.   I dn 0 ∈ Z, we have, Proof. For z = Y I dn

 σ (z) =

0 I dn −J t Y J I dn



(where recall that J is the anti-diagonal (1, · · · , 1) of size n × n). Hence, Z is σ stable, and the first part of the Lemma follows (in view of Fact above). We have, (†)

Z σ = {z ∈ Z | − J t Y J = Y }

Letting Y = J X, we have −J t Y J = Y ⇔ t X = −X The second part of Lemma follows from this. Consider H /Qn as the Grassmannian of n-dimensional subspaces of V := K 2n . It is well-known that the set of maximal totally isotropic subspaces (for the symmetric form ( , ) on V ) is a closed subvariety of H /Qn isomorphic to G/Pn , the orthogonal Grassmannian. Lemma 7.2.0.2 ψH induces an inclusion ψH : Z σ → G/Pn

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7 Orthogonal Grassmannian

  I dn 0 considered as a point of H /Qn corresponds to the Y I dn   I dn . Let us denote this nn-dimensional subspace spanned by the columns of Y dimensional subspace by Uz . We have, Proof. Now z =

    0J  I dn t −J Y J = Y ⇔ I dn Y =0 Y J 0 t

Hence we obtain in view of (†) in the proof of Lemma 7.2.0.1 that z ∈ Z σ ⇔ Uz is a maximal totally isotropic subspace. The required result follows from this.  Let us denote the restriction of the morphism ψH : Z → H /Qn , Y → I dn 0 − to Sk Mn by ψG . Let OG the opposite big cell in G/Pn . J Y I dn − 7.2.1 Identification of Sk Mn with OG

Let Ln denote the tautological line bundle on H /Qn (= Gn,2n ) for the Plücker embedding. Then H 0 (H /Qn , Ln ) is the dual Weyl H -module with highest weight ωn . Recall (cf. Chapter 5) Proposition 7.2.1.1 Let I d = (1, · · · , n)(∈ In,2n ); denote f := pI d . We have 1. f is a lowest weight vector in H 0 (H /Qn , Ln ). − is the principal open set (H /Qn )f . 2. OH 7.2.2 Canonical dual pair Let Y ∈ Mn . Given two s-tuples A := (a1 , · · · , as ), B := (b1 , · · · , bs ), in Is,n , s ≤ n, let us denote by p(A, B)(Y ), the s-minor of Y with row and column indices given by A, B respectively. Consider the identification (∗)

Mn (K) ∼ =

   I dn − , Y ∈ Mn (K) ∼ = OH JY

where J is as above. Let i := (i1 , · · · , in ) ∈ In,2n , and let pi be the associated Plücker co-ordinate on Gn,2n (= H /Qn ). Let fi denote the restriction of pi to Mn (under the identification (*)). Then fi (Y ) = p(i(A), i(B)), where i(A), i(B) are given as follows: Let r be such that ir ≤ n, ir+1 > n; let s = n − r. Then i(A) is the s-tuple given by (2n + 1 − in , · · · , 2n + 1 − ir+1 ), while i(B) is the s-tuple given by the complement of (i1 , · · · , ir ) in (1, · · · , n). Recall (cf. Chapter 6) that (i(A), i(B)) is the canonical dual pair associated to i.

7.2 The variety G/Pn

79

7.2.3 The bijection θ Define a partial order ≥ on the set of minors of Y ∈ Mn as follows: Given A, B, A , B  where A, B ∈ Is,n , and A , B  ∈ Is  ,n , p(A, B) > p(A , B  ) if s ≤ s  , and at ≥ at , bt ≥ bt , 1 ≤ t ≤ s. Then the map i → (i(A), i(B)) defines an orderreversing bijection θ between In,2n and minors of Mn . In the above bijection, the n-tuple (1, · · · , n) will correspond to the constant function 1 (corresponding to the minor with the set of row (resp. column) indices being the empty set). 7.2.4 The dual Weyl G-module with highest weight ωn Let Ln be the restriction of Ln to G/Pn . For i ∈ In,2n , let pi be the restriction of pi to G/Pn ; note that pi ∈ H 0 (G/Pn , Ln ). Let p(i(A), i(B)) be the restriction − of pi to Mn , Mn being identified with the opposite cell OH as above (cf. (*)); let  p (i(A), i(B)) be the restriction of p(i(A), i(B)) to Sk Mn . Let f = pI d , a lowest weight vector in H 0 (H /Qn , Ln ), and f  the restriction of f to G/Pn . Let Y ∈ Sk Mn . Consider i ∈ In,2n such that i(A) = i(B). Then p  (i(A), i(A))(Y ) is a principal minor of the skew symmetric matrix Y . Hence p  (i(A), i(A))(Y ) is a square; denoting qi (Y ), the corresponding Pfaffian, we obtain a regular function qi : Sk Mn → K Let r = #i(A); then we have that qi is non-zero if and only if r is even (since the determinant of a skew symmetric r × r matrix is zero, if r is odd). Proposition 7.2.4.1 Let notation be as above. We have 1. For i ∈ In,2n such that i(A) = i(B), we have a regular function qi : Sk Mn → K such that qi2 = p  (i(A), i(A)); further, qi is non-zero if and only if #i(A) is even. 2. Let L be the ample generator of Pic G/Pn . Then L2 = Ln . Proof. (1) follows from the discussion above. (2). Let f be a lowest weight vector in H 0 (H /Qn , Ln ); then the one-dimensional − − -stable, where BH is the Borel subgroup of H opposite to BH . Also, span Kf is BH − − σ − − ) (= G∩BH ) is precisely BG , the Borel one sees easily that BH is σ -stable, and (BH subgroup of G opposite to BG . This fact together with the fact that Kf is stabilized − implies that f  is a lowest weight vector in H 0 (G/Pn , Ln ); further, by BG weight of f  (= weight of f ) = −(1 + · · · + n ) = −2ωn (note that ωn = 12 (1 + · · · + n )). Hence we obtain that a highest weight vector in H 0 (G/Pn , Ln ) has weight w0 (G)(−2ωn ) (here, w0 (G) denotes the element of largest length in WG ). On the other hand, H 0 (G/P , L) is the dual Weyl module with highest weight i(ωn )(note that i = −w0 (G), the Weyl involution). These two facts together with (1) imply (2). As an immediate consequence we obtain

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7 Orthogonal Grassmannian

− Theorem 7.2.4.2 We have an identification Z σ ∼ . = OG − , we have that Z ∼ Proof. Under the identification ψ : Z ∼ = (H /Qn )f ; = OH thus Z is precisely the set of points in H /Qn where the lowest weight vector f (∈ H 0 (H /Qn , Ln )) does not vanish. Hence we obtain that

(G/Pn )f  = Z ∩ G/Pn 

 Id . Then z ∈ G/Pn if and only if the n-dimensional subspace JY   Id spanned by the columns of is totally isotropic, i.e, if and only if JY Let z ∈ Z, say z =

    0J  I dn I dn t Y =0 Y J 0 i.e, if and only if −J t Y J = Y Hence, letting Y = J X, we obtain that − (= (G/Pn )f  ) = Z ∩ G/Pn = Sk Mn = Z σ OG

This implies the required result. − − σ Corollary 7.2.4.3 OG = (OH ) = Sk Mn (Thus the opposite cell in H /Qn induces the opposite cell in G/Pn .) − , and Proof. This follows from the above Theorem, the identification Z ∼ = OH Lemma 7.2.0.1.

Corollary 7.2.4.4 dim G/Pn = 12 n(n − 1). 7.2.5 Identification of Dt (Sk Mn ) with YG (ϕ) Fix an odd integer t where 1 ≤ t ≤ n. Let Dt (Sk Mn ) denote the subscheme of Sk Mn defined by the vanishing of t minors in Sk Mn . In this subsection, we shall describe an identification of Dt (Sk Mn ) with the opposite cell YPn (ϕ) in a suitable Schubert variety XPn (ϕ) in G/Pn . Unlike the symplectic group case where we have that for w ∈ WSp(2n) , XSp(2n) (w) = XH (w) ∩ Sp(2n)/B, scheme-theoretically (cf. Chapter 6, Proposition 6.1.1.2), for G = SO(2n), the Schubert variety XG (w), w ∈ WG need not equal XH (w)∩G/BG ; in fact, the intersection XH (w)∩G/BG need not even be irreducible (for details, see [65]). Nevertheless it is true that for w ∈ W Pn , XPn (w) = XQn (w) ∩ G/Pn (scheme-theoretically) as given by the following

7.2 The variety G/Pn

81

Proposition 7.2.5.1 Let w ∈ W Pn . Let XPn (w) (resp. XQn (w)) be the associated Schubert variety in G/Pn (resp. H /Qn ). Then under the canonical inclusion G/Pn ⊂ H /Qn , we have, XPn (w) = XQn (w) ∩ G/Pn (scheme-theoretically). Proof. Denote Z := XQn (w)∩G/Pn . We have XPn (w) ⊆ XQn (w)∩G/Pn (clearly), and is in fact an irreducible component of Z (in view of Proposition 7.1.1.1). Let Z  be another irreducible component of Z. Then Z  = XPn (w  ), for some w  ∈ W Pn (Z being closed and BG -stable in G/Pn , is a union of Schubert varieties in G/Pn ). Now the inclusion XPn (w  ) ⊂ Z implies in particular that CPn (w  )(= CQn (w  )σ ) is contained in Z (here, CPn (w  ) (resp. CQn (w  )) is the Schubert cell in G/Pn (resp. H /Qn ) associated to w  ). Hence we obtain that CQn (w  )σ ⊆ XQn (w) ∩ CQn (w  ); this implies in particular that CQn (w  ) ⊆ XQn (w) (since XQn (w) ∩ CQn (w  ) is non-empty, and XQn is BH -stable), and hence we obtain that w  ≤ w in W Qn , and hence also in W Pn also (in view of the fact that the partial order on W Pn is induced by the partial order on W Qn (cf. Proposition 7.1.1.4)). Thus we get that Z  (= XPn (w  )) ⊂ XPn (w). Hence we obtain that (∗)

XPn (w) = XQn (w) ∩ G/Pn (set-theoretically)

The assertion that (*) is in fact a scheme-theoretic equality follows from the fact that for any τ ∈ W Pn , τ ≤ w, the tangent space to XQn (w) at the T -fixed point τ Qn is σ -stable, and its subspace of σ -fixed points is the tangent space to XPn (w) at the T -fixed point τ Pn (cf. Chapter 13, Theorem 13.2.1.8). Fix an odd integer t ≤ n. Let ϕ ∈ W Pn be defined by ϕ = (t, t + 1, · · · , n, 2n + 2 − t, 2n + 3 − t, · · · , 2n) (note that ϕ consists of two blocks [t, n], [2n + 2 − t, 2n] of consecutive integers here, for i < j, [i, j ] denotes the set {i, i + 1, · · · , j }; also note that mϕ (= t − 1) is − , the opposite cell in XG (ϕ). even). Let YPn (ϕ) := XPn (ϕ) ∩ OG − Theorem 7.2.5.2 The isomorphism Sk Mn ∼ induces an isomorphism = OG

Dt (Sk Mn ) ∼ = YPn (ϕ) Proof. We have scheme-theoretic identifications Dt (Sk Mn ) ∼ = Dt (Mn ) ∩ Sk Mn ∼ O− Sk Mn = G

− YPn (ϕ) ∼ = YQn (ϕ) ∩ OG

On the other hand, we have an isomorphism (cf. Chapter 5) Dt (Mn ) ∼ = YQn (ϕ) The required result now follows.

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7 Orthogonal Grassmannian

Corollary 7.2.5.3 Dt (Sk Mn ) is normal, Cohen-Macaulay of dimension n(t − 1) − 12 t (t − 1). Proof. The normality and Cohen-Macaulayness follow since Schubert varieties in G/Pn are normal and Cohen-Macaulay (cf. §A.12 of Appendix). The assertion on the dimension follows in view of Proposition 7.1.0.1 (note that for w ∈ W P , we have, dim XP (w) = l(w)). The bijection η: First observe that for τ := (a1 , · · · , ar , b1 , · · · , bs ) ∈ W Pn , where a1 < · · · < ar ≤ n < b1 < · · · < bs (s being even), we have, {b1 , · · · , bs } is just the complement in {n + 1, n + 2, · · · , 2n} of {a1 , · · · , ar } (arranged in ascending order). Hence, τ is completely determined by (a1 , · · · , ar ). Denoting Ieven :=

∪

{r|n−r is even}

Ir,n

the map τ → (a1 , · · · , ar ) defines a bijection bij

υ : W Pn → Ieven Set A := {the diagonal of Ieven × Ieven } Then we have an obvious bijection: bij

ρ : W Pn → A We shall refer to the elements of A as admissible elements in Ieven × Ieven . Set    i(A) = i(B) An,2n := i ∈ In,2n  #i(A) is even (here, recall that (i(A), i(B)) is the dual canonical pair associated to i). Using the bijection υ, we define η to be the bijection bij

η : An,2n → A, i → (i(A), i(A)) This defines an order-reversing bijection bij

An,2n → W Pn 7.2.6 A standard monomial basis for Dt (Sk Mn ) Fix an odd integer t, 1 ≤ t ≤ n. For A := (a1 , · · · , as ), s being even, let q(A) denote Pfaffian of the restriction of p(A, A)(∈ K[Mn ]) to Sk Mn ; if i is the element of In,2n with (A, A) as the associated dual canonical pair, then we shall denote q(A) also by q(i).

7.2 The variety G/Pn

83

Definition 7.2.6.1 A monomial q(i1 ) · · · q(ir ), it ∈ An,2n is standard if i1 ≥ · · · ≥ ir , i.e., if i1 (A) ≤ i2 (A) ≤ · · · ≤ ir (A). Theorem 7.2.6.2 Let notation be as above. K[Dt (Sk Mn )] has a basis consisting of standard monomials of the form q(A1 ) · · · q(Ar ), A1 ≤ · · · ≤ Ar , r ∈ N, A1 ≥ (1, · · · , t − 1), Ai ∈ An,2n . (Note that the condition that (1, · · · , t − 1) ≤ A amounts to the condition that #A ≤ t − 1, i.e., q(A)’s are Pfaffians of principal minors of size ≤ t − 1.) Proof. We have the following facts: • K[XPn (ϕ)] has a basis consisting of standard monomials on XPn (ϕ) of the form p(τ1 ) · · · p(τr ), r ∈ N, ϕ ≥ τ1 ≥ · · · ≥ τr , τi ∈ W Pn (cf. Chapter 8, Theorem 8.1.0.2; note that ωn is minuscule). • i → (i(A), i(A)) is an order-reversing bijection between An,2n and Pfaffians of even-sized principal minors of (a generic matrix) Y ∈ Sk Mn . bij

• We have a bijection, An,2n → W Pn . bij

• Under the bijection υ : W Pn → Ieven , we have, υ(ϕ) = [t, n](= (t, t + 1, · · · , n)) (cf. Theorem 6.2.5.2). • The canonical dual pair corresponding to ϕ(= ([t, n], [(t − 1) , 1 ])) is ((1, · · · , t − 1), (1, · · · , t − 1)). These facts together with Theorem 7.2.5.2 imply the required result. Remark 7.2.6.3. The basis described in Theorem 7.2.6.2 is precisely De ConciniProcesi’s basis for K[Dt (Sk Mn )] (cf. [22]). Thus our approach gives a different proof of DeConcini-Procesi’s basis for K[Dt (Sk Mn )]. Taking t > n, we have Dt (Sk Mn ) equals Sk Mn , and we obtain Theorem 7.2.6.4 K[Sk Mn ] has a basis consisting of standard monomials of the form q(A1 ) · · · q(Ar ), A1 ≤ · · · ≤ Ar , r ∈ N, As ∈ An,2n , 1 ≤ s ≤ r.

8 The standard monomial theoretic basis

Standard monomial theory (SMT) for Schubert varieties in the Grassmannian was presented in sections 4.3, 4.4, 4.5 of Chapter 4. Some of its applications, including the proof of normality and Cohen-Macaulayness of Schubert varieties, were presented in section 4.6 of Chapter 4. As we will see in Chapter 10, this is already good enough for deriving the first and second fundamental theorems of classical invariant theory (CIT) for the general linear group (cf. Example A of Chapter 1 and also Theorem 10.2.2.2, Chapter 10). The reason is that the ring generated by the basic invariants for the action of the general linear group is the ring of functions on a determinantal variety, and this variety is the opposite cell of a certain Schubert variety in the Grassmannian—the first of these facts appears in Theorem 10.3.1.1 of Chapter 10, and the second was already proved in Theorem 5.2.2.7 of Chapter 5. The rings generated by the basic invariants in the case of the symplectic and orthogonal group actions (Examples B and C respectively of Chapter 1) are rings of functions on skew-symmetric and symmetric determinantal varieties respectively— these facts will be proved respectively in Theorems 10.5.0.1 and 10.4.0.2 of Chapter 10. And, as was already seen in Theorem 6.2.5.2 of Chapter 6 and Theorem 7.2.5.2 of Chapter 7 respectively, these varieties are the opposite cells of certain Schubert varieties in the orthogonal and symplectic Grassmannians respectively. In order to derive the results of CIT in these cases in a fashion analogous to the case of the general linear group, we need an SMT for Schubert varieties in these Grassmannians. As pointed out in Chapter 1, this line of thought was a motivating factor in the development of SMT. SMT for the orthogonal Grassmannian is due originally to Seshadri [111], and for the symplectic Grassmannian to Lakshmibai-Musili-Seshadri [61, 62] and independently De Concini [19]. We state the theorems respectively in the two sections below. The proofs are postponed to the appendix, for giving them here would divert the flow of the book. The reader is encouraged to accept these statements without proof and go on.

86

8 The standard monomial theoretic basis

8.1 SMT for the even orthogonal Grassmannian The purpose of this section is to state the main theorem of SMT for the even orthogonal Grassmannian. The preparatory material below is dealt with more systematically in Chapter 7. See also Example A.3.2 of the Appendix. Let K be an algebraically closed field of characteristic not equal to 2. Fix a vector space V of finite dimension over K and a non-degenerate symmetric form  ,  on V . We will assume the dimension of V to be even, say 2n, for that is true in the case of the invariant theoretic application we have in mind. A linear subspace of V is said to be isotropic if the form  ,  vanishes identically on it. It is elementary to see that an isotropic subspace of V has dimension at most n and that every isotropic subspace is contained in one of dimension n. Denote by Gn (V ) the Grassmannian of n-dimensional subspaces of V and by I the set of all n-dimensional isotropic subspaces of V . Then I is a closed subvariety of Gn (V ). The orthogonal group O(V ) of linear automorphisms of V preserving  ,  acts transitively on I —this follows from Witt’s theorem that an isometry between subspaces can be lifted to one of the whole vector space. However, the special orthogonal group SO(V ) does not act transitively on I . There are two connected components of I . We take I to be one of the these components (to be specific, the one containing the point corresponding to the span of e1 , . . . , en in the notation below), and call it the even orthogonal Grassmannian. Then SO(V ) acts transitively on I. Using this we can identify I as the quotient of SO(V ) by the stabilizer of any point (for example, the span of e1 , . . . , en in the notation to be defined below)—one needs also to check that the differential at the identity of the orbit map defined by the point is surjective. These stabilizers are (conjugate) maximal parabolic subgroups of SO(V ). Thus, after fixing a Borel subgroup B of SO(V ), we can identify I as the quotient of SO(V ) by a certain maximal parabolic subgroup P containing B: I∼ = G/P , G = SO(V ). Further, the fundamental weight ω associated to P is minuscule (see §A.2.3 of the Appendix for details and also for the definition of a minuscule weight). The Schubert varieties of I are defined with respect to a Borel subgroup of SO(V )—they are the Borel orbit closures of I with the canonical reduced scheme structure. We now make some choices that are convenient for the study of Schubert varieties. For j an integer such that 1 ≤ j ≤ 2n, set j ∗ := 2n + 1 − j . Fix a basis e1 , . . . , e2n of V such that  1 if i = j ∗ ei , ej  = 0 otherwise The elements of SO(V ) that are diagonal with respect to the basis e1 , . . . , e2n form a maximal torus T of SO(V ). Similarly the elements of SO(V ) that are upper triangular with respect to e1 , . . . , e2n form a Borel subgroup B of SO(V )—a linear transformation is upper triangular with respect to e1 , . . . , e2n if for each j , 1 ≤ j ≤ 2n, the image of ej under the transformation is a linear combination of ei with i ≤ j .

8.1 SMT for the even orthogonal Grassmannian

87

To index the Schubert varieties, we choose a natural combinatorial indexing set for the T -fixed points. Denote by I (n, 2n) the set of all subsets of distinct entries of {1, . . . , 2n} of cardinality n. An element v of I (n, 2n) may be written as v = (v1 , . . . , vn ) where 1 ≤ v1 < . . . < vn ≤ 2n and v = {v1 , . . . , vn }. Given v = (v1 , . . . , vn ) and w = (w1 , . . . , wn ) in I (n, 2n), we say v ≤ w if v1 ≤ w1 , . . . , vn ≤ wn . Clearly, ≤ defines a partial order on I (n, 2n). Let J (n) denote the set of elements v of I (n, 2n) with the following properties: (1) exactly one of j , j ∗ belongs to v for every j , 1 ≤ j ≤ n; and (2) the number of elements in v that are bigger than n is even (possibly 0). The partial order on J (n) induced from the partial order ≤ on I (n, 2n) is also denoted ≤. The T -fixed points of I are parametrized by J (n): for v = (v1 , . . . , vn ) in J (n), the corresponding T -fixed point, denoted ev , is the span of ev1 , . . . , evn . These points lie in different B-orbits and the union of their B-orbits is all of I. Schubert varieties are thus indexed by the T -fixed points and so in turn by J (n). Given w in J (n), we denote by X(w) the closure of the B-orbit of the T -fixed point ew . We have the Bruhat (B-orbit) decomposition: , Bev . X(w) = v≤w

Bev

belongs to the closure of the orbit Bew if and only if In other words, the orbit v ≤ w. Let I ⊆ Gn (V ) → P(∧n V ) be the Plücker embedding (as described in Chapter 4). The pull-back to I of the line bundle O(1) on P(∧n V ) is the square of the ample generator of the Picard group of I—this follows from Proposition 7.2.4.1,(2) of Chapter 7 (or also the calculation done in Example A.3.2 of the Appendix). Letting L denote the ample generator, we want to describe the homogeneous coordinate rings of I and its Schubert subvarieties in the embedding defined by L–we have in fact that L is very ample (cf. Chapter 3, §3.3.2, Summary (4)). For θ in I (n, 2n), let pθ denote the corresponding Plücker coordinate. Consider the affine patch A of P(∧n V ) given by p = 1, where  := (1, . . . , n). The intersection A ∩ Gn (V ) of this patch with the Grassmannian is an affine space. Indeed the n-plane corresponding to an arbitrary point z of A ∩ Gn (V ) has a basis consisting of column vectors of a matrix of the form   I C= A where I is the identity matrix of size n × n and A is an arbitrary matrix of size n × n. The association z → A is bijective. The restriction of a Plücker coordinate pθ to A ∩ Gn (V ) is given by the determinant of a submatrix of size n × n of C, the entries of θ determining the rows to be chosen from C to form the submatrix. As can be readily verified, a point z of A ∩ Gn (V ) belongs to I if and only if the corresponding matrix A = (aij ) is skew-symmetric with respect to the antidiagonal: aij + aj ∗ i ∗ = 0, where the columns and rows of A are numbered 1, . . . , n and n + 1, . . . , 2n respectively. For example, if n = 4, then a matrix that is skewsymmetric with respect to the anti-diagonal looks like this:

88

8 The standard monomial theoretic basis

⎛

⎞ −d −c −b 0 ⎜ −g −f 0 b ⎟ ⎜ ⎟ ⎝ −i 0 f c ⎠ 0 i gd Since the set of these matrices is connected and contains the point that is spanned by e1 , . . . , en , it follows that this affine patch of I intersects only I. In other words, p vanishes everywhere on I \ I. Computing pθ /p as a function on the affine patch p = 0, we see that it is the determinant of a skew-symmetric matrix of even size, and therefore a square—the square root, which is determined up to sign, is called the Pfaffian. This suggests that pθ itself is a square: more precisely that there exists a section qθ of the line bundle L on I such that qθ2 = pθ . A weight calculation confirms this to be the case.1 In fact, we have that qθ ’s are precisely the extremal weight vectors (cf. Chapter 3, § 3.3.1) in H 0 (G/P , L). The qθ are also called Pfaffians. Definition 8.1.0.1 A standard monomial in J (n) is a totally ordered sequence v1 ≥ . . . ≥ vt (with repetitions allowed) of elements of J (n). Such a standard monomial is said to be w-dominated for w ∈ J (n) if w ≥ v1 . To a standard monomial v1 ≥ . . . ≥ vt in J (n) we associate the product qv1 · · · qvt , where the qv are sections of the line bundle L defined above. Such a product is also called a standard monomial and it is said to be dominated by w for w ∈ J (n) if the underlying monomial in J (n) is dominated by w. We can now state the main theorem of standard monomial theory for I and its Schubert subvarieties. Theorem 8.1.0.2 (Seshadri [111]) Standard monomials qv1 · · · qvr of degree r form a basis for the space of forms of degree r in the homogeneous coordinate ring of I in the embedding defined by the ample generator L of the Picard group. More generally, for w ∈ J (n), the w-dominated standard monomials of degree r form a basis for the space of forms of degree r in the homogeneous coordinate ring of the Schubert subvariety X(w) of I. Proof. The theorem will be derived as a special case of the more general theorem proved in the Appendix (see Theorem A.4.0.1 of the Appendix). More generally, we have results similar to Theorem 8.1.0.2 for Schubert varieties in a minuscule G/P (cf. (Seshadri [111])): A minuscule fundamental weight ω has the property (in fact, this is a defining property) that the weights (for the T -action) in H 0 (G/P , L) form one orbit for the action of the Weyl group. Hence the extremal 1 With notation as in Example A.3.2 of the Appendix, the weight of p , for θ in I (n, 2n), θ

is −(εθ1 + · · · + εθn )—of course, while trying to calculate the restriction to SO(2n), we would have to set εj + εj ∗ = 0, that is, εn+1 = −εn , . . . , ε2n = −ε1 . For θ in I (n, 2n), if μ1 , μ2 are weights of H 0 (G/P , L) such that μ1 + μ2 equals the weight of pθ , then μ1 = μ2 = −(εθ1 + · · · + εθn )/2.

8.2 SMT for the symplectic Grassmannian

89

weight vectors (cf. Chapter 3, § 3.3.1) in H 0 (G/P , L) form a basis for H 0 (G/P , L); further, for w ∈ W P , the standard monomials qv1 · · · qvr , w ≥ v1 ≥ · · · ≥ vr of degree r form a basis for H 0 (X(w), Lr ).

8.2 SMT for the symplectic Grassmannian The purpose of this section is to state the main theorem of SMT for the symplectic Grassmannian. The preparatory material below is dealt with again and more systematically in Chapter 6. See also Example A.3.3 of the Appendix. Let K be an algebraically closed field of arbitrary characteristic. Fix a vector space V of finite dimension over K and fix a non-degenerate alternating form  ,  on V . Then the dimension of V is necessarily even, say 2n. A linear subspace of V is said to be isotropic if the form  ,  vanishes identically on it. It is elementary to see that an isotropic subspace of V has dimension at most n and that every isotropic subspace is contained in one of dimension n. Denote by Gn (V ) the Grassmannian of n-dimensional subspaces of V and by Mn (V ) the set of all n-dimensional isotropic subspaces of V . Then Mn (V ) is a closed subvariety of Gn (V ), and is called the variety of maximal isotropic subspaces or the symplectic Grassmannian or also the Lagrangian Grassmannian. The group Sp(V ) of linear automorphisms of V preserving  ,  acts transitively on Mn (V )—this follows from Witt’s theorem that an isometry between subspaces can be lifted to one of the whole vector space. Using this we can identify Mn (V ) as the quotient of Sp(V ) by the stabilizer of any point (for example, the span of e1 , . . . , en in the notation to be defined below)—one needs also to check that the differential at the identity of the orbit map defined by the point is surjective. These stabilizers are (conjugate) maximal parabolic subgroups of Sp(V ). Thus, after fixing a Borel subgroup B of Sp(V ), we can identify Mn (V ) as the quotient of Sp(V ) by a certain maximal parabolic subgroup P containing B: Mn (V ) ∼ = G/P , G = Sp(V ) The Schubert varieties of Mn (V ) are defined with respect to a Borel subgroup of Sp(V )—they are the Borel orbit closures of Mn (V ) with the canonical reduced scheme structure. We now make some choices that are convenient for the study of Schubert varieties. For j an integer such that 1 ≤ j ≤ 2n, set j ∗ := 2n + 1 − j . Fix a basis e1 , . . . , e2n of V such that ⎧ ⎨ 1 if i = j ∗ and i < j ei , ej  = −1 if i = j ∗ and i > j ⎩ 0 otherwise The elements of Sp(V ) that are diagonal with respect to the basis e1 , . . . , e2n form a maximal torus T of Sp(V ). Similarly the elements of Sp(V ) that are upper triangular

90

8 The standard monomial theoretic basis

with respect to e1 , . . . , e2n form a Borel subgroup B of Sp(V )—a linear transformation is upper triangular with respect to e1 , . . . , e2n if for each j , 1 ≤ j ≤ 2n, the image of ej under the transformation is a linear combination of ei with i ≤ j . To index the Schubert varieties, we choose a natural combinatorial indexing set for the T -fixed points. Denote by I (n, 2n) the set of all subsets of distinct entries of {1, . . . , 2n} of cardinality n. An element v of I (n, 2n) may be written as v = (v1 , . . . , vn ) where 1 ≤ v1 < . . . < vn ≤ 2n and v = {v1 , . . . , vn }. Given v = (v1 , . . . , vn ) and w = (w1 , . . . , wn ) in I (n, 2n), we say v ≤ w if v1 ≤ w1 , . . . , vn ≤ wn . Clearly, ≤ defines a partial order on I (n, 2n). Let I (n) denote the set of elements v of I (n, 2n) with the property that exactly one of j , j ∗ belongs to v for every j , 1 ≤ j ≤ n. The partial order on I (n) induced from the partial order ≤ on I (n, 2n) is also denoted ≤. The T -fixed points of Mn (V ) are parametrized by I (n): for v = (v1 , . . . , vn ) in I (n), the corresponding T -fixed point, denoted by ev , is the span of ev1 , . . . , evn . These points lie in different B-orbits and the union of their B-orbits is all of Mn (V ). Schubert varieties are thus indexed by the T -fixed points and so in turn by I (n). Given w in I (n), we denote by X(w) the closure of the B-orbit of the T -fixed point ew . We have the Bruhat (B-orbit) decomposition: , X(w) = Bev . v≤w

Bev

belongs to the closure of the orbit Bew if and only if In other words, the orbit v ≤ w. Let Mn (V ) ⊆ Gn (V ) → P(∧n V ) be the Plücker embedding (as described in Chapter 4). The pull-back to Mn (V ) of the line bundle O(1) on P(∧n V ) is the ample generator of the Picard group of Mn (V ) (cf. Chapter 6, Proposition 6.2.4.1). We want to describe the homogeneous coordinate rings of Mn (V ) and its Schubert subvarieties in this embedding. For θ in I (n, 2n), let pθ denote the corresponding Plücker coordinate. Consider the affine patch A of P(∧n V ) given by p = 1, where  := (1, . . . , n). The intersection A ∩ Gn (V ) of this patch with the Grassmannian is an affine space. Indeed the n-plane corresponding to an arbitrary point z of A ∩ Gn (V ) has a basis consisting of column vectors of a matrix of the form   I C= A where I is the identity matrix of size n × n and A is an arbitrary matrix of size n × n. The association z → A is bijective. The restriction of a Plücker coordinate pθ to A ∩ Gn (V ) is given by the determinant of a submatrix of size n × n of C, the entries of θ determining the rows to be chosen from C to form the submatrix. As can be readily verified, a point z of A ∩ Gn (V ) belongs to Mn (V ) if and only if the corresponding matrix A = (aij ) is symmetric with respect to the antidiagonal: aij = aj ∗ i ∗ , where the columns and rows of A are numbered 1, . . . , n and n + 1, . . . , 2n respectively. For example, if n = 4, then a matrix that is symmetric with respect to the anti-diagonal looks like this:

8.2 SMT for the symplectic Grassmannian

⎛

d ⎜g ⎜ ⎝i j

c f h i

b e f g

91

⎞ a b⎟ ⎟ c⎠ d

Notation 8.2.0.1 For u = (u1 , . . . , un ) in I (n, 2n), set u∗ := (u∗n , . . . , u∗1 ). The association u → u∗ is an order reversing involution of I (n, 2n). There is another order reversing involution on I (n, 2n), namely u → {1, . . . , 2n} \ u. These two involutions commute with each other. Composing them, we obtain an order preserving involution on I (n, 2n): u → u# := {1, . . . , 2n} \ u∗ . Note that u ∈ I (n) if and only if u = u# . Lemma 8.2.0.2 Given any θ ∈ I (n, 2n), we have pθ = ±pθ # on Mn (V ). The sign factor can easily be determined explicitly but we will have no use for it2 . Proof. Since Mn (V ) is irreducible and its intersection A ∩ Mn (V ) with the affine patch is non-empty, it is enough to check that pθ = ±pθ # on A ∩ Mn (V ), and this follows from the symmetry property just mentioned of the matrix A. The relations pθ = pθ # do not span the space of all linear relations among the pθ — see Example 8.1 below. In order to describe a nice parametrizing set for a basis for the space of linear forms in the homogeneous coordinate ring of Mn (V )—in fact for describing bases for spaces of forms of any given degree—we make the following definition. Definition 8.2.0.3 3 Denote by  the element (1, . . . , n) of I (n). The -degree of an element x of I (n) is the cardinality of x \ {1, . . . , n} or equivalently that of {1, . . . , n} \ x. More generally, given any v ∈ I (n), the v-degree of an element x of 2 The sign factor is 1 or −1 according as

{(θ1 − 1) + · · · + (θk − k)} + {(θk+1 − (d + 1)) + · · · + (θd − (2d − k))} is even or odd and k is the cardinality of [d] ∩ θ . To prove this, let θ = (θ1 , . . . , θk , θk+1 , . . . , θd ) and ϕ = (ϕ1 , . . . , ϕd−k , ϕd−k+1 , . . . , ϕd ) be the complement of θ in {1, . . . , 2d}. Then, as can be easily seen, pθ is (−1){(θ1 −1)+···+(θk −k)} times the determinant of the d − k × d − k submatrix of the matrix A above obtained by taking the entries in rows θk+1 , . . . , θd and columns ϕ1 , . . . , ϕd−k . ∗ ∗ , . . . , ϕ ∗ ). Since u → u∗ commutes with , ϕd−k We have θ # = (ϕd∗ , . . . , ϕd−k+1 1 ∗ ) taking complements, it follows that the complement in {1, . . . , d} of (ϕd∗ , . . . , ϕd−k+1 ∗ ∗ is (θd , . . . , θk+1 ). By applying to pθ # the observation in the last paragraph about pθ , ∗

∗

we see that pθ # is (−1){(ϕd −1)+···+(ϕd−k+1 −k)} times the determinant submatrix of size ∗ , . . . , ϕ∗ d − k × d − k of the matrix A above obtained by taking the entries in rows ϕd−k 1 ∗ ∗ and columns θd , . . . , θk+1 . It follows from the symmetry of A about the anti-diagonal that the two submatrices of A considered above have the same determinant. And it is easy to see that (ϕd∗ − 1) + · · · + ∗ − k) equals (θk+1 − (d + 1)) + · · · + (θd − (2d − k)). (ϕd−k+1 3 Admissible pairs as defined here are a special case of the admissible pairs to be defined in the Appendix.

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8 The standard monomial theoretic basis

I (n) is the cardinality of x\v or equivalently that of v\x. An ordered pair Λ = (x, y) of elements of I (n) is called an admissible pair if x ≥ y and the -degrees of x and y are equal. Given admissible pairs Λ = (x, y) and Λ = (x  , y  ), we say Λ ≥ Λ if y ≥ x  , that is, if x ≥ y ≥ x  ≥ y  . An ordered sequence (Λ1 , . . . , Λt ) of admissible pairs is called a standard tableau if Λi ≥ Λi+1 for 1 ≤ i < t. We often write Λ1 ≥ . . . ≥ Λt to denote the standard tableau (Λ1 , . . . , Λt ). Given w ∈ I (n), we say that a standard tableau Λ1 ≥ . . . ≥ Λt is w-dominated if w ≥ x1 , where Λ1 = (x1 , y1 ). Proposition 8.2.0.4 There is a bijective map (x, y) → θ from the set of admissible pairs (x, y) to elements θ of I (n, 2n) such that θ ∩ [n] ≥ θ # ∩ [n], where [n] := {1, . . . , n}. Proof. Let [n]c := {n + 1, . . . , 2n}. Given an admissible pair (x, y), set   θ = (x ∩ [n]) ∪ y ∩ [n]c That θ satisfies the required condition is readily verified. For the map in the other direction, set       and y = θ # ∩ [n] ∪ θ ∩ [n]c x = (θ ∩ [n]) ∪ θ # ∩ [n]c It is easy to verify that (x, y) is an admissible pair. Remark 8.2.0.5. An element θ of I (n, 2n) can be specified by giving θ ∩ [n] and θ # ∩ [n]. The ordered pairs (θ ∩ [n], θ # ∩ [n]) associated to the admissible pair (x, y) by the above proposition were called admissible elements in Chapter 6. Example 8.1. The relations pθ = pθ # do not span the linear space of relations. For example, let n = 4,

θ1 = (1, 2, 7, 8),

θ2 = (1, 4, 5, 8)

and

θ3 = (1, 3, 6, 8).

Then θ1# = (3, 4, 5, 6),

θ2# = (2, 3, 6, 7)

and

θ3# = (2, 4, 5, 7).

From the form displayed above of a matrix that is symmetric with respect to the anti-diagonal, we get pθ1 = pθ # = df − cg, 1

pθ2 = pθ # = cg − bi 2

and

pθ3 = pθ # = bi − df, 3

and consequently, pθ1 + pθ2 + pθ3 = 0. The association of the previous proposition is not a bijection if the condition on θ is dropped. In the case n = 4 for example there are 42 admissible pairs (x, y) and 43 pairs (θ, θ # ). If we try the procedure in the proof above for recovering (x, y) on an arbitrary (θ, θ # ), the two resulting elements may not be comparable. Taking for example θ = (2, 3, 6, 7), θ # = (1, 4, 5, 8), we recover (2, 3, 5, 8) and (1, 4, 6, 7).

8.2 SMT for the symplectic Grassmannian

93

Definition 8.2.0.6 Given an admissible pair Λ = (x, y), we define the associated Plücker co-ordinate pΛ to be pθ where (x, y) → (θ, θ # ) by the association of Proposition 8.2.0.4. Observe that pθ = pθ # by Lemma 8.2.0.2. To formal products of admissible pairs we associate the product of the associated Plücker coordinates. In particular, to a standard tableau Λ1 ≥ . . . ≥ Λt we associate pΛ1 · · · pΛt . Such monomials associated to standard tableaux are called standard monomials. Given w in I (n), we say that a standard monomial is w-dominated if the corresponding standard tableau is w-dominated. We can now state the main theorem of standard monomial theory for Mn (V ) and its Schubert subvarieties. Theorem 8.2.0.7 (De Concini [19], Lakshmibai-Musili-Seshadri [61, 62]) Standard monomials pΛ1 · · · pΛt of degree r form a basis for the space of forms of degree r in the homogeneous coordinate ring of Mn (V ) in the Plücker embedding. More generally, for w ∈ I (n), the w-dominated standard monomials of degree r form a basis for the space of forms of degree r in the homogeneous coordinate ring of the Schubert subvariety X(w) of Mn (V ). Proof. The theorem will be derived as a special case of the more general theorem to be proved in the Appendix (see Theorem A.4.0.1 of the Appendix).

9 Review of GIT

In this chapter, we do a brief review of GIT - Geometric Invariant Theory. Our treatment follows that of [96]. For further details, we refer the reader to [87] & [110]. We fix a base field K which we assume to be algebraically closed of arbitrary characteristic. In this chapter, we shall work just with the closed points of an algebraic variety.

9.1 G-spaces Consider an algebraic variety X together with an algebraic group action, i.e., we are given an algebraic group G and a morphism G × X → X. For x ∈ X, we shall denote the orbit through x by O(x). We shall denote the fixed points set by X G , namely, X G = {x ∈ X | g · x = x, ∀g ∈ G} Defining “quotients” (which are again algebraic varieties) is the central theme in GIT. It turns out that actions by “reductive groups” do admit some natural notion of “quotients”. We shall define such quotients for actions by reductive Groups. We first start with recalling the definition of reductive groups (and definitions of related notions). After pointing out (with an example) that the orbit space (whose underlying set is the set of orbits) need not have a variety structure, we recall the definitions of various quotients, and then explore these in the case of affine and projective varieties. Throughout G will denote an affine algebraic group defined over K. 9.1.1 Reductive groups We first recall the definitions of reductive groups, linearly reductive groups, geometrically reductive groups, and completely reducible groups. For generalities on reductive groups, we refer the reader to Borel [7] & Humphreys [43]. Definition 9.1.1.1 A G-module is a finite-dimensional K-vector space V together with an algebraic group morphism ρ : G → GL(V )(= the group of K-linear automorphisms of V , also denoted AutK V or just Aut V ). We refer to ρ as “a rational representation” of G.

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9 Review of GIT

Remark 9.1.1.2. Note that a rational representation ρ : G → GL(V ), determines a morphism G × V → V , (g, v) → ρ(g)(v), and hence an action of G on V . We refer to this as a linear action (of G on V ). Definition 9.1.1.3 A G-module V is irreducible, if it has no proper, non-zero Gstable subspaces. Definition # 9.1.1.4 A G-module is V is completely reducible, if it can be written as V = i Vi , where each Vi is an irreducible G-module. Definition 9.1.1.5 An algebraic group G is completely reducible if every (finite dimensional) G-module V is completely reducible. Definition 9.1.1.6 An algebraic group G is linearly reductive if given a finite dimensional G-module V , and a non-zero v ∈ V G , there exists a G-invariant linear form F ∈ V ∗ such that F (v) = 0. Fact: G is completely reducible if and only if G is linearly reductive. For a proof, see Nagata [92] Definition 9.1.1.7 An algebraic group G is geometrically reductive if given a finite dimensional G-module V , and a non-zero v ∈ V G , there exists a G-invariant form F ∈ Sym(V ∗ ) with deg(F ) > 0 such that F (v) = 0. Before defining reductive groups, we first recall the radical of an algebraic group G. Denote by Gu the unipotent part of G. Recall that a Borel subgroup of an algebraic group is a maximal connected solvable subgroup. The radical of an algebraic group G, denoted R(G), is the connected component through e (the identity element in G) of the intersection of all Borel subgroups of G, i.e. R(G) := (∩B∈B B)◦ B being the set of all Borel subgroups in G. Definition 9.1.1.8 G is reductive if R(G)u = (e) (the trivial subgroup). (Note: This is equivalent to saying that R(G) is a torus, i.e, a direct product of copies of K ∗ .) Theorem 9.1.1.9 (H. Weyl) In characteristic 0, a reductive group is linearly reductive. For a proof see [29]. Remark 9.1.1.10. In positive characteristics, a reductive group need not be linearly reductive as explained in the example below:

9.1 G-spaces

97

Example: Let char(K) = p > 0, and G = GL(V ) = GL(n) where V is a vector space over K of dimension n and p|n. Let X = Mn (K), the space of n × n matrices with entries in K, which we identify with End(V ). Let G act on X by conjugation. We then have that v := I dV ∈ X is G-invariant. Fact: Under the action of GL(n) on Mn (K) by conjugation, a G-invariant polynomial form is a symmetric function of the eigenvalues of a matrix (for details, see §9.3.1 below). Therefore if there exists a linear G-invariant form F , then it has to be a multiple of the sum of the eigenvalues, that is, F = λtr(A), for some λ ∈ K (here, tr(A) is the trace of A = sum of the diagonal entries of A). Therefore for v = I d, F (v) = λn = 0 (since p|n). Therefore G cannot be linearly reductive. Let G be an algebraic group acting (say, on the right) on an algebraic variety X. Let Y := X/G be the orbit space of X under the action of G; a point in Y is a G-orbit, i.e., O(x), for some x ∈ X. In general, Y need not have a variety structure; if Y has a variety structure, then it forces each orbit to be closed, which may not happen in general. As an example, consider X = {upper triangular matrices in M2 (K)} G = {invertible upper triangular matrices in M2 (K)}     11 ab Let G act on X by conjugation. Consider A = ∈ X and g = ∈ G. 01 0d Then       1 d −b 1 a/d ab = gAg −1 = A 0 1 0d ad 0 a    1t ∗ ∗ where a/d ∈ K . Thus G · A = O(A) = ,t ∈ K . 01   10 Now O(A) = O(A) ∪ , so O(A) is not closed! 01 Let X = Spec R where R is a finitely generated K-algebra. Each f ∈ R corresponds to a map f : X → K. A group action G  X (say, on the left) induces an action G  R defined by g · f (x) := f (g −1 x) for g ∈ G, f ∈ R, and x ∈ X. Let R G = {f ∈ R | g · f = f, ∀g ∈ G}. If R G is a finitely generated K-algebra, then we can use the variety Spec R G for defining a quotient. But R G need not be a finitely generated K-algebra in general: Hilbert’s Fourteenth Problem: Given a finitely generate K-algebra R and an algebraic group action G  R, is R G also a finitely generated K-algebra? Nagata answered this in the negative in 1958 (cf. [91, 94]) by providing a counterexample. However, if G is reductive (such as is the case with GLn , SLn , Sp2n , On , or any of the exceptional groups), then we shall see (thanks to Mumford’s conjecture) below that for the action of G on an affine variety Spec R, R G is finitely generated.

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Mumford’s conjecture: If G is reductive, then G is geometrically reductive (in any characteristic). Mumford’s conjecture was proved in 1975 by Haboush [36]. In fact, we have, Theorem 9.1.1.11 The following are equivalent: 1. G is reductive. 2. G is geometrically reductive. 3. If G  R where R is a finitely generated K-algebra, then R G is finitely generated. The implication (1) ⇒ (2) was proved by Haboush (cf. [36]). The implication (2) ⇒ (3) was proved by Nagata (cf. [93]). The implication (3) ⇒ (1) was proved by Popov (cf. [98]). The proof of the above Theorem may be found in GIT [87, Appendix 1].

9.2 Affine quotients We start with a fundamental result concerning geometrically reductive groups. Proposition 9.2.0.1 Let G be a geometrically reductive algebraic group acting on an affine variety X = Spec R. Given two disjoint G-stable closed subsets W1 , W2 ⊂ X, there exists f ∈ R G such that F (W1 ) = 0 and F (W2 ) = 1. Proof. Since W1 ∩ W2 = ∅, we have I1 + I2 = R, Ij being the defining ideal of Wj , j = 1, 2. Hence there exist fj ∈ Ij , j = 1, 2 such that f1 + f2 = 1. Then h := f1 has the property that h(W1 ) = 0 and h(W2 ) = 1. Let V = the span of {g · h, g ∈ G}. Note that V is the smallest G-submodule of R containing h. Then V is G-stable and finite dimensional (recall the well-known fact that any finite dimensional subspace U of R is contained in a finite dimensional G-stable subspace of R (for a proof, see Borel [7] for example); in the present case, U = Kh). Let {h1 = h, · · · , hn } be a basis for V . Write  g · hi = aij (g)hj , aij ∈ K[G] j

This gives rise to G → Aut V , g → (aij (g))1≤i,j ≤n The morphism ϕ : X → V (= K n ), x → (h1 (x), · · · , hn (x)) is a G-morphism, i.e., ϕ(g · x) = g · ϕ(x). Further ϕ(W1 ) = 0, ϕ(W2 ) = (h1 (W2 ), · · · , hn (W2 )) = v, say (note that writing cij gij h, we have, hi (W1 ) = 0, hi (W2 ) = a constant, since W1 , W2 are Ghi = j

stable, and h(W1 ) = 0, h(W2 ) = 1). Write v = (h1 (x), · · · , hn (x)), where x is any point in W2 . Note that v = 0 (since h1 (W2 ) = 1); further, for g ∈ G, we have,

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g · v = (g · h1 (x), · · · , g · hn (x)) = (h1 (g −1 · x), · · · , hn (g −1 · x)) = (h1 (x), · · · , hn (x)) =v Thus v ∈ V G . Hence the geometric reductivity of G implies that there exists a F in K[x1 , · · · , xn ] such that F (0) = 0, F (v) = 1. Then f := F ◦ ϕ has the required properties. 9.2.1 Affine actions Let G be a reductive group acting on an affine variety X = SpecR where R is a finitely generated K-algebra. Let Y := SpecR G . Let ϕ : X → Y be the morphism induced by the inclusion R G → R. We first list some properties of the morphism ϕ. Proposition 9.2.1.1 1. ϕ is surjective. 2. ϕ is G-invariant, i.e., ϕ is constant on orbits: ϕ(g · x) = ϕ(x), g ∈ G, x ∈ X. 3. ϕ is an affine morphism, i.e. the inverse image of an open affine set is an open affine set. 4. If W ⊂ X is G-stable and closed, then ϕ(W ) is closed. 5. If W1 , W2 are two disjoint G-stable closed subsets of X, then ϕ(W1 ) ∩ ϕ(W2 ) = ∅. 6. Given open U ⊂ Y , the map ϕ ∗ : K[U ] → K[ϕ −1 (U )]G is an isomorphism. Proof. (1) Let y be a (closed) point in Y . Let my be the maximal ideal in K[Y ](= R G ) R = corresponding to y. Let f1 , · · · , fr be generators for my . Then we have that G fi m R) ∩ R , f ∈ R (in view of [96], Lemma 3.4.2, we have that for any f ∈ ( f i fi R. Let x be the fi R G ). Hence there exists a maximal ideal m containing point corresponding to m. Then fi (x) = 0, ∀i, and it follows that ϕ(x) = y. Thus we obtain that ϕ is surjective. (2) To show ϕ(g · x) = ϕ(x), g ∈ G, x ∈ X, we should show that (∗)

f (ϕ(g · x)) = f (ϕ(x)), f ∈ K[Y ](= R G ), g ∈ G, x ∈ X.

Now for any f ∈ R G , g ∈ G, x ∈ X, we have f (gx) = f (ϕ(g · x)), f (x) = f (ϕ(x)) f (g · x) = g −1 · f (x) = f (x) from which (*), and hence the G-invariance of ϕ follows.

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Assertion (3) follows trivially from the fact that a morphism f : X → Y is an affine morphism if and only if there exists an open affine cover Y = ∪ Vi such that f −1 (Vi ) is open affine. (4) Assume if possible that ϕ(W ) is not closed in Y . Let y ∈ ϕ(W ) \ ϕ(W ). Then ϕ −1 (y), W are two G-stable, disjoint, closed subsets of X. Hence by Proposition 9.2.0.1, there exists a f ∈ R G such that f (W ) = 0, f (ϕ −1 (y)) = 1. Hence we obtain f ( ϕ(W ) ) = 0, f (y) = 1, a contradiction (note that y ∈ ϕ(W )). Hence our assumption is wrong and (4) follows. (5) Since W1 , W2 are two G-stable, disjoint, closed subsets of X, we have by Proposition 9.2.0.1 that there exists a f ∈ R G such that f (W1 ) = 0, f (W2 ) = 1. Hence writing W1 , W2 as unions of G-orbits and using the G-invariance of ϕ, we obtain f (ϕ(W1 )) = 0, f (ϕ(W2 )) = 1. Assertion (5) follows from this. (6) It suffices to prove assertion (6) for principal open subsets, that is, for Yf := {y ∈ Y |f (y) = 0}, where f ∈ K[Y ] is a regular function (since the collection {Yf , f ∈ K[Y ]} is a base for the Zariski topology on Y ). Let Xf := ϕ −1 (Yf ). Take U = Yf , −1 then K[U ] = K[X]G f . On the other hand, ϕ (U ) = Xf and K[Xf ] = K[X]f . It is seen easily that the localization of the G-invariants equals the G-invariants of the localization, (K[X]G )f = (K[X]f )G . This proves (6). As an important consequence, we obtain that ϕ has the universal mapping property: Proposition 9.2.1.2 (Universal mapping property) With X, Y, ϕ as above, given an algebraic variety Z, and a G-invariant morphism F : X → Z, there exists a unique morphism χ : Y → Z such that the diagram X



ϕ

 Y



F  χ



  Z

is commutative. Proof. If possible, let χ1 , χ2 be two such morphisms. Let y ∈ Y , say, y = ϕ(x), for some x ∈ X (cf. Proposition 9.2.1.1, (1)). For i = 1, 2, we have χi (y) = χi (ϕ(x)) = F (x). Hence we obtain χ1 = χ2 , and the uniqueness of χ follows. We shall now prove the existence of χ. We first Claim 1: For y ∈ Y, F is constant on ϕ −1 (y). If possible, assume that z1 , z2 ∈ F (ϕ −1 (y)), z1 = z2 . Write zi = F (xi ), i = 1, 2, where x1 , x2 are two (distinct) points of ϕ −1 (y). Let W1 = F −1 (z1 ), W2 = F −1 (z2 ) Then W1 , W2 are two disjoint, G-stable, closed subsets of X. Hence by Proposition 9.2.1.1, (5), we obtain that ϕ(W1 ) ∩ ϕ(W2 ) = ∅; but y ∈ ϕ(W1 ) ∩ ϕ(W2 ) (note that

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y = ϕ(xi ) ∈ ϕ(Wi ), i = 1, 2)), a contradiction. Hence our assumption is wrong, and Claim 1 follows. Take an affine cover Z := ∪ Vi for Z. Let Ui = {y ∈ Y | F (ϕ −1 (y)) ∈ Vi }. Denote by ϕi , Fi , the restriction of ϕ, F respectively, to ϕ −1 (Ui ). Define χi : Ui → Vi , y → F (ϕ −1 (y)) (note that χi is well-defined in view of Claim 1). Thus we have the following commutative diagram: ϕ −1 (Ui ) ϕi

 Ui





 Fi  χi



   Vi

Claim 2: The map χi : Ui → Vi is in fact a morphism of varieties. The comorphism Fi∗ maps K[Vi ] inside K[Fi−1 Vi ]G : (∗)

Fi∗ : K[Vi ] → K[Fi−1 Vi ]G

(since Fi is G-invariant). Also, by Proposition 9.2.1.1, (6), the comorphism ϕi∗ maps K[Ui ] isomorphically onto K[ϕi−1 (Ui )]G : (∗∗) Now

ϕi∗ : K[Ui ] ∼ = K[ϕi−1 (Ui )]G K[ϕi−1 (Ui )]G = K[ϕi−1 χi−1 (Vi )]G = K[Fi−1 (Vi )]G

This together with (**) implies (∗ ∗ ∗)

K[Ui ] ∼ = K[Fi−1 Vi ]G

Hence from (*), (***), we obtain that χ∗ : K[Vi ] → K[Ui ] is a K-algebra homomorphism. Claim 2 follows from this (in view of the well-known fact that if U, V are two varieties with V affine, then the map Mor(U, V ) → H omK-alg (K[V ], K[U ]), f → f ∗ is a bijection. See for example [86] for a proof.) Clearly, the χi , χj agree on Ui ∩ Uj , and hence may be glued to define a morphism χ : Y → Z. This completes the proof of Proposition 9.2.1.2.

9.3 Categorical quotients For the remaining sections of this chapter, we shall assume G to be a reductive group. Let X = SpecR be an affine variety on which G acts. Let Y := SpecR G (note that R G is a finitely generated K-algebra). Let ϕ : X → Y be the morphism induced by the inclusion R G → R. Consider the following two properties of ϕ (seen in the previous section):

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(i) ϕ is G-invariant, i.e, constant on G-orbits. (ii) ϕ has the universal mapping property. (As above, given f : X → Z which is constant on G-orbits, there exists a unique g : Y → Z such that f = g ◦ ϕ: X



ϕ

 Y



f  g



  Z

We make an abstraction of (i) and (ii), and define categorical quotients: Definition 9.3.0.1 Let X be an algebraic variety with an action by an algebraic group G. A pair (Y, ϕ) is called a categorical quotient if (i) (ii) (iii) (iv)

Y is an algebraic variety. ϕ is morphism of varieties from X to Y . ϕ is G-invariant, i.e., ϕ is constant on orbits ϕ has the universal mapping property.

Remark 9.3.0.2. It may be seen easily that if a categorical quotient exists, then it is unique up to isomorphism. Notation: We shall denote the categorical quotient (of X by G) by X//G ( if G acts on the right; a similar notation if G acts on the left). Immediately from the definition, we have Fact: If X is affine, then X//G = Spec(K[X]G ). 9.3.1 Examples Let X = Mn (K) = Spec R where R = K[x11 , . . . , xnn ]. Let G = GLn (K), and let G act on X by conjugation. Description of R G : Let A ∈ X. Up to conjugation, A is determined by its Jordan blocks. For f ∈ R G , f being constant on conjugacy classes, f is completely known by specifying its values on Jordan blocks. In the case of A being a diagonal matrix, its Jordan blocks are given by the diagonal entries. Further, denoting by Dn the set of diagonal matrices, we have ∪ gDn g −1 is dense in X. Hence, f ∈ R G is uniquely j

determined by knowing its values on diagonal matrices. Thus we obtain an injective homomorphism R G → K[t1 , · · · , tn ], f → f |Dn . Since two diagonal matrices with permuted diagonal entries are conjugate, f must be a symmetric function in the ti ’s, and we obtain an identification of R G with the space of symmetric polynomials in the ti ’s. Now a symmetric polynomial in the ti ’s is just a polynomial in the elementary symmetric functions ej ’s in the ti ’s, and the ej ’s are simply the coefficients of the characteristic polynomial.

9.4 Good quotients

Thus we obtain

103

RG ∼ = K[e1 , · · · , en ]

where ei is the elementary symmetric polynomial of degree i in the eigenvalues of A, i.e., the coefficients of the characteristic polynomial x n + a1 x n−1 + · · · + an−1 x + an of A. Hence, X//G = An , and the map X → An , A → (a1 , · · · , an ) is the corresponding (categorical) quotient map. Let X = Mn,d (K) = Spec R = Adn , where d ≤ n. Let G = SLd (K), and let G act on X by multiplication on the right. Consider the d-minors of a matrix A ∈ X. Let i = (i1 , . . . , id ) be an ordered sequence of indices such that 1 ≤ i1 < . . . < id ≤ n. Let Id,n denote the set of all such sequences. For each i ∈ Id,n , let Ai denote the d × d submatrix of A with rows given by the indices pi

{i1 , . . . , id }, and define the map pi : X → K by A → det (Ai ). Then pi ∈ R G for all i ∈ Id,n . In fact, we have, R G = K[pi , i ∈ Id,n ]  We shall see later (in Chapter 10) that the categorical quotient X//G = G d,n = Spec(R G ), the cone over the Grassmannian.

9.4 Good quotients We now make an abstraction of Proposition 9.2.1.1 to define a good quotient. Definition 9.4.0.1 Given an algebraic variety X and an algebraic group G acting on X, a good quotient is defined as a pair (Y, ϕ) satisfying the following: (i) ϕ : X → Y is a surjective morphism. (ii) ϕ is G-invariant, i.e., ϕ is constant on orbits: ϕ(g · x) = ϕ(x), g ∈ G, x ∈ X. (iii) ϕ is an affine morphism, i.e. the inverse image of an open affine set is again an open affine set. (iv) If W ⊂ X is G-stable and closed, then ϕ(W ) is closed. (v) If W1 , W2 are two disjoint G-stable closed subsets of X, then ϕ(W1 ) ∩ ϕ(W2 ) = ∅. (vi) Given open U ⊂ Y , the map ϕ ∗ : K[U ] → K[ϕ −1 (U )]G is an isomorphism.

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In addition, if the underlying set of Y is the orbit space X/G, then (Y, ϕ) is called a geometric quotient. The following facts are immediate from the definition. Facts: (1) A good quotient is a categorical quotient. (2) For X = Spec R, X//G = Spec(R G ) is a good quotient. (3) Consider any closed subgroup H ⊂ G (any algebraic group). Then the coset space G/H is a geometric quotient for the action of H on G by right multiplication. In [107], Rosenlicht describes the variety structure of G/H . (4) If the action of G on X is free, i.e., we have an inclusion G → Aut X and a geometric quotient exists, then X is a principal fiber bundle over Y with group G; recall (cf. [87]) that X is a principal fiber bundle over Y with group G if there exists a morphism ϕ : X → Y such that (i) ϕ is a flat morphism of finite type. (ii) The canonical map ψ : G × X → X × X is an isomorphism. (5) Just as Pn may be realized as a (geometric) quotient of Cn+1 \ {0} for the natural action of C∗ , any toric variety Z (see [30] for the definition of a toric variety) may be realized as a categorical quotient X//G for a suitable X open in an affine space and a suitable group G; if in addition, Z is smooth, then it is a good quotient (cf. [18]). Further properties of good & geometric quotients: We list below some properties of good & geometric quotients which are immediate from the definition. Proposition 9.4.0.2 (1) If (Y, ϕ) is a good (resp. geometric) quotient of X by G, then for any open U ⊂ Y , res(ϕ) : ϕ −1 (U ) → U is a good (resp. geometric) quotient of ϕ −1 (U ) by G (here, res(ϕ) denotes " the restriction). (2) Suppose Y has an affine cover, i.e. Y = Ui where each Ui is affine, and suppose each res(ϕ) : ϕ −1 (Ui ) → Ui is a good (resp. geometric) quotient of ϕ −1 (Ui ) by G. Then (Y, ϕ) is a good (resp. geometric) quotient of X by G. (3) Let (Y, ϕ) be a good (resp. geometric) quotient of X by G. For x1 , x2 ∈ X, ϕ(x1 ) = ϕ(x2 ) ⇔ O(x1 ) ∩ O(x2 ) = ∅. (This follows from (2) and (5) of Proposition 9.2.1.1) (4) Each fiber contains a unique closed orbit. (Clearly, each fiber cannot have two or more closed orbits by (5) of Proposition 9.2.1.1; also, each fiber ϕy being G-stable has a closed G-orbit (by “closed orbit lemma” stated below) which is in fact closed in X, since ϕy is closed in X. Lemma 9.4.0.3 Closed orbit Lemma: Let an algebraic group G act on an algebraic variety X. Then each orbit is a smooth variety which is open in its closure in X. Its boundary is a union of orbits of strictly smaller dimensions. In particular, the orbits of minimal dimension are closed. See Borel [7] for a proof of the closed orbit Lemma.)

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(5) If the action of G on X is closed, i.e., each orbit is closed, then (Y, ϕ)is a geometric quotient. (If the action of G on X is closed, then Y is clearly the orbit space.) 9.4.1 Some results on good quotients In this subsection, we shall prove some results on good quotients to be used for our discussion in the following sections. We first recall (the well-known) upper semi-continuity Theorem: Theorem 9.4.1.1 Let f : X → Y be a morphism of finite type of algebraic varieties X, Y . Then for any integer n, {x ∈ X | dim(f −1 (f (x))) ≥ n} is a closed subset of X. (An equivalent formulation: For any integer m, {x ∈ X | dim(f −1 (f (x))) ≤ m} is an open subset of X.) Let X be an algebraic variety together with an action by an algebraic group G. As a first application of “upper semi-continuity” we have Lemma 9.4.1.2 (1) For any integer n, {x ∈ X | dim Gx ≥ n} is a closed subset of X. (2) For any integer m, {x ∈ X | dim O(x) ≥ m} is an open subset of X. Proof. Consider the morphism φ : X × G → X × X,

(x, g) −→ (x · g, x).

(Here, we have supposed the action of G on the right). We see that the fiber of φ through (x, g) can be identified with the isotropy subgroup of G at x. Applying Theorem 9.4.1.1 (rather, its equivalent formulation) to φ, we obtain that the set V of points (x, g) such that dim Gx < n, is open in X × G. Now the canonical projection X × G → X, (x, g) −→ x is open. Hence, {x ∈ X | dim Gx < n} is open in X; assertion (1) follows from this. Assertion (2) follows from the fact dim Gx + dim O(x) = dim G. Lemma 9.4.1.3 Let d be the maximum of the orbit dimensions. Let Xmax = {x ∈ X | dim O(x) = d}. Then Xmax is a G-stable open subset of X. Proof. The G-stability of Xmax is clear. The assertion that Xmax is open follows from Lemma 9.4.1.2, (2) (note that by the maximality of d, we have Xmax = {x ∈ X | dim O(x) ≥ d}). Lemma 9.4.1.4 Let (Y, ϕ) be a good quotient of X by G. Let x ∈ X be such that O(x) is a closed orbit of dimension d (here, d is as in Lemma 9.4.1.3). Let q : Xmax → Y denote the restriction of ϕ to Xmax . Then O(x) is the fiber of q at q(x).

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Proof. Let x  ∈ Xmax be such that q(x  ) = q(x). We are required to show that x  belongs to O(x). Assume if possible that O(x) = O(x  ). We have that ϕ(x  ) = ϕ(x) (since q(x  ) = q(x)), and hence O(x) ∩ O(x  ) = ∅ (by Definition 9.4.0.1,(v)). But by hypothesis, O(x) = O(x). Hence we obtain that O(x) ⊆ O(x  ); this together with the assumption that O(x) = O(x  ), and the “closed orbit Lemma” implies that dim O(x) < d (since dim O(x  ) = d), which is a contradiction. Hence our assumption is wrong and the result follows. Lemma 9.4.1.5 Let (Y, ϕ) be a good quotient of X by G. Let X1 = {x ∈ Xmax | dim( fiber at q(x)) = d}, q being the restriction of ϕ to Xmax , and d being as in Lemma 9.4.1.3. Then X1 is a G-stable open subset of X. Proof. The G-stability of X1 is clear. Now G-orbits in X map to points in Y , so that this is also the case for q : Xmax → Y . Thus we have that the fibers of q are of dimension ≥ d (d being as in Lemma 9.4.1.3). Hence we obtain by Theorem 9.4.1.1 applied to q (we take m = d in the equivalent formulation of Theorem 9.4.1.1) that X1 is an open subset of Xmax and hence of X (since Xmax is an open subset of X (cf. Lemma 9.4.1.3)). Corollary 9.4.1.6 Let (Y, ϕ) be a good quotient of X by G. Let d be as in Lemma 9.4.1.3. Suppose that there exists a closed orbit of dimension d. Then dim Y = dim X − d. Proof. Let x ∈ X be such that O(x) is closed. Let X1 be as in Lemma 9.4.1.5. By Lemma 9.4.1.4, the orbit O(x) is the fiber at q(x). Thus the fiber at q(x) has dimension d, and hence x ∈ X1 ; in particular, X1 is non-empty. Now denoting the restriction of q to X1 also by just q, we have, q : X1 → Y is dominant and equidimensional with fiber dimension = d (that q is dominant follows from the fact that ϕ is surjective, in particular dominant, and X1 is a non-empty open subset of X). The required result follows from this. Lemma 9.4.1.7 Let (Y, ϕ) be a good quotient of X by G.   Let Xmax = {x ∈ Xmax | O(x) is closed }. Then Xmax is a G-stable open subset of X.  ⊆ X1 . The GProof. First observe that in view of Lemma 9.4.1.4, we have, Xmax  stability of Xmax is clear. Now X \ Xmax being a G-stable closed subset of X, we have, ϕ(X \ Xmax ) is a closed subset of Y . Hence Y  := Y \ ϕ(X \ Xmax ) is an open subset of Y .  = ϕ −1 (Y  ). Claim: Xmax

The inclusion ⊆:  (⊆ X   Let x ∈ Xmax max ). Then ϕ(x) ∈ ϕ(X \ Xmax )(= Y \ Y ). Hence ϕ(x) ∈ Y . The inclusion ⊇:  . Let us assume Let x ∈ X be such that ϕ(x) ∈ Y  . We need to show that x is in Xmax  if possible that x ∈ Xmax . Then either x ∈ X \ Xmax , or x ∈ Xmax , but O(x) is not closed.

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In the former case, ϕ(x) ∈ ϕ(X \ Xmax )(= Y \ Y  ). Hence ϕ(x) ∈ Y  , a contradiction (note that by hypothesis, ϕ(x) ∈ Y  ). In the latter case, choose x1 ∈ O(x) \ O(x). Then “closed orbit Lemma” implies that dim O(x1 ) < dim O(x) = d. Hence x1 ∈ Xmax ; this implies that ϕ(x1 ) ∈ ϕ(X \ Xmax )(= Y \ Y  ). Hence ϕ(x1 ) ∈ Y  . This implies that ϕ(x) ∈ Y  (since x1 ∈ O(x)), a contradiction (note that by hypothesis, ϕ(x) ∈ Y  ). Thus in both cases, we arrive at contradictions. Hence our assumption is wrong and the inclusion ⊇ follows. Corollary 9.4.1.8 Let notation be as in Lemma 9.4.1.7. Let Y  be as in the proof of  → Y  is a geometric quotient. Lemma 9.4.1.7. Then ϕ : Xmax   Proof. The facts that Y  is open in Y , and Xmax = ϕ −1 (Y  ) imply that ϕ : Xmax →  Y  is a good quotient (cf. Proposition 9.4.0.2,(1)). Further, each orbit O(x), x ∈ Xmax  being closed, we have that Y  is the orbit space of Xmax for the action of G.

We next recall the relationship between closed orbits and proper morphisms. We shall first recollect the definition of proper morphisms, complete varieties and some basic properties of complete varieties. Definition 9.4.1.9 A morphism f : X → Y of schemes is proper if it is separated (i.e., Δ : X → X ×Y X is a closed immersion), of finite type, and universally closed (by “universally closed” one means the following: f is closed, and for any morphism Y  → Y , the corresponding morphism f  : X  → Y  obtained by base extension is also closed). Definition 9.4.1.10 A variety X is complete if the structure morphism f : X → Spec K is proper (note that this is equivalent to f being universally closed). Proposition 9.4.1.11 (Some basic properties of complete varieties) (1) (2) (3) (4) (5) (6) (7)

Image of a complete variety is complete. A complete subvariety is closed. A closed subvariety of a complete variety is complete. A complete quasi-projective variety is projective. Constants are the only regular functions on a connected complete variety. A connected complete affine variety is a point. Inverse image of a complete variety under a proper morphism is complete.

For a proof, see Borel [7]. The following Proposition relates closed orbits and proper morphisms. Proposition 9.4.1.12 Let X be an algebraic variety together with an action by an (affine) algebraic group G. Let x ∈ X. Let σx be the morphism G → X, g → g · x The following are equivalent: (1) O(x) is closed and dim O(x) = dim G. (2) σx is a proper morphism.

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Proof. The implication ⇒: Let σx be the morphism G → O(x), g → g · x. The hypothesis that dim O(x) = dim G implies that fibers of σx are finite. Claim: σx is a finite morphism (recall that a morphism f : V → W is finite, if for every open affine U of W , f −1 (U ) is affine, and K[f −1 (U )] is integral over K[U ]). To prove the claim, clearly it suffices to find a non-empty open set U in O(x) such that σx : σ  −1 x (U ) → U is finite; for, we can then use the action of G to cover O(x) with such open sets. But now the existence of such a U follows from the following Sublemma: Let f : V → W be a morphism with finite fibers. Then there exists a non-empty open set U in W such that f : f −1 (U ) → U is a finite morphism. For a proof of the sublemma, see Newstead [96]. Claim now implies that σx is proper (since a finite morphism is proper). Further, the inclusion i : O(x) → X is a closed immersion (since O(x) is closed), and hence is proper. Now σx being i ◦ σx (a composition of two proper morphisms) is proper. The implication ⇐: The hypothesis that σx is proper implies that σx (G) = O(x) is closed in X; further, σx−1 (x)(= Gx ) is complete (in view of Proposition 9.4.1.11, (7)). Thus Gx is complete and affine (note that Gx being a closed subgroup of G is affine); hence Gx is finite (in view of Proposition 9.4.1.11, (6)). This implies that dim O(x) = dim G.

9.5 Stable and semi-stable points Projective quotients are not as straightforward as affine quotients. To motivate our definition, note for comparison that in the case of an action by a reductive group G on an affine variety X = Spec R ⊂ An , the categorical quotient of X by G exists, and is defined on all of X. However, if X is a projective variety, then in general there may not exist any of the three types of quotients (defined so far) on all of X. The best we can do in general is find a quotient on some nice open subvariety of X as described below. Consider a projective variety X = P roj R, together with an action " by an algebraic group G. Suppose X has an open affine G-stable cover X = α∈I Spec Rα . Then Spec RαG , α ∈ I could be glued to arrive at a quotient. Again, in general such a nice affine cover does not exist for a given X. Nevertheless we are now going to get a “good quotient” on an open G-stable subset of X, namely X ss := {the semistable points of X} which will admit a G-stable open affine cover. 9.5.1 Stable, semistable, and polystable points These concepts are defined for G-linear actions: X is an algebraic variety (affine or projective), i.e., X ⊆ V or X ⊆ P(V ), and X is a closed subset of V , P(V ) respectively, V a G-module (i.e., we are given an algebraic group morphism G → AutK V ), and the action of G on X is induced from the action of G on V , P(V ) respectively.

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For the rest of this section, we shall suppose that we have a linear action on X by a reductive group G. We first consider the affine case. Let Z ⊂ K n (= An ) be a closed subset and let G act linearly on Z. Definition 9.5.1.1 A point z ∈ Z is semistable if 0 ∈ G · z. Z ss := {z ∈ Z|z is semistable}. Definition 9.5.1.2 A point z ∈ Z is polystable if G · z is a closed orbit. Z ps := {z ∈ Z|z is polystable}. Definition 9.5.1.3 A point z ∈ Z is strongly stable (or just stable) if (i) G · z is a closed orbit. (ii) dim (G · z) = dim G. Z s := {z ∈ Z|z is stable}. Proposition 9.5.1.4 Let Z ⊂ K n (= An ) be a closed subset and let a reductive group G act linearly on Z. Let z ∈ Z. We have, (i) z ∈ Z ss ⇔ there exists a homogeneous G-invariant form F of degree > 0 on K n such that F (z) = 0 (ii) z ∈ Z s ⇔ σz : G → K n , g → g · z is proper. Proof. (i) Let z ∈ Z ss . Then 0, O(z) are two disjoint G-stable closed subsets of K n . Hence by Proposition 9.2.0.1, there exists a G-invariant polynomial F such that F (0) = 0, F ( O(z) ) = 1. Now F (0) = 0 implies that the constant term in F is zero. Hence, some homogeneous part of F is non-zero at z (note that any homogeneous part of F is again G-invariant, in view of the fact that if G acts on a graded K-algebra R, then R G is also graded). Conversely, given a homogeneous G-invariant form F of degree > 0 on K n such that F (z) = 0, clearly 0 ∈ O(z) (since F is homogeneous of degree > 0, we have, F (0) = 0). (ii) follows from the definition of stability and Proposition 9.4.1.12. Next, we define the notions of semistability, polystability, and stability for actions on a projective variety X. For the following definitions, we suppose X = P roj (R) → P(V ), V a G-module, G reductive, and the G-action on X is induced from that on P(V ).  We denote by X(→ An = K n = V ), the cone Spec R over X.

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Definition 9.5.1.5 A point x ∈ X is semistable, polystable, or stable if there exists a point xˆ lying over x such that xˆ is semistable, polystable, or stable respectively. Denote X ss :={x ∈ X|x is semistable}. X ps :={x ∈ X|x is polystable}. X s :={x ∈ X|x is stable}.

(9.1) (9.2) (9.3)

Note that X s ⊂ X ps ⊂ X ss . 9.5.2 Other characterizations of stability, semistability We now describe few other characterizations of stability, semistability. We begin with the following Lemma: Lemma 9.5.2.1 Let X be an algebraic variety (affine or projective) together with a linear action by a reductive group G, i.e., X ⊆ V or X ⊆ P(V ), and V a G-module (and X is a closed G-stable subset of V , P(V ) respectively). Let f be a G-invariant form on V of degree > 0. Then Xf is G-stable. Proof. Let x ∈ Xf . Let x(∈ ˆ V ) lie over x, in case X is projective. Denote y = x or xˆ according as X is affine or projective. Then f (y) = 0. For g ∈ G, we have, f (g · y) = (g −1 f )(y) = f (y) = 0 (the second equality holds in view of G-invariance of f ). This implies g · x ∈ Xf , and the G-stability of Xf follows. Similar to Proposition 9.5.1.4,(i) we have the following: Proposition 9.5.2.2 Let X = P roj R ⊂ P(V ), V = K n be a closed subset and let a reductive group G act linearly on X. Let x ∈ X. Then x ∈ X ss ⇔ there exists a homogeneous G-invariant form F of degree > 0 on V such that F (x) = 0. The result is immediate from Proposition 9.5.1.4,(i) and the definition of semistable points on X. Next, we describe three other characterizations of stable points on projective varieties (for linear actions by reductive groups). Proposition 9.5.2.3 (First characterization of stability) Let X = P roj R ⊂ P(V ), V = K n be a closed subset and let a reductive group G act linearly on X. Let x ∈ X. Then x ∈ X s ⇔ σxˆ : G → K n , g → g · xˆ is proper, xˆ being a point lying over x. The result is immediate from Proposition 9.5.1.4,(ii). Before describing the other two characterizations for stability of points in X = P roj R, we consider the following:

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Lemma 9.5.2.4 Let X be a projective variety together with a linear action by a reductive group G, i.e., X ⊆ P(V ), and V a G-module (and X is a closed subset of P(V )). Let x ∈ X. Let f be a G-invariant form on V of degree d > 0 such that f (x) = 0. Let xˆ be a point in V lying over x. Let σxˆ , (σx )f be the morphisms ˆ (σx )f : G → P(V )f , g → g · x σxˆ : G → V , g → g · x, Then (σx )f is proper if and only if σxˆ is. Proof. Let f (x) ˆ = α( = 0), Zα = {y ∈ V | f (y) = α}, and let iα be the inclusion Zα → V ; note that iα is a closed immersion. Let (σxˆ )f be the morphism (σxˆ )f : G → Zα , g → g · xˆ Consider the commutative diagram G



(σxˆ )f



 Zα

 σxˆ 



iα

  V

This implies a factorization σxˆ = iα ◦ (σxˆ )f

(∗) Let πf be the morphism

πf : Zα → P(V )f , z → [z] We have, πf is a finite morphism (note that fibers of πf are the d-th roots of 1, d being deg f ). Consider the commutative diagram G



(σxˆ )f

 Zα



 (σx )f   πf

   P(V )f

This implies a factorization (∗∗)

(σx )f = πf ◦ (σxˆ )f

Consider the following (easy) facts: • Composite of two proper morphisms is proper. • Let F be a proper morphism of algebraic varieties. Further, let F = θ ◦ δ, with θ a separated morphism. Then δ is a proper morphism; in particular, if θ is proper, then δ is a proper morphism (θ ◦ δ being a proper morphism). • A closed immersion is proper. (In particular, iα is proper) • A finite morphism is proper. (In particular, πf is proper.) These facts together with (*), (**) imply the required result.

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Lemma 9.5.2.5 (Second characterization of stability) Let X, G be as in Lemma 9.5.2.4. Let x ∈ X. The following are equivalent: (i) x ∈ X s . (ii) (a) There exists a G-invariant form f on V of degree > 0 such that f (x) = 0. (b) O(x) is closed in Xf . (c) dim O(x) = dim G. Note: Condition (ii) is equivalent to the condition x ∈ Xfs , for some G-invariant form f on V of degree > 0. Hence we have: Reformulation: x ∈ X s if and only if there exists a G-invariant form f on V of degree > 0 such that x ∈ Xfs . Proof. Suppose that there exists a G-invariant form f on V of degree > 0, such that x ∈ Xf ; note that Xf is G-stable (cf. Lemma 9.5.2.1). Let if be the closed immersion: if : Xf :→ P(V )f . Let (φx )f be the map

(φx )f : G → Xf , g → g · x

Then (σx )f : G → P(V )f , g → g · x, factors as G



(φx )f

 Xf



 (σx )f   if

   P(V )f

Thus we obtain (σx )f = if ◦ (φx )f Hence as in the proof of Lemma 9.5.2.4, we have (∗)

(σx )f is proper ⇔ (φx )f is

(i) ⇒ (ii): The hypothesis x ∈ X s in particular implies that x ∈ X ss , and assertion ii(a) follows from Proposition 9.5.2.2. Let xˆ be a point in V lying over x. Then the hypothesis that x ∈ X s implies that xˆ is stable, and hence σxˆ : G → V , g → g · xˆ is proper (cf. Lemma 9.4.1.12). This implies (in view of Lemma 9.5.2.4) that (σx )f is proper; hence (φx )f is proper (cf. (*)). This together with Proposition 9.4.1.12 (applied to Xf ) imply assertions (ii)(b),(c). (ii) ⇒ (i): The hypotheses (ii)(b),(c) imply that (φx )f : G → Xf is proper (in view of Proposition 9.4.1.12 applied to Xf ), and hence (σx )f is proper (cf. (*)). This implies (in view of Lemma 9.5.2.4) that σxˆ : G → V is proper. It follows (in view of Proposition 9.4.1.12) that xˆ (and hence x) is stable. Lemma 9.5.2.6 (Third characterization of stability) Let X, G be as in Lemma 9.5.2.4. Let x ∈ X. The following are equivalent:

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(i) x ∈ X s . (ii) (a) There exists a G-invariant form θ on V of degree d > 0 such that θ (x) = 0. (b) G-action on Xθ is closed. (c) dim O(x) = dim G Proof. The implication ⇐ follows from Lemma 9.5.2.5. The implication ⇒: The hypothesis that x ∈ X s implies that there exists a Ginvariant form f of degree > 0 such that x ∈ Xfs (by Lemma 9.5.2.5, rather its reformulation). Let d = dim G, Z = {y ∈ Xf | dim O(y) < d}. Then in view of Lemma 9.4.1.2,(2), Z is a closed subset of Xf . Now, O(x), Z are two disjoint Gstable closed subsets of the affine variety Xf = Spec R(f ) (here, R is given by X = G such that P roj R). Hence by (geometric reductivity of G), there exists a F ∈ R(f ) F (Z) = 0, F (O(x)) = 1) (cf. Proposition 9.2.0.1). Writing F = fHr (for suitable H, r), we have H (∈ R G ). Let S = S(V ∗ ), R = S/I ; then H t ∈ S G /(I ∩ S G ) for a suitable t (cf. Newstead [96], Lemma 3.4.1). Taking a lift h for H t in S G , we obtain F t = fhs , for a suitable s. Let θ = f h. We have h(Z) = 0 (since F (Z) = 0). Therefore θ (z) = 0, for z ∈ Z. Hence (∗)

Z ∩ Xθ = ∅

Claim: For any y ∈ Xθ , dim O(y) = dim G. Let y ∈ Xθ . Then f (y) = 0 (since θ (y) = 0). Hence y ∈ Xf , and therefore Xθ ⊆ Xf . This together with (*) implies Xθ ⊆ Xf \ Z Claim follows (by the definition of Z). Claim together with “closed orbit Lemma” implies that the G-action on Xθ is closed. Now x ∈ Xθ (since f (x) = 0, F (x) = 0, we have h(x) = 0, and therefore θ (x) = 0), the G-action on Xθ is closed, and dim O(x) = dim G (cf. Lemma 9.5.2.5,ii(c)). This completes the proof of the implication ⇒, and hence of the Lemma. Combining Proposition 9.5.2.3, Lemmas 9.5.2.5, 9.5.2.6, we obtain the following: Proposition 9.5.2.7 Let X be a projective variety together with a linear action by a reductive group G, i.e., X ⊆ P(V ), and V (= K n ) a G-module (and X is a closed subset of P(V )). Let x ∈ X. The following are equivalent. (i) x ∈ X s (ii) σxˆ : G → K n , g → g · xˆ is proper, xˆ being a point lying over x. (iii) dim O(x) = dim G, and there exists a G-invariant form f on V of degree > 0 such that x ∈ Xf and O(x) is closed in Xf . (iv) dim O(x) = dim G, and there exists a G-invariant form θ on V of degree > 0 such that x ∈ Xθ and the G-action on Xθ is closed. As a consequence of Propositions 9.5.1.4, 9.5.2.2, we have the following:

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Proposition 9.5.2.8 Let X be an algebraic variety (affine or projective) together with a linear action by a reductive group G, i.e., X ⊆ V or X ⊆ P(V ), and V a G-module (and X is a closed subset of V , P(V ) respectively). Then X ss is a G-stable open subset of X. Proof. In view of Propositions 9.5.1.4, 9.5.2.2, we have, X ss = ∪ XF where the union is taken over all G-invariant forms of degree > 0 on V . Hence it follows that X ss is open. The G-stability of X ss follows from that of XF (cf. Lemma 9.5.2.1). Proposition 9.5.2.9 Let X be an algebraic variety (affine or projective) together with an action of a reductive group G. Then X s is a G-stable open subset of X. Proof. This requires a proof only when X s is non-empty. If X is affine, then the result follows from Lemma 9.4.1.7 (note that since X s is  = X s ). non-empty, we have d = dim G, Xmax Let X be projective, say, X = P roj R ⊆ P(V ), and V a G-module (and X is a closed subset of P(V )). Let Z = ∪ Xf the union being taken over the set of all G-invariant homogeneous forms f on V of degree > 0 such that the action of G on Xf is closed. Then Z is G-stable and open. Further, X s ⊆ Z (in view of Lemma 9.5.2.6); hence the maximum of the orbit dimensions on Z equals dim G. In fact, we have X s = Zmax ; for, if x ∈ Zmax , then dim O(x) = dim G, x ∈ Xf for some G-invariant homogeneous form f , and action of G on Xf is closed; hence x ∈ X s (cf. Proposition 9.5.2.7). Thus X s (= Zmax ) is G-stable and open in Z (cf. Lemma 9.4.1.3), and hence open in X.

9.6 Projective quotients In the previous sections we have described several types of quotients. The class of categorical quotients includes the class of good quotients, which in turn includes the class of geometric quotients. Let X = P roj R ⊂ P(V ), where X is closed in Pn , and V = K n . A good quotient for X ss : We will now see that X ss (= {all the semistable points of X}) admits a good quotient. As seen in the proof of Proposition 9.5.2.8, we have XF X ss =

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115

the union being taken over the set of all G-invariant homogeneous forms f on V of degree > 0. Now XF = Spec R(F ) , where R(F ) is the homogeneous localization G , and in fact is a good quotient of R at F . Then XF //G exists, namely, Spec R(F ) (recall that a categorical affine quotient is a good quotient). Hence by Proposition 9.4.0.2,(2), we have, X ss admits a good quotient; in fact, X ss //G is nothing but P roj R G . A geometric quotient for X s : Let Y = P roj R G , and ϕ : X ss → Y , the quotient map. Let W = Yf the union being taken over the set of all G-invariant homogeneous forms f on V of degree > 0 such that the action of G on Xf is closed. Let Z = ϕ −1 (W ) = ∪ Xf where the union is taken over the same set of f ’s as above. As in the proof of Proposition 9.5.2.9, we have, X s = Zmax In particular, the maximum orbit dimension on Z equals dim G. Denoting   := {x ∈ Zmax | O(x) is closed}, we have that the inclusion Zmax ⊆ Zmax is Zmax s in fact an equality; for, if x ∈ Zmax (= X ), say x ∈ Xf , for some G-invariant homogeneous f , then O(x) is closed in Xf (by the definition of Z); hence, O(x) is closed in Z in view of the following easy observation: Suppose Z = ∪ Uα is an open cover, and T is a subset of Z such that Uα ∩ T is closed in Uα for all α. Then T is closed in Z. Thus we obtain  = Zmax (= X s ) Zmax Also, by the Claim in the proof of Lemma 9.4.1.7, we have, (∗)

 ) = ϕ −1 (W \ ϕ(Z \ Zmax )) = ϕ −1 (Y s ), say X s (= Zmax

where, Y s = W \ ϕ(Z \ Zmax ). Now Y s being open in W , we get (cf. Proposition 9.4.0.2, (1)) ϕ : Xs → Y s is a good quotient. It is in fact a geometric quotient since the G-action on X s (=  ) is closed, and hence Y s is the orbit space of X s (for the action of G). Zmax Remark 9.6.0.1. Now Y s being open in Y (= X ss //G = P roj R G ), X ss //G may be regarded as a compactification of X s /G. From the discussion above, we obtain the following: Lemma 9.6.0.2 Let X, G be as in Lemma 9.5.2.4. Let x ∈ X ss . The following are equivalent: (i) x ∈ X s .

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(ii) O(x) is closed in X ss , and dim O(x) = dim G. Proof. The implication (i) ⇒ (ii): Let x ∈ X s . We have (cf. Proposition 9.5.2.7) dim O(x) = dim G. Now O(x) ⊆ ϕ −1 (ϕ(x)). Also, since ϕ(x) is in Y s , we have, ϕ −1 ϕ(x) ⊆ −1 ϕ (Y s ) = X s (cf. (*) above). Thus we obtain O(x) ⊆ X s

(∗∗) Hence

O(x) ∩ X ss ⊆ X s ∩ X ss = X s Thus, denoting O(x) ∩ X ss (the orbit closure in X ss ) by C(x), we have, C(x) ⊆ X s . This together with the “closed orbit Lemma” and the fact that dim O(y) = dim G, y ∈ X s implies that C(x) = O(x), i.e., O(x) is closed in X ss . The implication (ii) ⇒ (i): The hypothesis that O(x) is closed in X ss implies that for every G-invariant homogeneous form f such that f (x) = 0, we have, O(x) is closed in Xf (since Xf ⊆ X ss ). This together with the hypothesis that dim O(x) = dim G implies that x ∈ X s (cf. Proposition 9.5.2.7). We now give some examples: Example 1: Let X = Pn−1 . Consider the natural action of G := SLn (K) on X, we have X ss = ∅ (since constants are the only invariant polynomials). Example 2: Let G = GLn (K), X = Mn (K), G  X by conjugation. Then X ss = {non-nilpotent matrices} X s = {non-nilpotent diagonalizable matrices}. Example 3: Let G = SLd (K), X = Md×n (K), G  X by multiplication on the left. Then X s = X ss = {d × n matrices of rank d}. Example 4: Let X = {quadrics in Pn−1 }. Elements of X are of the form aij xi xj , where i and j range over 1, . . . , n and aij ∈ K. We may identify each quadric with a symmetric n × n matrix. In other words, denoting S := Sym Mn , we have a bijection bij

X ←→ P(S),



aij xi xj ←→ A = (aij ) ∈ S.

Thus we may identify X with P(S). Let G = SLn (K). Consider the action of G on X induced by the action of G on S by conjugation. The function Δ := det (i.e. the determinant) on S is G-invariant. We have, X ss = P(S)Δ (= {quadrics such that |A| = 0}), X s = ∅

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9.7 L-linear actions More generally, One defines stability, semistability on a projective variety X with respect to a L-linear action of G on X, i.e., L is a line bundle over X, together with a G-action such that under the natural projection π : L → X, the following hold: (i) π(g · y) = g · π(y), y ∈ L, g ∈ G. (ii) For x ∈ X, g ∈ G, the natural map Lx → Lg·x is linear. Let G be reductive, and L an ample line bundle on X. Definition 9.7.0.1 For a L-linear action of G on X, a point x ∈ X is semistable if there exists a G-invariant section f of Lr (for some positive integer r) such that f (x) = 0, and Xf is affine. Definition 9.7.0.2 For a L-linear action of G on X, a point x ∈ X is stable if (i) dim O(x) = dim G. (ii) There exists a G-invariant section f of Lr (for some positive integer r) such that f (x) = 0, Xf is affine, and the action of G on Xf is closed. Remark 9.7.0.3.Note that Definition 9.5.1.5, (1),(3) coincide with Definitions 9.7.0.1, 9.7.0.2, with L being the restriction to X of the tautological line bundle on P(V ). Denote X ss (L), X s (L) the set of points given by Definitions 9.7.0.1, 9.7.0.2 respectively. We have results for X ss (L), X s (L) similar to the results for X ss , X s as seen in §9.6: Theorem 9.7.0.4 (i) There exists a good quotient (YL , ϕL ) of X ss (L) by G, and YL is quasi-projective. (ii) There exists an open subset YLs of YL , such that ϕ −1 (YLs ) = X s (L), and (YLs , ϕ) is a geometric quotient of X s (L) by G. (iii) For x1 , x2 ∈ X ss (L), ϕ(x1 ) = ϕ(x2 ) ⇔ O(x1 ) ∩ O(x2 ) ∩ X ss (L) = ∅ (iv) Let x ∈ X ss (L). The following are equivalent: a) x ∈ X s (L). b) O(x) is closed in X ss (L), and dim O(x) = dim G. For a proof of Theorem 9.7.0.4, see Newstead [96], Theorem 3.21.

9.8 Hilbert-Mumford criterion In this section, we recall Hilbert-Mumford criterion. In practice it is a hard problem to find (calculate) X ss and X s for a general X and G. The Hilbert-Mumford criterion gives a numerical criterion for the determination of X ss and X s in terms of one-parameter subgroups of G given by algebraic

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group morphisms λ : K ∗ → G. In the sequel, we shall abbreviate “one-parameter subgroup” as 1-PS.  ⊂ V (= K n ) be the cone over X; the actions of Let X = P roj R ⊆ P(V ) and X  X respectively. Fix x ∈ X. Given a 1-PS G on V , P(V ) induce (linear) actions on X, ϕ λ : K ∗ → G, define ϕλ : K ∗ → K n \{0} by t −→ λ(t) · x. ˆ Question: Does ϕλ extend to all of P1 (= K ∪ {∞}) ? In general, a morphism f : K ∗ → Y may not extend to all of P1 . Fact: If Y is projective, then f does extend to all of P1 , and the extension is unique. For a proof of the above Fact, see Borel [7]. In the present situation, with f = ϕλ , we have, Y = K n \{0}; hence, ϕλ extends to all of P1 and the extension is unique. Let ϕ0 = ϕλ (0), and ϕ∞ = ϕλ (∞). One denotes ϕ0 , ϕ∞ , also by limt→0 ϕλ (t), limt→∞ ϕλ (t) respectively. The next question is how to compute ϕ0 and ϕ∞ . We have λ : K ∗ → G → Aut V , where V = K n . Then λ(K ∗ ) is a family of commuting semisimple endomorphisms of V , and so the actions of λ(K ∗ ) on V can be simultaneously diagonalized. We ri denote the corresponding basis by {ei }. We have, λ(t)ei = t ei for some ri ∈ Z. Let xˆ = ci ei . Then  (∗) λ(t) · xˆ = ci t r i ei Define μ(λ, x) := − min{ri }. Theorem 9.8.0.1 (Hilbert-Mumford criterion) We have (i) x ∈ X ss if and only if μ(λ, x) ≥ 0, for every one-parameter subgroup λ of G. (ii) x ∈ X s if and only if μ(λ, x) > 0, for every one-parameter subgroup λ of G. Sketch of proof of Hilbert-Mumford criterion: we first note the following: (i) μ(λ, x) > 0 if and only if there exists a term t ri with ri < 0 on the right hand side of (*). (ii) μ(λ, x) = 0 if and only if there are no negative exponents (for t) on the right hand side of (*), and there is a term not involving t on the right hand side of (*) (iii) μ(λ, x) < 0 if and only if all exponents (for t) on the right hand side of (*) are > 0. We may reformulate these in terms of lim λ(t) · x: ˆ • μ(λ, x) > 0 if and only if lim λ(t) · xˆ does not exist in K n . t→0

• μ(λ, x) ≥ 0 if and only if lim λ(t) · xˆ either does not exist in K n , or the limit t→0

exists in K n and is non-zero.

We shall first see a proof of Hilbert-Mumford criterion for G = K ∗ :

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Theorem 9.8.0.2 Let G = K ∗ . Then (i) x ∈ X ss if and only if μ(λ, x) ≥ 0, for every one-parameter subgroup λ of G. (ii) x ∈ X s if and only if μ(λ, x) > 0, for every one-parameter subgroup λ of G. Proof. We first prove (ii). We have, by definition, x ∈ X s if and only if (i) G · xˆ is closed, and (ii) dim G · xˆ = dim G. We have ϕxˆ : K ∗ → G · xˆ ⊂ K n defined by t → t · x. ˆ Since dim K ∗ = 1, dim ϕxˆ (K ∗ ) is either 0 or 1. If dim ϕxˆ (K ∗ ) = 0, then xˆ is G-fixed. Otherwise, dim ϕxˆ (K ∗ ) = 1, in which case x is stable if and only if G · xˆ is closed in K n . Now the closure of G · xˆ in P(V ) is G · xˆ ∪ {ϕ0 , ϕ∞ }. Hence for the closure of G · xˆ in K n , we have four cases, depending on whether either or both of ϕ0 and ϕ∞ belong to K n or not: ⎧ if ϕ0 , ϕ∞ ∈ K n ⎨ G · xˆ G · xˆ = G · xˆ ∪ {ϕ0 , ϕ∞ } if ϕ0 , ϕ∞ ∈ K n ⎩ G · xˆ ∪ {ϕ } if only ϕ ∈ K n ,  = 0 or ∞ Now G · xˆ is closed in K n if and only if neither ϕ0 nor ϕ∞ is in K n , which is to say that the limits lim t · xˆ and lim t · xˆ do not exist in K n . This holds if and only if t→0

t→∞

there are both negative and positive exponents of t in formula (*). Now if we define λ1 : K ∗ → K ∗ by t → t and λ2 : K ∗ → K ∗ by t → t −1 , then any one-parameter subgroup λ : K ∗ → K ∗ is given by t → t r for some r ∈ Z. Hence, λ = λri for some i ∈ {1, 2} and r ≥ 0. Thus there are both negative and positive exponents of t in formula (*) if and only if μ(x, λ1 ) > 0 and μ(x, λ2 ) > 0, which happens if and only if μ(x, λ) > 0 for every one-parameter subgroup λ. This completes the proof of (ii). ˆ which The proof of (i) is similar. We have that x ∈ X ss if and only if 0 ∈ G · x, happens if and only if the limits lim t · xˆ and lim t · xˆ are not zero. This happens if t→∞

t→0

and only if μ(x, λ1 ) ≥ 0 and μ(x, λ2 ) ≥ 0, which happens if and only if μ(x, λ) ≥ 0 for every one-parameter subgroup λ. This completes the proof of (i). Proof of Hilbert-Mumford criterion: Now we let G be any reductive group. We first prove the implication (⇒) for (i) and (ii). Let xˆ be a point in K n lying over x. (i) Let x ∈ X ss . Then 0 ∈ G · x. ˆ This implies that 0 ∈ K ∗ · xˆ for every one∗ parameter subgroup λ : K → G. Hence x ∈ X ss for the K ∗ action on X, and this implies that μ(x, λ) ≥ 0 for every one-parameter subgroup λ by Theorem 9.8.0.2, (i). (ii) Let x ∈ X s . Let λ : K ∗ → G be a (non-trivial one-parameter subgroup). Consider the commutative diagram λ

K∗ → G ↓ ↓σxˆ ∗ K · xˆ → G · xˆ → K n ϕxˆ

The hypothesis that x ∈ X s implies that dim G · x = dim G, and hence Gxˆ is finite. This implies

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9 Review of GIT

Kx∗ˆ is finite

Further λ(K ∗ ) is closed in G (since the image of an algebraic group homomorphism ˆ σxˆ (λ(K ∗ ))) is closed in K n (since x ∈ X s , σxˆ is is closed), and hence, K ∗ · x(= proper (cf. Proposition 9.4.1.12), in particular closed). This together with (*) implies that x is stable for the K ∗ action, and the required result follows from this (in view of Theorem 9.8.0.2,(ii)). The implication (⇐): We are required to show the following: (i)if x ∈ X s , then there exists a 1-PSλ such that μ(x, λ) ≤ 0, (ii)if x ∈

X ss , then there exists a 1-PSλ such that μ(x, λ) < 0. In view of Theorem 9.8.0.2, it suffices to show that if x is non-stable (resp. nonsemistable), then there exists a 1-PS λ such that x is still non-stable (resp. nonsemistable) for the induced action of K ∗ . Such a λ is constructed using the “valuative criterion” for proper morphisms (see [37] for example), and Iwahori decomposition G(L) = G(A)λL (L∗ )G(A) {λL }

of L-valued points of (the reductive group) G, in terms of the A-valued points of G, where A is a valuation ring and L is the quotient field of A; the union is taken over all 1-PS’s λ. Also, λL denotes the 1-PS L∗ → G(L) induced by λ : K ∗ → G(K). We skip the details (and refer the reader to [110]).

10 Classical Invariant Theory

In this chapter, we describe classical invariant theory (cf. [115]) for the classical group actions. We give GIT-theoretic proofs of the results in [115]. By relating them to certain G/P ’s (and their Schubert varieties), we describe “standard monomial bases” for these rings of invariants.

10.1 Preliminary lemmas In this section, we first prove some Lemmas concerning quotients leading to the main Lemma (cf. Lemma 10.1.0.4), to be applied to the following situation: Suppose, we have an action of a reductive group G on an affine variety X = SpecR. Suppose that S is a subalgebra of R G . We give below (cf. Lemma 10.1.0.4) a set of sufficient conditions for the equality S = R G . We start with recalling Theorem 10.1.0.1 (Zariski Main Theorem, [86],III.9) Let ϕ : X → Y be a morphism such that (i) (ii) (iii) (iv)

ϕ is surjective fibers of ϕ are finite ϕ is birational Y is normal

Then ϕ is an isomorphism. Let X = SpecR and a reductive group G act linearly on X, i.e., we have a linear action of G on an affine space Ar and we have a G-equivariant closed immersion X → Ar . Further, let R be a graded K-algebra. Let X ss be the set of semi-stable points of X (i.e., points x such that 0 ∈ G · x). Let X1 = P roj R, and X1ss , the set of semi-stable points of X1 (i.e., points y ∈ X1 such that if xˆ is any point in K n+1 \ 0 lying over y, then xˆ is in X ss ). Let f1 , · · · , fN be homogeneous G-invariant elements in R. Let S = K[f1 , · · · , fN ]. Then for the morphism Spec R G → Spec S, the hypothesis (2) in Theorem 10.1.0.1 may be concluded if {f1 , · · · , fN } is base-point free on X1ss as given by the following

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10 Invariant theory

Lemma 10.1.0.2 Suppose f1 , · · · , fN are homogeneous G-invariant elements in R such that for any x ∈ X ss , fi (x) = 0, for at least one i. Then Spec R G → Spec S has finite fibers. Proof. Case 1: Let f1 , · · · , fN be of the same degree, say, d. Let Y = SpecR G (= X//G, the categorical quotient) and ϕ : X → Y be the canonical quotient map. Let X1 = P roj R, and X1ss be the set of semi-stable points of X1 . Let Y1 = P roj R G (= X1ss //G), and ϕ1 : X1ss → Y1 be the canonical quotient map. Consider ψ : X → AN , x → (f1 (x), · · · , fN (x)). This induces a map ρ : Y → AN (since f1 , · · · , fN are G-invariant). The commutative diagram X



ϕ



 Y

ψ 



ρ

   AN

induces the commutative diagram X1ss



ϕ1

 Y1



 ψ1   ρ1

   PN −1

PN −1

Note that ψ1 : → is defined in view of the hypothesis that for any x ∈ X ss , fi (x) = 0, for at least one i. Note also that f1 , . . . , fN are sections of the ample line bundle OX1 (d) as well as the basic fact from GIT that this line bundle descends to an ample line bundle on Y1 , which we denote by OY 1 (d). Claim 1: ρ1 is a finite morphism. Proof of Claim 1: Since f1 , · · · , fN are G-invariant, we get that fi ∈ H 0 (Y1 , OY1 (d)). Hence we obtain that X1ss

ρ1∗ (OPN−1 (1)) = OY1 (d) Thus, ρ1∗ (OPN−1 (1)) is ample, and hence ρ1 is finite (over any fiber (ρ1 )z , z ∈ PN −1 , ρ1∗ (OPN−1 (1)) | (ρ1 )z is both ample and trivial, and hence dim(ρ1 )z is zero), and Claim 1 follows. Claim 2: ρ is a finite morphism.  . We have PN −1 = ProjK[x , . . . , Proof of Claim 2: Let S  = R G . Let S  (d) = ⊕n Snd 1 xN ]. Since ρ1 is finite, OY1 is a coherent OPN−1 -module. We see that H 0 (PN −1 , OY1 ⊗ OPN−1 (n)) ) H 0 (Y1 , ρ1∗ (OPN−1 (n)) since the direct image of ρ1∗ (OPN−1 (n)) by ρ1 is OY1 ⊗ OPN−1 (n) and ρ1 is a finite morphism. On the other hand we have ρ1∗ (OPN−1 (n)) ) OY1 (nd).

10.1 Preliminary lemmas

123

Thus we have  . H 0 (PN −1 , OY1 ⊗ OPN−1 (n)) ) H 0 (Y1 , OY1 (nd)) ) Snd

Thus the graded K[x1 , . . . , xN ]-module associated to the coherent sheaf OY1 on PN −1 is S  (d) and by the basic theorems of Serre, S  (d) is of finite type over K[x1 , . . . , xN ]. Now a d-th power of any homogeneous element of S  is in S  (d) and thus S  is integral over K[x1 , . . . , xN ], which proves that ρ is finite. Claim 2 and hence the required result follows from this. Case 2: Let f1 , · · · , fN be homogeneous possibly of different degrees, say, degfi = di . Let d = l.c.m.{di }, ei = ddi . Set gi = fiei , 1 ≤ i ≤ N. Then {g1 , · · · , gN } is again base-point free on (P roj R)ss . Hence by Case 1, we have that R G is a finite K[g1 , · · · , gN ]-module, and hence a finite K[f1 , · · · , fN ]-module (note that K[g1 , · · · , gN ] → K[f1 , · · · , fN ] → R G ). In the Lemma below, we describe a set of sufficient conditions for (3) of Theorem 10.1.0.1, namely, birationality. Lemma 10.1.0.3 Suppose F : X → Y is a surjective morphism of (irreducible) algebraic varieties, and U is an open subset of X such that (i) F | U : U → Y is an immersion (ii) dim U = dim Y . Then F is birational. Proof. Hypothesis (1) implies that F (U ) is locally closed in Y . This fact together with Hypothesis (2) implies that F (U ) is open in Y , and the result follows. We now return to the situation of a linear action of a reductive group G on an affine variety X = SpecR with R a graded K-algebra. Let f1 , · · · , fN be homogeneous G-invariant elements in R. Let S = K[f1 , · · · , fN ]. Combining Theorem 10.1.0.1 with Lemmas 10.1.0.2, 10.1.0.3, we arrive at the following Lemma which gives a set of sufficient conditions for the equality S = R G . Before stating the lemma, let us observe the following. Suppose that U is a non-empty G-stable open subset in X. Since ϕ : X −→ Spec R G is surjective, ϕ(U ) contains a non-empty open subset. Hence by shrinking U , if necessary, we can suppose that ϕ(U ) is open. We suppose that this is the case and denote ϕ(U ) by U //G. Lemma 10.1.0.4 Let notation be as above. Let ψ : X → AN be the map, x → (f1 (x), · · · , fN (x)). Denote D = SpecS. Then D is the categorical quotient of X by G and ψ : X → D is the canonical quotient map, if the following conditions are satisfied: (i) For x ∈ X ss , ψ(x) = (0). (ii) There is a G-stable open subset U of X such that ψ | U //G : U //G → D is an immersion. (iii) dim D = dim U //G. (iv) D is normal.

124

10 Invariant theory

Remark 10.1.0.5. Suppose that U is a (non-empty) G-stable open subset of X, G operates freely with U/G as quotient, and ψ induces an immersion of U/G in AN . Then (ii) is satisfied: This assertion is immediately seen, for we have U/G −→ U //G −→ AN and the fact that U/G −→ AN is an immersion implies that U //G −→ AN immersion. In the following sections, using Lemma 10.1.0.4, we give a GIT-theoretic proof of the First and Second Fundamental Theorems of Classical invariant theory (cf. Weyl [115]) for the classical group actions.

10.2 SLd (K)-action In this section, we give a GIT-theoretic proof of the results for SLd (K)-actions appearing in classical invariant theory (cf. Weyl [115]). Let X = Mn,d , the space of n × d matrices with entries in K; further, let n > d. Let R = K[xij , 1 ≤ i ≤ n, 1 ≤ j ≤ d] be the coordinate ring of X. 10.2.1 The functions fτ Let A ∈ X. Consider the d × d-submatrices of A; these are indexed by Id,n := {i = (i1 < · · · < id )} namely, for i ∈ Id,n , let Ai denote the d × d-submatrix of A with row indices given by {i1 , . . . , id }. Then {Ai , i ∈ Id,n } give all the d × d-submatrices of A. For τ ∈ Id,n , let fτ be the regular function on X: fτ : X → K, A → |Aτ | (determinant of Aτ ) SLd (K)-action: Let G = SLd (K). Consider the action of G on X by right multiplication. For τ ∈ Id,n , A ∈ X, g ∈ G, we have, fτ (A · g) = |Aτ ||g| = |Aτ |. Thus (∗)

fτ ∈ R G , τ ∈ Id,n

Proposition 10.2.1.1 Let X, G be as above. Let X ◦ be the set of matrices in X of rank d (the maximum possible rank). Then X s = X ss = X ◦ Proof. We first Claim 1: We have (i)

X ss = X ◦

10.2 SLd (K)-action

125

The inclusion X ss ⊆ X ◦ : Let A ∈ X ◦ . We shall show that A ∈ X ss . Let r = rank(A)(< d). Then there exists a g ∈ GLd (K) such that Ag is in reduced row echelon form: ⎞ ⎛ 1 0 ··· 0 0 ··· 0 ⎜0 1 · · · · · · 0 · · · 0⎟ ⎟ ⎜ ⎜ .. .. .. .. . .⎟ ⎜ . . . . 0 .. .. ⎟ ⎟ ⎜ ⎟ Ag = ⎜ ⎜0 0 · · · 1 0 · · · 0⎟ ⎜0 0 · · · 0 0 · · · 0⎟ ⎟ ⎜ ⎜. . . . . . .⎟ ⎝ .. .. .. .. .. .. .. ⎠ 0 0 ··· 0 0 ··· 0 with precisely r non-zero columns (which appear as the first r columns). Let c = |g|; pick α, an n-th root of c. Then h := α1 g is in G. Then Ag = A(αh); replacing A by Ah(∈ O(A)), the SLd (K)-orbit of A), we may suppose that ⎞ ⎛ 1 0 ··· 0 0 ··· 0 ⎜0 1 · · · · · · 0 · · · 0⎟ ⎟ ⎜ ⎜ .. .. .. .. . .⎟ ⎜ . . . . 0 .. .. ⎟ ⎟ 1⎜ 0 0 · · · 1 0 · · · 0⎟ A= ⎜ ⎟ α⎜ ⎜0 0 · · · 0 0 · · · 0⎟ ⎟ ⎜ ⎜. . . . . . .⎟ . . . . . . . ⎝. . . . . . .⎠ 0 0 ··· 0 0 ··· 0 Choose integers {a1 , · · · , ad } such that  (∗∗) ai = 0, and ai > 0, 1 ≤ i ≤ r Consider the 1-PS Γ = {Dt ∈ G, t = 0}, where Dt is the diagonal matrix diag(t a1 , · · · , t ad ). Then ⎛ a1 ⎞ t 0 ··· 0 0 ··· 0 ⎜ 0 t a2 · · · · · · 0 · · · 0⎟ ⎜ ⎟ ⎜ .. .. .. .. . .⎟ ⎜ . . . . 0 .. .. ⎟ ⎟ 1⎜ 0 0 · · · t ar 0 · · · 0⎟ A · Dt = ⎜ ⎜ α ⎜ 0 0 · · · 0 0 · · · 0⎟ ⎟ ⎜ ⎟ ⎜ . . . . . . .⎟ . . . . . . . ⎝ . . . . . . .⎠ 0 0 ··· 0 0 ··· 0 Hence we obtain (in view of (**)) lim A · Dt = 0(∈ X).

t→0

Thus the origin 0 is in the closure of O(A), and consequently A is not semi-stable, and the inclusion X ss ⊆ X ◦ follows.

126

10 Invariant theory

The inclusion X ◦ ⊆ X ss : Let A ∈ X ◦ . We have that for some τ ∈ Id,n , fτ (A) = 0. This together with the G-invariance of fτ (cf. (*) above) implies that A ∈ X ss . This completes the proof of Claim 1. Claim 2: Let A ∈ X ◦ . Then the stabilizer of A equals e = I dd×d . Let τ ∈ Id,n be such that |Aτ | = 0. For g ∈ G, we have, (A · g)τ = Aτ g. Hence if A · g = A, then we have, Aτ = (A · g)τ = Aτ g This together with the invertibility of Aτ implies that stabilizer of A is just {e}, and Claim 2 follows. We have (by Claim 2) that O(A) (the G-orbit through A) is isomorphic to G; in particular, we have, (ii)

dim O(x) = dim G, x ∈ X ◦

Claim 3: Let x ∈ X ◦ . Then (iii)

O(x) = O(x),

i.e, O(x) is closed. If possible, assume O(x)  O(x). Pick a point y ∈ O(x) \ O(x). Then (ii) together with “closed orbit Lemma” (cf. Chapter 9, Lemma 9.4.0.3) implies that dim O(y) < dim G; hence y ∈ X ◦ (again by (ii)). This together with Claim 1 (cf. (i) above) implies that y is not semi-stable, and therefore 0 ∈ O(y)(⊆ O(x)), a contradiction (note that in view of (i), x is semi-stable). Now (ii) and (iii) imply X s = X ◦ . This together with (i) implies X s = X ss = X ◦ 10.2.2 The First and Second Fundamental Theorems of Classical Invariant Theory (cf. Weyl [115]) for the SLd (K)-action:   Let N = dn (= #Id,n ), and let ψ be the map ψ : X → AN , A → (|Aτ |, τ ∈ Id,n ) Theorem 10.2.2.1 Let X, G be as above. The categorical quotient X//G is isomord,n denotes the cone over the Grassmannian Gd,n . d,n , where, G phic to G Proof. Let ψ : X → AN be as above. We shall verify the hypotheses of Lemma 10.1.0.4 for X and ψ. d,n (since, K[Gd,n ] is generated First observe that D := image of ψ = G by |Aτ | (cf. Chapter 4)). Hence (iv) of Lemma 10.1.0.4 is satisfied. As above let X ◦ be the set of n × d matrices of rank d. Then we have (cf. Proposition 10.2.1.1) that if A ∈ X ss , then A ∈ X ◦ ; hence, ψ(A) = 0. Thus (i) of Lemma 10.1.0.4 is satisfied.

10.2 SLd (K)-action

127

Next we show that (ii) of Lemma 10.1.0.4 holds. Denote G = GLd (K). Let U = {A ∈ X | Aid ∈ G } where id is the d-tuple (1, 2, · · · , d), i.e., Aid is invertible. Clearly, U is a G-stable open subset of X (note that U = {A ∈ X | |Aid | = 0}). Claim: G operates freely on U , U → U mod G is a G-principal fiber space, and ψ induces an immersion U/G → AN . Proof of Claim: We have a G-equivariant isomorphism (†)

U∼ = G × Ms,d

where s = n − d. (Here, Ms,d denotes the space of s × d matrices with entries in K.) From this it is clear that and G operates freely on U . Further, we see that U mod G may be identified with the fiber space with base G mod G, and fiber Ms,d , associated to the principal fiber space G → G / G. It remains to show that ψ induces an immersion U/G → AN , i.e., to show that the map ψ : U/G → AN and its differential dψ are both injective. We first prove that ψ : U/G → AN is injective. Let x, x  in U/G be such that ψ(x) = ψ(x  ). Let η, η ∈ U be lifts for x, x  respectively. Using the identification (†) above, we may write η = (A, M), A ∈ G , M ∈ Ms,d , η = (A , M  ), A ∈ G , M  ∈ Ms,d The fact that ψ(x) = ψ(x  ) implies |A| = |A |. Hence g := A−1 A is in G(= SLd (K)). Thus we obtain A = A · g. Hence on U/G, we may suppose that x = (A, M), x  = (A, M  ) Let |A| = c( = 0); pick α, an nth root of c. Then A = αg, where g := Hence we may suppose (on U/G)

1 αA

is in G.

x = (αI dd×d , M), x  = (αI dd×d , M  ) Thus, we have lifts B, B  for x, x  respectively, where the n × d matrices B, B  are given by αI dd×d sitting on the top of the s × d matrices M, M  respectively:     αI dd×d αI dd×d  B= , B = M M Let i > d and 1 ≤ j ≤ d; let τ be the d-tuple (1, 2, · · · , j − 1, j + 1, · · · , d, j ). We have, |Bτ | = |Bτ | (since ψ(x) = ψ(x  )); but  , ε = ±1 |Bτ | = εα d−1 bij , |Bτ | = εα d−1 bij

Hence we obtain

 , i > d, 1 ≤ j ≤ d bij = bij

128

10 Invariant theory

This implies M = M  and hence x = x  which proves the injectivity of ψ : U/G → AN . To prove that the differential dψ is injective, we merely note that the above argument remains valid for the points over K[], the algebra of dual numbers (= K ⊕K, the K-algebra with one generator , and one relation  2 = 0), i.e., it remains valid if we replace K by K[], or in fact by any K-algebra. Thus condition (ii) of Lemma 10.1.0.4 holds. d,n . We have It remains to show dim U//G = dim G d,n dim U//G = dim U − dim G = nd − (d 2 − 1) = d(n − d) + 1 = dim G Thus all of the conditions of Lemma 10.1.0.4 hold for X, ψ, and hence Lemma d,n . 10.1.0.4 implies that X//G = G d,n and the results from Chapter 4, we have Using the identification X//G = G the following Theorem: Theorem 10.2.2.2 (i) First Fundamental Theorem The ring of invariants R G is generated by {fτ , τ ∈ Id,n }. (ii) Second Fundamental Theorem The ideal of relations among the generators in (i) is generated by the Plücker quadratic relations. Defining standard monomials in fτ ’s as in Chapter 4, we have (cf. Chapter 4) the following: Theorem 10.2.2.3 A standard monomial basis for the ring of invariants: The ring of invariants R G has a basis consisting of standard monomials in the regular functions fτ , τ ∈ Id,n .

10.3 GLn (K)-action: In this section, we give a GIT-theoretic proof of the results for GLn (K)-actions appearing in classical invariant theory (cf. DeConcini-Procesi [22], Weyl [115]). ∗ · · ⊕ V ⊕ V · · ⊕ V ∗ . Let V = K n , X = V  ⊕ ·  ⊕ · m copies

q copies

The GL(V )-action on X: Writing u = (u1 , u2 , ..., um ) with ui ∈ V and ξ = (ξ1 , ξ2 , ..., ξq ) with ξi ∈ V ∗ , we shall denote the elements of X by (u, ξ ). The (natural) action of GL(V ) on V induces an action of GL(V ) on V ∗ , namely, for ξ ∈ V ∗ , g ∈ GL(V ), denoting g · ξ by ξ g , we have ξ g (v) = ξ(g −1 v), v ∈ V The diagonal action of GL(V ) on X is given by g

g

g

g · (u, ξ ) = (gu, ξ g ) = (gu1 , gu2 , ..., gum , ξ1 , ξ2 , ..., ξq ), g ∈ G, (u, ξ ) ∈ X

10.3 GLn (K)-action:

129

The induced action on K[X] is given by (g · f )(u, ξ ) = f (g −1 (u, ξ )), f ∈ K[X], g ∈ GL(V ) Consider the functions ϕij : X −→ K defined by ϕij ((u, ξ )) = ξj (ui ), 1 ≤ i ≤ m, 1 ≤ j ≤ q. Each ϕij is a GL(V )-invariant: For g ∈ GL(V ), we have, (g · ϕij )((u, ξ )) = ϕij (g −1 (u, ξ )) −1

= ϕij ((g −1 u, ξ g ) g −1

= ξj (g −1 ui ) = ξj (ui ) = ϕij ((u, ξ )) It is convenient to have a description of the above action in terms of coordinates. So with respect to a fixed basis, we write the elements of V as row vectors and those of V ∗ as column vectors. Thus denoting by Ma,b the space of a × b matrices with entries in K, X can be identified with the affine space Mm,n × Mn,q . The action of GLn (K) (= GL(V )) on X is then given by A · (U, W ) = (U A, A−1 W ), A ∈ GLn (K), U ∈ Mm,n , W ∈ Mn,q And the action of GLn (K) on K[X] is given by     (A · f )(U, W ) = f A−1 (U, W ) = f U A−1 , AW , f ∈ K[X] Writing U = (uij ) and W = (ξkl ) we denote the coordinate functions on X, by uij and ξkl . Further, if ui denotes the i-th row of U and ξj the j -th column of W, the invariants ϕij described above are nothing but the entries ui , ξj  (= ξj (ui )) of the product U W. In the sequel, we shall denote ϕij (u, ξ ) also by ui , ξj . The Function p(A, B): For A ∈ I (r, m), B ∈ I (r, q), 1 ≤ r ≤ n, let p(A, B) be the regular function on X: p(A, B)((u, ξ )) is the determinant of the r × r-matrix . ( ui , ξj )i∈A, j ∈B . Let S be the subalgebra of R G generated by {p(A, B)}. We shall now show (using Lemma 10.1.0.4) that S is in fact equal to R G . 10.3.1 The First and Second Fundamental Theorems of Classical Invariant Theory (cf. [115]) for the action of GLn (K): Theorem 10.3.1.1 Let G = GLn (K). Let X be as above. The morphism ψ : X → Mm,q , (u, ξ ) → ϕij (u, ξ ) (= (ui , ξj )) maps X into the determinantal variety Dn+1 (Mm,q ), and the induced homomorphism ψ ∗ : K[Dn+1 (Mm,q )] → K[X] between the coordinate rings induces an isomorphism ψ ∗ : K[Dn+1 (Mm,q )] → K[X]G , i.e. the determinantal variety Dn+1 (Mm,q ) is the categorical quotient of X by G.

130

10 Invariant theory

Proof. Clearly, ψ(X) ⊆ Dn+1 (Mm,q ) (since, ψ(X) = Spec S, and clearly Spec S ⊆ Dn+1 (Mm,q ) (since any n + 1 vectors in V are linearly independent)). We shall prove the result using Lemma 10.1.0.4. To be very precise, we shall first check the conditions (i)-(iii) of Lemma 10.1.0.4 for ψ : X → Mm,q , deduce that the inclusion Spec S ⊆ Dn+1 (Mm,q ) is in fact an equality, and hence conclude the normality of Spec S (condition (iv) of Lemma 10.1.0.4). (i) Let x = (u, ξ ) = (u1 , . . . , um , ξ1 , . . . , ξq ) ∈ X ss . Let Wu be the subspace of V spanned by xi ’s and Wξ the subspace of V ∗ spanned by ξj ’s. Assume if possible that ψ((u, ξ )) = 0, i.e. ui , ξj  = 0 for all i, j . Case (a): Wξ = 0, i.e., ξj = 0 for all j . Consider the one parameter subgroup Γ = {gt , t = 0} of GL(V ), where gt = tIn , In being the n × n identity matrix. Then gt · x = gt · (u, 0) = (tu, 0), so that gt · x → (0) as t → 0. Thus the origin 0 is in the closure of G · x, and consequently x is not semi-stable, which is a contradiction. Case (b): Wξ = 0. Since the case Wu = 0 is similar to Case (a), we may assume that Wu = 0. Also the fact that Wξ = 0 together with the assumption that xi , ξj  = 0 for all i, j implies that dimWu < n. Let r = dim Wu so that we have 0 < r < n. Hence, we can choose a basis {e1 , . . . , en } of V such that Wu = e1 , . . . , er , r < n, and ∗ , . . . , e∗ , where {e∗ , . . . , e∗ } is the dual basis in V ∗ . Consider the one Wξ ⊂ er+1 n n 1 parameter subgroup Γ = {gt , t = 0} of GL(V ), where   0 tIr gt = . 0 t −1 In−r We have gt · (u, ξ ) = (tu, tξ ) → 0 as t → 0. Thus, by the same reasoning as in Case (a), the point (u, ξ ) is not semi-stable, which leads to a contradiction. Hence we obtain ψ((u, ξ )) = 0. Thus (i) of Lemma 10.1.0.4 holds. (ii) Let U = {(u, ξ ) ∈ X | {u1 , . . . , un }, {ξ1 , . . . , ξn } are linearly independent} Clearly, U is a G-stable open subset of X. Claim: G operates freely on U , U → U mod G is a G-principal fiber space, and ψ induces an immersion U/G → Mm,q . Proof of Claim: We have a G-equivariant identification (†)

· · × V ∗ U∼ · · × V × V ∗ × · = G × G × V × · (m−n) copies

(q−n) copies

from which it is clear that and G operates freely on U . Further, we see that U mod G may be identified with the fiber space with base G×G mod G (G acting on G×G as ∗ · · × V × V · · × V ∗ g ·(g1 , g2 ) = (g1 g, g −1 g2 ), g, g1 , g2 ∈ G), and fiber V × ·  × · (m−n) copies

(q−n) copies

associated to the principal fiber space G×G → (G×G) / G. It remains to show that

10.3 GLn (K)-action:

131

ψ induces an immersion U/G → AN , i.e., to show that the map ψ : U/G → Mm,q and its differential dψ are both injective. We first prove that ψ : U/G → Mm,q is injective. Let x, x  in U/G be such that ψ(x) = ψ(x  ). Let η, η ∈ U be lifts for x, x  respectively. Using the identification (†) above, we may write η = (A, un+1 , · · · , um , B, ξn+1 , · · · , ξq ), A, B ∈ G  η = (A , un+1 , · · · , um , B  , ξn+1 , · · · , ξq ), A , B  ∈ G 

(here, ui , 1 ≤ i ≤ n are given by the rows of A, while ξi , 1 ≤ i ≤ n are given by the columns of B; similar remarks on ui , ξi ). The hypothesis that ψ(x) = ψ(x  ) implies in particular that ui , ξj  = ui , ξj , 1 ≤ i, j ≤ n which may be written as

AB = A B 

This implies that A = A · g, where g = BB  −1 . Hence on U/G, we may suppose that x = (u1 , · · · , un , un+1 , · · · , um , ξ1 , · · · , ξq ) x  = (u1 , · · · , un , un+1 , · · · , um , ξ1 , · · · , ξq ) where {u1 , . . . , un } is linearly independent. For a given j , we have, ui , ξj  = ui , ξj , 1 ≤ i ≤ n, implies ξj = ξj (since {u1 , . . . , un } is linearly independent). Thus we obtain ξj = ξj , for all j

(∗)

On the other hand, we have (by definition of U ) that {ξ1 , . . . , ξn } is linearly independent. Hence fixing an i, n + 1 ≤ i ≤ m, we get ui , ξj  = ui , ξj (= ui , ξj ), 1 ≤ j ≤ n, implies ui = ui Thus we obtain ui = ui , for all i

(∗∗)

The injectivity of ψ : U/G → Mm,q follows from (*) and (**). To prove that the differential dψ is injective, we merely note that the above argument remains valid for the points over K[], the algebra of dual numbers (= K ⊕K, the K-algebra with one generator , and one relation  2 = 0), i.e., it remains valid if we replace K by K[], or in fact by any K-algebra. Thus (ii) of Lemma 10.1.0.4 holds. (iii) We have dim U/G = dim U − dim G = (m + q)n − n2 = dim Dn+1 (Mm,q ).

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10 Invariant theory

The immersion U/G → Spec S(⊆ Dn+1 (Mm,q )) together with the fact above that dim U/G = dim Dn+1 (Mm,q ) implies that Spec S in fact equals dim Dn+1 (Mm,q ). Thus (iii) of Lemma 10.1.0.4 holds. (iv) The normality of Spec S(= Dn+1 (Mm,q )) follows from Chapter 5, Corollary 5.2.2.8. Combining the above Theorem with Chapter 5, Theorem 5.2.4.2, we obtain the following Corollary 10.3.1.2 Let X and G be as above. Let ϕij denote the regular function (u, ξ ) → ui , ξj  on X, 1 ≤ i ≤ m, 1 ≤ j ≤ q, and let f denote the m × q matrix   ϕij . The ring of invariants K[X]G has a basis consisting of standard monomials in the regular functions pλ,μ (f ) with #λ ≤ n, where #λ = t is the number of elements in the sequence λ = (λ1 , . . . , λt ) and pλ,,μ (f ) is the t-minor with row indices λ1 , . . . , λt and column indices μ1 , . . . , μt . As a consequence of the above Theorem and the Corollary, we obtain the first and Second Fundamental Theorems of classical invariant theory (cf. [115]). Let notation be as above. Theorem 10.3.1.3 (i) First Fundamental Theorem The ring of invariants K[X]GL(V ) is generated by ϕij , 1 ≤ i ≤ m, 1 ≤ j ≤ q. (ii) Second Fundamental Theorem The ideal of relations among the generators in (1) is generated by the (n + 1)minors of the m × q-matrix (ϕij ). Further we have Theorem 10.3.1.4 A standard monomial basis for the ring of invariants: The ring of invariants K[X]GL(V ) has a basis consisting of standard monomials in the regular functions p( A, B), A ∈ I (r, m), B ∈ I (r, q), r ≤ n.

10.4 On (K)-action In this section, we give a GIT-theoretic proof of the results for On (K)-actions appearing in classical invariant theory (cf. [22], [115]). Let V = K n , together with a non-degenerate, symmetric bilinear form , . Let G = O(V ) (the orthogonal group consisting of linear automorphisms of V preserving , ). We shall take the matrix of the form ,  to be Jn := anti-diagonal(1, · · · , 1). Then a matrix A ∈ GL(V ) is in O(V ) if and only if t AJ A = J , i.e, J −1 (t A−1 )J = A, i.e, J (t A−1 )J = A (note that J −1 = J ) n n n n n n n n

Remark 10.4.0.1. In particular, note that a diagonal matrix A = diagonal(t1 , · · · , tn ) is in O(V ) if and only if tn+1−i = ti−1 .

10.4 On (K)-action

133

Let X denote the affine space V ⊕m = V ⊕ · · · ⊕ V , where m > n. For u = (u1 , u2 , ..., um ) ∈ X, writing ui = (ui1 , · · · , uin ) (with respect to the standard basis for K n ), we shall identify u with the m × n matrix U := (uij )m×n . Thus, we identify X with Mm,n (K), the space of m × n matrices (with entries in K). Consider the diagonal action of O(V ) on X. The induced action on K[X] is given by (A · f )(u) = f (A · u) = f (U A), i ∈ X, f ∈ K[X], A ∈ On (K)(= O(V )) The Functions ϕij : Consider the functions ϕij : X → K defined by ϕ ij ((u)) = ui , uj , 1 ≤ i, j ≤ m. Each ϕij is clearly in K[X]O(V ) . Let S be the subalgebra of K[X]O(V ) generated by {ϕij }. We shall now show (using Lemma 10.1.0.4) that S is in fact equal to K[X]O(V ) . Theorem 10.4.0.2 The morphism ψ : X → Sym Mm , u → (ui , uj ) is O(V )invariant. Further, it maps X onto Dn+1 (Sym Mm ) and identifies the categorical quotient X//O(V ) with Dn+1 (Sym Mm ) (here, Dn+1 (Sym Mm ) is as in Chapter 6, namely the determinantal variety contained in Sym Mm , the space of symmetric m × m matrices, consisting of matrices of rank ≤ n). Proof. Clearly, ψ(X) ⊆ Dn+1 (Sym Mm ) (since, ψ(X) = Spec S, and clearly Spec S ⊆ Dn+1 (Sym Mm ) (since any n + 1 vectors in V are linearly independent)). We shall prove the result using Lemma 10.1.0.4. To be very precise, we shall first check the conditions (i)-(iii) of Lemma 10.1.0.4 for ψ : X → Sym Mm , deduce that the inclusion Spec S ⊆ Dn+1 (Sym Mm ) is in fact an equality, and hence conclude the normality of Spec S (condition (iv) of Lemma 10.1.0.4). (i) Let x = u = (u1 , . . . , um ) ∈ X ss . Let Wx be the subspace of V spanned by ui ’s. Let r = dim Wx . Then r > 0 (since x ∈ X ss ). Assume if possible that ψ(x) = 0, i.e. ui , uj  = 0 for all i, j . This implies in particular that Wx is totally isotropic; hence, r ≤ [ n2 ], integral part of n2 . Hence we can choose a basis {e1 , · · · , en } of V such that Wx = the K-span of {e1 , · · · , er }. Writing each vector ui as a row vector (with respect to this basis), we may represent u by the m × n matrix U given by ⎞ ⎛ u11 u12 . . . u1r 0 . . . 0 ⎜ u21 u22 . . . u2r 0 . . . 0⎟ ⎟ ⎜ U := ⎜ . .. .. .. .. .. ⎟ ⎝ .. . . . . .⎠ um1 um2 . . . umr 0 . . . 0 Choose integers a1 , · · · , ar , ar+1 , · · · , an , so that ai > 0, i ≤ [ n2 ], and an+1−i = −ai , i ≤ [ n2 ] (if n is odd, say, n = 2 + 1, then, we take a +1 to be 0). Let gt be the diagonal matrix gt = diag(t a1 , · · · t ar , t ar+1 , · · · , t an ) (note that gt ∈ O(V ) (cf. Remark 10.4.0.1)). Consider the one parametric subgroup {gt , t ∈ K ∗ }. We have, gt x = gt · U = Ugt = Ut , where ⎞ ⎛ a t 1 u11 t a2 u12 . . . t ar u1r 0 . . . 0 ⎜ t a1 u21 t a2 u22 . . . t ar u2r 0 . . . 0⎟ ⎟ ⎜ Ut = ⎜ . .. .. .. .. .. ⎟ ⎝ .. . . . . .⎠ t a1 um1 t a2 um2 . . . t ar umr 0 . . . 0

134

10 Invariant theory

Hence gt x → 0 as t → 0 (note that r ≤ [ n2 ], and hence ai > 0, i ≤ r), and this implies that 0 ∈ G · x (G being On (K)) which is a contradiction to the hypothesis that x is semi-stable. Therefore our assumption that ψ(x) = 0 is wrong and (i) of Lemma 10.1.0.4 is satisfied. (ii) Let U = {u ∈ X | {u1 , . . . , un }are linearly independent} Clearly, U is a G-stable open subset of X. Claim: G operates freely on U , U → U mod G is a G-principal fiber space, and ψ induces an immersion U/G → Sym Mm . Proof of Claim: Let H = GLn (K). We have a G(= On (K))-equivariant identification U∼ · · × V = H × F, say = H × V × ·

(∗)

(m−n) copies

where F = V × · · · × V. From this it is clear that G operates freely on U . Further, (m−n) copies

we see that U mod G may be identified with the fiber space with base H mod G, and fiber V × · · · × V associated to the principal fiber space H → H / G. It re(m−n) copies

mains to show that ψ induces an immersion U/G → AN , i.e., to show that the map ψ : U/G → AN and its differential dψ are both injective. We first prove the injectivity of ψ : U/G → AN . Let x, x  in U/G be such that ψ(x) = ψ(x  ). Let η, η ∈ U be lifts for x, x  respectively. Using the identification (*) above, we may write η = (A, un+1 , · · · , um ), A ∈ H η = (A , un+1 , · · · , um ), A ∈ H (here, ui , 1 ≤ i ≤ n, are given by the rows of A, while ui , 1 ≤ i ≤ n, are given by the rows of A . The hypothesis that ψ(x) = ψ(x  ) implies in particular that ui , uj  = ui , uj , 1 ≤ i, j ≤ n which may be written as

AJn t A = A Jn t A

where Jn is the matrix of the form ,  (note that since we are writing a vector v ∈ V as a row vector, vi , vj  = vi Jn t vj ). Hence we obtain (A

−1

A)Jn t (A

−1

A) = Jn , i.e., A

−1

This implies that (∗∗)

A = A · g, for some g ∈ G.

Hence on U/G, we may suppose that

A∈G

10.4 On (K)-action

135

x = (u1 , · · · , un , un+1 , · · · , um ) x  = (u1 , · · · , un , un+1 , · · · , um ) where {u1 , · · · , un } is linearly independent. For a given j, n + 1 ≤ j ≤ m, we have, ui , uj  = ui , uj , 1 ≤ i ≤ n, implies, uj = uj (since, {u1 , · · · , un } is linearly independent and the form ,  is non-degenerate). Thus we obtain uj = uj , for all j.

(†)

The injectivity of ψ : U/G → AN follows from (†). To prove that the differential dψ is injective, we merely note that the above argument remains valid for the points over K[], the algebra of dual numbers (= K ⊕K, the K-algebra with one generator , and one relation  2 = 0), i.e., it remains valid if we replace K by K[], or in fact by any K-algebra. Thus (ii) of Lemma 10.1.0.4 holds. (iii) We have 1 dim U/G = dim U − dim G = mn − n(n − 1) = dim Dn+1 (Sym Mm ) 2 (cf. Chapter 6, Corollary 6.2.5.3 with t = n+1). The immersion U/G → Spec S(⊆ Dn+1 (Sym Mm )) together with the fact above that dim U/G = dim Dn+1 (Sym Mm ) implies that Spec S in fact equals Dn+1 (Sym Mm ). Thus (iii) of Lemma 10.1.0.4 holds. (iv) The normality of Spec S(= Dn+1 (Sym Mm )) follows from Chapter 6, Corollary 6.2.5.3. Using the identification X//O(V ) ∼ = Dn+1 (Sym Mm ), we get Theorem 10.4.0.3 (i) First Fundamental Theorem The ring of invariants K[X]O(V ) is generated by ϕij = ui , uj , 1 ≤ i, j ≤ m. (ii) Second Fundamental Theorem The ideal of relations among the generators in (i) is generated by the (n + 1)minors of the symmetric m × m-matrix (ϕij ). Further, using the results of Chapter 6 (especially §6.2.9 of Chapter 6), with notations as in loc.cit, we obtain Theorem 10.4.0.4 (A Standard Monomial Basis for K[X]O(V ) ) The ring of invariants K[X]O(V ) has a basis consisting of standard monomials in the regular functions p(A, B), A, B ∈ I (r, m), A ≥ B, r ≤ n.

136

10 Invariant theory

10.5 Sp2 (K)-action In this section, we give a GIT-theoretic proof of the results for SP2 (K)-actions appearing in classical invariant theory (cf. [22], [115]). Let V = K n , n = 2 , together with a non-degenerate, skew-symmetric bilinear form , . Let G = Sp(V ) (the symplectic group consisting of linear automorphisms of V preserving , ) on X. We shall take the matrix of the form ,  to be Jn := anti-diagonal(1, · · · , 1, −1, · · · , −1). Then a matrix A ∈ GL(V ) is in Sp(V )       l times

l times

if and only if t AJ A = J , i.e, J −1 (t A−1 )J = A, i.e, J (t A−1 )J = −A (note that J −1 = −J ) n n n n n n n n

Let X denote the affine space V ⊕m = V ⊕ · · · ⊕ V , where m > n. For u = (u1 , u2 , ..., um ) ∈ X, writing ui = (ui1 , · · · , uin ) (with respect to the standard basis for K n ), we shall identify u with the m × n matrix U := (uij )m×n . Thus, we identify X with Mm,n , the space of m × n matrices (with entries in K). Consider the diagonal action of Sp(V ) on X. The induced action on K[X] is given by (A · f )(u) = f (A · u) = f (U A), i ∈ X, f ∈ K[X], A ∈ Spn (K)(= Sp(V )) The Functions ϕij : Consider the functions ϕij : X → K defined by ϕ ij ((u)) = ui , uj , 1 ≤ i, j ≤ m. Each ϕij is clearly in K[X]Sp(V ) . Theorem 10.5.0.1 Let Sk Mm denote the space of skew symmetric matrices with entries in K. The morphism ψ : X → Sk Mm , u → (ui , uj ) is Sp(V )invariant. Further, it maps X onto Dn+1 (Sk Mm ) and identifies the categorical quotient X//Sp(V ) with Dn+1 (Sk Mm ) (here, Dn+1 (Sk Mm ) is the determinantal variety inside Sk Mm , the space of skew symmetric m × m matrices, consisting of matrices of rank ≤ n). The proof is exactly same as that of Theorem 10.4.0.2 (with appropriate modifications). Using the identification X//Sp(V ) ∼ = Dn+1 (Sk Mm ), and the results of Chapter 7 (especially, §7.2.6) of Chapter 7, we obtain, with notations as in loc.cit Theorem 10.5.0.2 (i) First Fundamental Theorem The ring of invariants K[X]Sp(V ) is generated by ϕij = ui , uj , 1 ≤ i, j ≤ m. (ii) Second Fundamental Theorem The ideal of relations among the generators in (i) is generated by the (n + 1)minors of the skew symmetric m × m-matrix (ϕij ). (iii) A Standard Monomial Basis for K[X]Sp(V ) The ring of invariants K[X]Sp(V ) has a basis consisting of standard monomials q(A1 ) · · · q(Ar ), A1 ≤ · · · ≤ Ar , r ∈ N, A1 ≥ (1, · · · , t − 1), Ai ∈ An,2n in the Pfaffians (notations as in Chapter 7, §7.2.6).

11 SLn (K)-action

∗ In this chapter, we consider the SLn (K)-action on X = V · · ⊕ V⊕V · · ⊕ V ∗ ,  ⊕ ·  ⊕ · m copies

q copies

V = K n . We present a proof (cf. [72]) of the Cohen-Macaulayness of K[X]SLn (K) . While Spec K[X]GLn (K) (being a certain determinantal variety (cf. Chapter 10)) gets identified with the opposite cell in a certain Schubert variety, Spec K[X]SLn (K) does not have a direct relationship to Schubert varieties. Nevertheless, certain “Schubert variety connections” are used in a crucial way (cf. §11.3). As mentioned in the Introduction, in recent times, among the several techniques of proving the CohenMacaulayness of algebraic varieties, two techniques have proven to be quite effective, namely, Frobenius-splitting technique and deformation technique. Frobenius-splitting technique is used in [82], for example, for proving the (arithmetic) Cohen-Macaulayness of Schubert varieties. Frobenius-splitting technique is also used in [80, 81], [84] for proving the Cohen-Macaulayness of certain varieties. The deformation technique consists in constructing a flat family over A1 , with the given variety as the generic fiber (corresponding to t ∈ K invertible). If the special fiber (corresponding to t = 0) is Cohen-Macaulay, then one may conclude the Cohen-Macaulayness of the given variety. Hodge algebras (cf. [20]) are typical examples where the deformation technique affords itself very well. Deformation technique is also used in [12, 17, 21, 33, 46]. The philosophy behind these works is that if there is a “standard monomial basis” for the co-ordinate ring of the given variety, then the deformation technique will work well in general (using the “straightening relations”). It is this philosophy that is adopted in [72] in proving the CohenMacaulayness of K[X]SLn (K) . To make this more precise, the proof of the CohenMacaulayness of K[X]SLn (K) is accomplished in the following steps: • One first constructs a K-subalgebra S of K[X]SLn (K) by prescribing a set of algebra generators {fα , α ∈ H }, H being a finite partially ordered set and fα ∈ K[X]SLn (K) . • One then constructs a “standard monomial” basis for S by (i) defining “standard monomials” in the fα ’s

138

11 SLn (K)-action

(ii) writing down the straightening relation for a non-standard (degree 2) monomial fα fβ (iii) proving linear independence of standard monomials (by relating the generators of S to certain determinantal varieties) (iv) proving the generation of S (as a vector space) by standard monomials (using (ii)). In fact, to prove the generation for S, one first proves generation for a “graded version” R(D) of S, where D is a distributive lattice obtained by adjoining 1, 0 (the largest and the smallest elements of D) to H . One then deduces the generation for S. In fact, one constructs a “standard monomial” basis for R(D). While the generation by standard monomials for S is deduced from the generation by standard monomials for R(D), the linear independence of standard monomials in R(D) is deduced from the linear independence of standard monomials in S (cf. (iii) above). • One proves the normality and Cohen-Macaulayness of R(D) by showing that Spec R(D) flatly degenerates to the toric variety associated to the distributive lattice D. • One deduces the normality and Cohen-Macaulayness of S from the normality and Cohen-Macaulayness of R(D) (cf. Theorem 11.4.4.4). • Using the normality of S and a crucial Lemma concerning GIT (cf. Chapter 10, Lemma 10.1.0.4 which gives a set of sufficient conditions for a normal sub algebra of K[X]SLn (K) to equal K[X]SLn (K) ), one shows that S is in fact K[X]SLn (K) , and hence concludes that K[X]SLn (K) is Cohen-Macaulay. As a consequence, one obtains the following: • First Fundamental Theorem for SLn (K)-invariants, i.e., describing algebra generators for K[X]SLn (K) . • Second Fundamental Theorem for SLn (K)-invariants, i.e., describing generators for the ideal of relations among these algebra generators for K[X]SLn (K) . • A Standard Monomial Basis for K[X]SLn (K) As a by-product one recovers Theorem 3.3 of DeConcini-Procesi [22] (which describes a set of algebra generators for K[X]SLn (K) ).

11.1 Quadratic relations In this section, we want to derive certain quadratic relations on a determinantal variety. As in Chapter 5, let Dt be the determinantal variety in Mr,d (K) (∼ = Ard ), defined by the vanishing of t-minors. As in Chapter 5, we have an identification of Dt with the opposite cell Y (φ) in Gd,n , n = r + d, where φ is the d-tuple ([t, d], [n + 2 − t, n]). Given s ≤ min{r, d}, and a pair of s-tuples, A, B, we shall denote by p(A, B), the regular function on Dt which is just the s-minor (of the r × d generic matrix (xij )) with row and column indices given by A, B respectively. Recall that a monomial p(A1 , B1 ) · · · p(Am , Bm ) is standard if A1  B1  · · ·  Am  Bm , the partial order  being as in Chapter 5, §5.2.4. Recall the following Theorem 11.1.0.1 Standard monomials in p(A, B)’s with # A ≤ t − 1 form a basis for K[Dt ], the algebra of regular functions on Dt .

11.1 Quadratic relations

139

Also, using the bijection θ (cf. Chapter 5, §5.2.3) between the minors of (xij ) and Plücker co-ordinates (via the identification of Mr,d (K) with the opposite cell in Gd,n , n = r + d), we have (in view of Chapter 4, Proposition 4.3.3.3), the following Proposition 11.1.0.2 Let p(A1 , A2 ), p(B1 , B2 ) (in K[Dt ]) be not comparable. Let  ai p(Ci1 , Ci2 )p(Di1 , Di2 ), ai ∈ K ∗ (∗) p(A1 , A2 )p(B1 , B2 ) = be the straightening relation in K[Dt ]. Then for every i, Ci1 , Ci2 , Di1 , Di2 have cardinalities ≤ t − 1; further, (i) Ci1 ≥ both A1 and B1 ; Di1 ≤ both A1 and B1 . (ii) Ci2 ≥ both A2 and B2 ; Di2 ≤ both A2 and B2 . (iii) The term p((A1 , A2 ) ∨ (B1 , B2 ))p((A1 , A2 ) ∧ (B1 , B2 )) occurs in (*) with coefficient 1. 11.1.1 The partially ordered set Hr,d Let Hr,d =

∪

I (s, r) × I (s, d)

0≤s≤min {r,d}

where our convention is that (∅, ∅) is the element of Hr,d corresponding to s = 0. We define a partial order  on Hr,d as follows: • We declare (∅, ∅) as the largest element of Hr,d . • For (A, B), (A , B  ) in Hr,d , say, A = {a1 , · · · , as }, B = {b1 , · · · , bs }, A = {a1 , · · · , as  }, B  = {b1 , · · · , bs  } for some s, s  ≥ 1, we define (A, B)  (A , B  ) if s ≤ s  , aj ≥ aj , bj ≥ bj , 1 ≤ j ≤ s. Note that via the bijection θ (cf. Chapter 5, §5.2.3), join and meet of two noncomparable elements (A1 , A2 ), (B1 , B2 ) of Hr,d exist, and in fact are given by (A1 , A2 )∨(B1 , B2 ) = (A1 ∨B1 , A2 ∨B2 ), (A1 , A2 )∧(B1 , B2 ) = (A1 ∧B1 , A2 ∧B2 ). Remark 11.1.1.1. On the R.H.S. of (*), Ci1 , Ci2 could both be the empty set (in which case p(Ci1 , Ci2 ) is understood as 1). For example, with X being a 2 × 2 matrix of indeterminates, we have p1,2 p2,1 = p2,2 p1,1 − p∅,∅ p12,12 Remark 11.1.1.2. In the sequel, while writing a straightening relation as in Proposition 11.1.0.2, if for some i, Ci1 , Ci2 are both the empty set, we keep the corresponding p(Ci1 , Ci2 ) on the right hand side of the straightening relation (even though its value is 1) in order to have homogeneity in the relation. Taking t = d + 1 (in which case Dt = Z = Mr,d (K)) in Theorem 11.1.0.1 and Proposition 11.1.0.2, we obtain Theorem 11.1.1.3 (i) Standard monomials in p(A, B)’s form a basis for K[Z] (∼ = K[xij , 1 ≤ i ≤ r, 1 ≤ j ≤ d]). (ii) Relations similar to those in Proposition 11.1.0.2 hold on Z.

140

11 SLn (K)-action

Remark 11.1.1.4. Note that Theorem 11.1.1.3,(1) recovers the result of DoubiletRota-Stein (cf. [26], Theorem 2): Remark 11.1.1.5. Theorem 11.1.0.1 is also proved in DeConcini-Procesi [22] (Theorem 1.2 in [22]). But we had taken the Schubert variety theoretic approach to deduce Theorem 11.1.0.1 from the results on Schubert varieties as discussed in chapter 4 in order to derive the straightening relations as given by Proposition 11.1.0.2(which are crucial for the discussion in §11.3).

11.2 The K-algebra S ∗ Let V = K n , X = V · · ⊕ V ⊕ V · · ⊕ V ∗ .  ⊕ ·  ⊕ · m copies

q copies

The GL(V )-action on X: Writing u = (u1 , u2 , ..., um ) with ui ∈ V and ξ = (ξ1 , ξ2 , ..., ξq ) with ξi ∈ V ∗ , we shall denote the elements of X by (u, ξ ). The (natural) action of GL(V ) on V induces an action of GL(V ) on V ∗ , namely, for ξ ∈ V ∗ , g ∈ GL(V ), denoting g · ξ by ξ g , we have ξ g (v) = ξ(g −1 v), v ∈ V The diagonal action of GL(V ) on X is given by g

g

g

g · (u, ξ ) = (gu, ξ g ) = (gu1 , gu2 , ..., gum , ξ1 , ξ2 , ..., ξq ), g ∈ G, (u, ξ ) ∈ X The induced action on K[X] is given by (g · f )(u, ξ ) = f (g −1 (u, ξ )), f ∈ K[X], g ∈ GL(V ) Consider the functions ϕij : X −→ K defined by ϕij ((u, ξ )) = ξj (ui ), 1 ≤ i ≤ m, 1 ≤ j ≤ q. Each ϕij is a GL(V )-invariant: For g ∈ GL(V ), we have, (g · ϕij )((u, ξ )) = ϕij (g −1 (u, ξ )) −1

= ϕij ((g −1 u, ξ g ) g −1

= ξj (g −1 ui ) = ξj (ui ) = ϕij ((u, ξ )) It is convenient to have a description of the above action in terms of coordinates. So with respect to a fixed basis, we write the elements of V as row vectors and those of V ∗ as column vectors. Thus denoting by Ma,b the space of a × b matrices with entries in K, X can be identified with the affine space Mm,n × Mn,q . The action of GLn (K) (= GL(V )) on X is then given by A · (U, W ) = (U A, A−1 W ), A ∈ GLn (K), U ∈ Mm,n , W ∈ Mn,q

11.2 The K-algebra S

141

As seen in chapter 10, we have an action of GLn (K) on K[X] given by     (A · f )(U, W ) = f A−1 (U, W ) = f U A−1 , AW , f ∈ K[X]   Writing U = uij and W = (ξkl ) we denote the coordinate functions on X, by uij and ξkl . Further, if ui denotes the i-th row of U and ξj the - j -th. column of W, the invariants ϕij described above are nothing but the entries ui , ξj (= ξj (ui )) of the product U W. . In the sequel, we shall denote ϕij (u, ξ ) also by ui , ξj . 11.2.1 The SLn (K)-action Let X be as above. We shall denote K[X] by R so that R = K[uij , ξkl 1 ≤ i ≤ m, 1 ≤ j, k ≤ n, 1 ≤ l ≤ q]. We consider the action of SLn (K) on X induced from the GLn (K)-action. The Function p(A, B): For A ∈ I (r, m), B ∈ I (r, q), 1 ≤ r ≤ n, let p(A, B) be the regular function on X: p(A, B)((u, ξ )) is the determinant of the r × r-matrix . ( ui , ξj )i∈A, j ∈B .   The Functions u(I ), ξ(J ): As above, let U = uij 1≤i≤m, 1≤j ≤n and W = (ξkl )1≤k≤n, 1≤l≤q . For I ∈ I (n, m), J ∈ I (n, q), let u(I ), ξ(J )) denote the following regular functions on X: u(I )((u, ξ )) = the n-minor of U with row indices given by I . ξ(J )((u, ξ )) = the n-minor of W with column indices given by J . Note that for the diagonal action of SLn (K) (= SL(V )) on X, we have, u(I ), ξ(J ) are in R SLn (K) . The K-algebra S: Let S be the K-subalgebra of R generated by {u(I ), ξ(J ), p(A, B), I ∈ I (n, m), J ∈ I (n, q), A ∈ I (r, m), B ∈ I (r, q), 1 ≤ r ≤ n}. We shall denote the set I (n, m) indexing the u(I )’s by Hu and the set I (n, q) indexing the ξ(J )’s by Hξ . Also, we shall denote Hp := ∪ (I (r, m) × I (r, q)), and 1≤r≤n

set ˙ ξ ∪ Hp H = Hu ∪H ˙ (n, q) ∪ = I (n, m)∪I

∪ (I (r, m) × I (r, q)),

1≤r≤n

where ∪˙ denotes a disjoint union. (If m = q, then Hu , Hξ are to be considered as two disjoint copies of I (n, m).) Then the algebra generators {u(I ), ξ(J ), p(A, B), I ∈ I (n, m), J ∈ I (n, q), A ∈ I (r, m), B ∈ I (r, q), 1 ≤ r ≤ n} of S are indexed by the set H . Clearly S ⊆ R SL(V ) . Remark 11.2.1.1. The K-algebra- S could have been simply defined as the K. subalgebra of R G generated by { ui , ξj } (i.e., by {p(A, B), #A = #B = 1}) and {u(I ), ξ(J )}. But we have a purpose in defining it as above, namely, the standard monomials (in S) will be built out of the p(A, B)’s with #A ≤ n, the u(I )’s and ξ(J )’s.

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Our goal is to show that S equals R SL(V ) . A Partial Order on H : Define a partial order on H as follows: (i) (ii) (iii) (iv) (v)

The partial order on Hp is as in §11.1.1 (note that Hp ⊂ Hm,q ) The partial order on Hu and Hξ are as in Chapter 4, §4.1.1. Any element of Hu and any element of Hξ are not comparable. No element of Hu , Hξ is greater than any element of Hp . For I ∈ Hu and (A, B) ∈ Hp , we define I ≤ (A, B) if I ≤ A (the partial order being as in Chapter 4, §4.1.1). Similarly, for J ∈ Hξ and (A, B) ∈ Hp , we define J ≤ (A, B) if J ≤ B.

Lemma 11.2.1.2 H is a ranked poset of rank d := (m + q)n − n2 , i.e., all maximal chains in H have the same cardinality = (m + q)n − n2 + 1. Proof. Clearly, H is a ranked poset (since it is composed of ranked posets). To compute the rank of H , we consider the maximal chain consisting of τ1 , · · · , τN , where the first q of them are given by (m, q), (m, q − 1), · · · , (m, 1) (of Hp ), the next (m − 1) of them are given by (m − 1, 1), (m − 2, 1), · · · , (1, 1) (of Hp ). (thus contributing m + q − 1 to the cardinality of the chain). This is now followed by the q − 1 elements of Hp : ((1, m), (1, q)), ((1, m), (1, q − 1)), · · · , ((1, m), (1, 2)), followed by the m − 2 elements of Hp : ((1, m − 1), (1, 2)), ((1, m − 2), (1, 2)), · · · , ((1, 2), (1, 2)) (thus contributing m + q − 3 to the cardinality of the chain). Thus proceeding, finally, we end up with ((1, 2, · · · , n), (1, 2, · · · , n)) (in Hp ). This is now followed by either (1, 2, · · · , n) of Hu or (1, 2, · · · , n) of Hξ . The number of elements in the above chain equals [m + q − 1 + (m + q − 3) + · · · + m + q − (2n − 1)] + 1 = (m + q)n − n2 + 1.

11.3 Standard monomials in the K-algebra S Definition 11.3.0.1 A monomial F in the p(A, B)’s, u(I )’s, and ξ(J )’s, is said to be standard if F satisfies the following conditions: (i) If F involves u(I ), for some I (resp. ξ(J ) for some J ), then F does not involve ξ(J  ) for any J  (resp. u(I  ), for any I  ). (ii) If F = p(A1 , B1 ) · · · p(Ar , Br )u(I1 ) · · · u(Is ) (resp. p(A1 , B1 ) · · · p(Ar , Br )ξ(J1 ) · · · ξ(Jt )), where r, s, t are integers ≥ 0, then A1 ≥ · · · ≥ Ar ≥ I1 ≥ · · · ≥ Is ( resp. B1 ≥ · · · ≥ Br ≥ J1 ≥ · · · ≥ Jt )

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11.3.1 Quadratic relations In this subsection, we describe certain straightening relations to be used while proving the linear independence of standard monomials and generation (of S as a Kvector space) by standard monomials. Theorem 11.3.1.1

(i) Let I ∈ Hu , J ∈ Hξ . We have u(I )ξ(J ) = p(I, J )

(ii) Let I, I  ∈ Hu be not comparable. We have,  br u(Ir )u(Ir ), br ∈ K ∗ u(I )u(I  ) = r

where for all r, Ir ≥ both I, I  , and Ir ≤ both I, I  . (iii) Let J, J  ∈ Hξ be not comparable. We have, ξ(J )ξ(J  ) =



cs ξ(Js )ξ(Js ), cs ∈ K ∗

s

where for all s, Js ≥ both J, J  , and Js ≤ both J, J  . (iv) Let (A1 , A2 ), (B1 , B2 ) ∈ Hp be not comparable. Then we have p(A1 , A2 )p(B1 , B2 ) =



ai p(Ci1 , Ci2 )p(Di1 , Di2 ), ai ∈ K ∗ ,

i

where (Ci1 , Ci2 ), (Di1 , Di2 ) belong to Hp ; further, for every i, we have a) Ci1 ≥ both A1 and B1 ; Di1 ≤ both A1 and B1 . b) Ci2 ≥ both A2 and B2 ; Di2 ≤ both A2 and B2 . (v) Let I ∈ Hu , (A, B) ∈ Hp be such that A ≥ I . We have, p(A, B)u(I ) =

 dt p(At , Bt )u(It ), dt ∈ K ∗ t

where for every t, we have, At ≥ (resp. It ≤) both A and I , and Bt ≥ B. (vi) Let J ∈ Hξ , (A, B) ∈ Hp be such that B ≥ J . We have, p(A, B)ξ(J ) =

 el p(Al , Bl )ξ(Jl ), el ∈ K ∗ l

where for every l, we have, Al ≥ A, and Bl ≥ (resp. Jl ≤) both B and J . Proof. In the course of the proof, we will be repeatedly using the fact that the subalgebra generated by {p(A, B), A ∈ I (r, m), B ∈ I (r, q) 1 ≤ r ≤ n} being R GL(V ) (∼ = K[Dn+1 (Mm,q )] (cf. Chapter 10, §10.3)), the results given in Proposition 11.1.0.2 apply to this subalgebra. (i) is clear from the definitions of u(I ), ξ(J ) and p(I, J ).

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(ii). We shall denote a minor of U (= (uij )1≤i≤m, 1≤j ≤n ) with rows and columns given by I, J (where I, J ∈ I (r, m) for some r ≤ n) by Δ(I, J ). Observe that if #I = n, then J = (1, 2, · · · , n) necessarily (since U has size m × n). Thus for I ∈ Hu , we have that u(I ) = Δ(I, In ), u(I  ) = Δ(I  , In ) (as minors of U ), where In = (1, 2, · · · , n), we have, in view of Proposition 11.1.0.2, (ii), u(I )u(I  ) = Δ(I, In )Δ(I  , In ) =



bi Δ(Ci1 , Ci2 )Δ(Di1 , Di2 ), ai ∈ K ∗ ,

i

where we have for every i, Ci1 ≥ both I and I  ; Di1 ≤ both I and I  ; Ci2 ≥ In ; Di2 ≤ In which forces #Di2 = n (in view of the partial order (cf. §11.1.1); note that Di2 being the column indices of a minor of the m × n matrix U , we have that #Di2 ≤ n). Hence we obtain that Di2 = In , for all i. In particular, we obtain that #Di1 (= #Di2 ) = n. This in turn implies (by consideration of the degrees in uij ’s of the terms in the above sum) that #Ci1 = #Ci2 = n. Hence Ci2 = In (again note that Ci2 gives the column indices of the n-minor Δ(Ci1 , Ci2 ) of the m × n matrix U ). Thus the above relation becomes  bi u(Ci1 )u(Di1 ), u(I )u(I  ) = i

with Ci1 ≥ both I and I  ; Di1 ≤ both I and I  . This proves (ii). Proof of (iii) is similar to that of (ii). (iv) is a direct consequence of Proposition 11.1.0.2. (v). If #A = n = #B, then p(A, B)u(I ) = u(A)u(I )ξ(B). By (ii), u(A)u(I ) =

 di u(Ci )u(Di ), di ∈ K ∗ i

where Ci ≥ both A, I , and Di ≤ both A, I . Hence   di u(Ci )u(Di )ξ(B) = di p(Ci , B)u(Di ) p(A, B)u(I ) = i

i

where Ci ≥ both A, I , and Di ≤ both A, I , and the result follows. Let then #A < n. By (i), we have u(I )ξ(In ) = p(I, In ). Hence, p(A, B)u(I )ξ(In ) = p(A, B)p(I, In ). The hypothesis that A ≥ I implies that p(A, B)p(I, In ) is not standard (note that the facts that #A < n, #I = n implies that I ≥ A). Hence (iv) implies that p(A, B)p(I, In ) =



ai p(Ci1 , Ci2 )p(Di1 , Di2 ), ai ∈ K ∗ ,

where (Ci1 , Ci2 ), (Di1 , Di2 ) belong to Hp ; further, for every i, Ci1 ≥ both A and I ; Di1 ≤ both A and I ; Ci2 ≥ both B and In ; Di2 ≤ both B and In which forces Di2 = In (note that in view of Theorem 10.3.1.4, all minors in the above relation have size ≤ n); and hence #Di1 = n, for all i. Hence p(Di1 , Di2 ) = u(Di1 )ξ(In ),

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145

for all i. Hence cancelling ξ(In ), we obtain  p(A, B)u(I ) = ai p(Ci1 , Ci2 )u(Di1 ), where Ci1 ≥ both A and I , Di1 ≤ both A and I , and Ci2 ≥ B. This proves (v). Proof of (vi) is similar to that of (v). 11.3.2 Linear independence of standard monomials In this subsection, we prove the linear independence of standard monomials. Lemma 11.3.2.1 Let (A, B) ∈ Hp , I ∈ Hu , J ∈ Hξ . (i) The set of standard monomials in the p(A, B)’s is linearly independent. (ii) The set of standard monomials in the u(I )’s is linearly independent. (iii) The set of standard monomials in the ξ(J )’s is linearly independent. Proof. (i) follows from Theorem 11.1.0.1. (ii), (iii) follow from Theorem 11.1.1.3,(1) applied to K[uij , 1 ≤ i ≤ m, 1 ≤ j ≤ n], K[ξkl , 1 ≤ k ≤ n, 1 ≤ l ≤ q] respectively. Proposition 11.3.2.2 Standard monomials are linearly independent. Proof. For a monomial M, by u-degree (resp. ξ -degree) of M, we shall mean the degree of M in the variables uij ’s (resp. ξkl ’s ). We have u-degree of p(A1 , B1 ) · · · p(Ar , Br ) = ξ -degree of p(A1 , B1 ) · · · p(Ar , Br ) =



#Ai

i

u-degree of u(I1 ) · · · u(Is ) = ns, ξ -degree of u(I1 ) · · · u(Is ) = 0 ξ -degree of ξ(J1 ) · · · ξ(Jt ) = nt, u-degree of ξ(J1 ) · · · ξ(Jt ) = 0 By considering the u-degree and the ξ -degree, and using Lemma 11.3.2.1 we see that {p(A1 , B1 ) · · · p(Ar , Br ), u(I1 ) · · · u(Is ), ξ(J1 ) · · · ξ(Jt ), r, s, t ≥ 0} is linearly independent. Let (∗)

F := R + S = 0

be a relation among standard monomials, where R = a i Mi , S = bi Ni such that each Mi (resp. Ni ) is a standard monomial of the form p(A1 , B1 ) · · · p(Ari , Bri ) (resp. p(A1 , B1 ) · · · p(Aqi , Bqi )u(I1 ) · · · u(Isi )ξ(J1 ) · · · ξ(Jti ), qi ≥ 0, and at least one of {si , ti } > 0 ). Let g be in GLn (K) such that (det g)si +ti = 1, for all i. Then using the facts that g · p(A, B) = ) = (det g)u(I ), g · ξ(J ) = p(A, B), g · u(I (det g)ξ(J ), we have, F − gF = bi (1 − (det g)si +ti )Ni = 0. Hence if we show that the Ni ’s are linearly independent, then (in view of Lemma 11.3.2.1,(1)), we would obtain that (*) is the trivial relation. Thus we may suppose that  (∗∗) F = bi Ni = 0,

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where each Ni is a standard monomial of the form p(A1 , B1 ) · · · p(Ar , Br )u(I1 ) · · · u(Is )ξ(J1 ) · · · ξ(Jt ) where r ≥ 0 and at least one of {s, t} > 0; in fact, Ni ’s being standard, in any Ni , precisely one of {si , ti } is non-zero. We first multiply (**) by u(In )N (In being (1, 2, · · · , n)), for a sufficiently large N (N could be taken to be any integer greater than all of the t’s, appearing in the ξ(J1 ) · · · ξ(Jt )’s); we then replace a ξ(J )u(In ) by p(In , J ) (cf. Theorem 11.3.1.1, (1)). Then in the resulting sum, any monomial will involve only the p(A, B)’s and the u(I )’s. Thus we may suppose that (**) is of the form  (∗ ∗ ∗) G := ci Gi = 0 where each Gi is of the form p(A1 , B1 ) · · · p(Ar , Br )u(I1 ) · · · u(Is ). Note that for each standard monomial M = p(A1 , B1 ) · · · p(Ar , Br )u(I1 ) · · · u(Is ) (resp. p(A1 , B1 ) · · · p(Ar , Br )ξ(J1 ) · · · ξ(Jt )) appearing in (**), M u(In )N is again standard. Again, considering G − gG, g ∈ GLn (K), with det g = a root of unity, as above, we may suppose that in each monomial p(A1 , B1 ) · · · p(Ar , Br )u(I1 ) · · · u(Is ) appearing in (***), s > 0. Further, in view of Lemma 11.3.2.1,(2), we may suppose that for at least one monomial r > 0. Now considering the ξ -degree of the monomials, we may suppose (in view of Lemma 11.3.2.1,(2)) that in each monomial p(A1 , B1 ) · · · p(Ar , Br )u(I1 ) · · · u(Is ) appearing in (***), r > 0. Thus, for each monomial p(A1 , B1 ) · · · p(Ar , Br )u(I1 ) · · · u(Is ) appearing in (***), we have, r, s > 0. Now the ξ -degree (as well as the u-degree) being the same for all of the monomials in (***), for any two monomials Gi , Gi  , say Gi = p(A1 , B1 ) · · · p(Ar , Br )u(I1 ) · · · u(Is ), Gi  = p(A1 , B1 ) · · · p(Ar  , Br  )u(I1 ) · · · u(Is ) #Ai = #Ai . This together with the fact that the u-degree is the we have 1≤i≤r

1≤i≤r 

same for all of the terms Gk ’s in (***) implies that s = s  . Thus we obtain that in all of the monomials p(A1 , B1 ) · · · p(Ar , Br )u(I1 ) · · · u(Is ) in (***), the integer s is the same (and s > 0). Now we multiply (***) through out by ξ(In )s (where In = (1, 2, · · · , n)) to arrive at a linear sum  di Hi = 0 where each Hi is a standard monomial in the p(A, B)’s (note that Hi = p(A1 , B1 ) · · · p(Ar , Br )p(I1 , In ) · · · p(Is , In ) is standard). Now the required result follows from the linear independence of p(A, B)’s (cf. Lemma 11.3.2.1,(1)). 11.3.3 The algebra S(D) To prove the generation of S (as a K-vector space) by standard monomials, we define a K-algebra S(D), construct a standard monomial basis for S(D) and deduce the

11.3 Standard monomials in the K-algebra S

147

results for S (in fact, it will turn out that S(D) ∼ = S). We first define the K-algebra R(D) as follows: Let D = H ∪ {1} ∪ {0} H being as in the beginning of §11.2.1. Extend the partial order on H to D by declaring {1} (resp. {0}) as the largest (resp. smallest) element. Let P (D) be the polynomial algebra P (D) := K[X(A, B), Y (I ), Z(J ), X(1), X(0), (A, B) ∈ Hp , I ∈ Hu , J ∈ Hξ ] Let a(D) be the homogeneous ideal in the polynomial algebra P (D) generated by the six relations of Theorem 11.3.1.1 (X(A, B), Y (I ), Z(J ) replacing p(A, B), u(I ), ξ(J ) respectively), with relations (1) and (4) homogenized as follows: (1) is homogenized as X(I )Y (J ) = X(I, J )X(0)

(∗) while (4) is homogenized as

X(A1 , A2 )X(B1 , B2 ) =



ai X(Ci1 , Ci2 )X(Di1 , Di2 )

where X(Ci1 , Ci2 ) is to be understood as X(1) if both Ci1 , Ci2 equal the empty set (cf. Remark 11.1.1.2). Let R(D) = P (D)/a(D) We shall denote the classes of X(A, B), Y (I ), Z(J ), X(1), X(0) in R(D) by x(A, B), y(I ), z(J ), x(1), x(0) respectively. The Algebra M(D): Set M(D) = R(D)(x(0)) , the homogeneous localization of x(A,B) y(I ) z(J ) R(D) at x(0). We shall denote x(1) x(0) , x(0) , x(0) , x(0) (in M(D)) by q(1), r(A, B), s(I ), t (J ) respectively. A Grading for M(D): We give a grading for M(D) by assigning degree one to s(I ), t (J ), and degree 2 to q(1), r(A, B), where as above I ∈ Hu , J ∈ Hξ , (A, B) ∈ Hp . The Algebra S(D): Set S(D) = M(D)(q(1)) , the homogeneous localization of s(I ) t (J ) M(D) at q(1). We shall denote r(A,B) q(1) , q(1) , q(1) (in S(D)) by c(A, B), d(I ), e(J ) respectively. Let ϕD : S(D) → S be the map, ϕD (c(A, B)) = p(A, B), ϕD (d(I )) = u(I ), ϕD (e(J )) = ξ(J ). Consider the canonical maps θD : R(D) → M(D), δD : M(D) → S(D) Denote γD : R(D) → S as the composite γD = ϕD ◦ δD ◦ θD .

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11.3.4 A standard monomial basis for R(D) We define a monomial in x(A, B), y(I ), z(J ), x(1), x(0) (in R(D)) to be standard in exactly the same way as in Definition 11.3.0.1 (we declare x(1) (resp. x(0)) as the largest (resp. smallest)). Proposition 11.3.4.1 The standard monomials in the x(A, B)’s, y(I )’s, z(J )’s, x(1)’s, x(0)’s are linearly independent. Proof. The result follows by considering γD : R(D) → S, and using the linear independence of standard monomials in S (cf. Proposition 11.3.2.2). Generation of R(D) by Standard Monomials: We shall now show that any nonstandard monomial F in R(D) is a linear sum of standard monomials. Observe that if M is a standard monomial, then x(1)l M (resp. Mx(0)l ) is again standard; hence we may suppose F to be: F = x(A1 , B1 ) · · · x(Ar , Br )y(I1 ) · · · y(Is )z(J1 ) · · · z(Jt ) Using the relations y(I )z(J ) = x(I, J )x(0), we may suppose that F = x(A1 , B1 ) · · · x(Ar , Br )y(I1 ) · · · y(Is ) or F = x(A1 , B1 ) · · · x(Ar , Br )z(J1 ) · · · z(Jt ), say, F = x(A1 , B1 ) · · · x(Ar , Br )y(I1 ) · · · y(Is ). Fix an integer N sufficiently large. To each element A ∈ ∪nr=1 I (r, m), we associate an (n+1)-tuple as follows: Let A ∈ I (r, m), for some r, say, A = (a1 , · · · , ar ). To A, we associate the n + 1-tuple A := (a1 , · · · , ar , m, m, · · · , m, 1) Similarly, for B ∈ ∪nr=1 I (r, q), say, B = (b1 , · · · , br ), we associate the n + 1-tuple B := (b1 , · · · , br , q, q, · · · , q, 1) To F , we associate the integer nF (and call it the weight of F ) which has the entries of A1 , B1 , A2 , B2 , · · · , Ar , Br , I1 , · · · , Is as digits (in the N-ary presentation). The hypothesis that F is non-standard implies that either x(Ai , Bi )x(Ai+1 , Bi+1 ) is non-standard for some i ≤ r −1, or, x(Ar , Br )y(I1 ) is non-standard or y(Ij )y(Ij +1 ) is non-standard for some j ≤ s − 1. Straightening these using Theorem 11.3.1.1, we obtain that F = ai Fi where nFi > nF , ∀i, and the result follows by decreasing induction on nF (note that while straightening a degree 2 relation using Theorem 11.3.1.1, (4), if x(1) occurs in a monomial G, then the digits in nG corresponding to x(1) are taken to be (m, m · · · , m, q, q · · · , q ). Also note that the largest F of       n+1 times

n+1 times

degree r in x(A, B)’s and degree s in the y(I )’s is x({m}, {q})r u(I0 )s (where I0 is the n-tuple (m + 1 − n, m + 2 − n, · · · , m)) which is clearly standard (the starting point of the decreasing induction). Hence we obtain Proposition 11.3.4.2 Standard monomials in x(A, B), y(I ), z(J ), x(1), x(0) generate R(D) as a K-vector space.

11.3 Standard monomials in the K-algebra S

149

Combining Propositions 11.3.4.1, 11.3.4.2, we obtain Theorem 11.3.4.3 Standard monomials in x(A, B), y(I ), z(J ), x(1), x(0) give a basis for the K-vector space R(D). 11.3.5 Standard monomial bases for M(D), S(D) Standard monomials in r(A, B), s(I ), t (J ), q(1)) in M(D) (resp. c(A, B), d(I ), e(J )) in S(D)) are defined in exactly the same way as in Definition 11.3.0.1. Proposition 11.3.5.1 Standard monomials in r(A, B), s(I ), t (J ), q(1) give a basis for the K-vector space M(D). Proof. The linear independence of standard monomials follows as in the proof of Prop 11.3.4.1 by considering ϕD ◦ δD : M(D) → S, and using the linear independence of standard monomials in S (cf. Proposition 11.3.2.2). To see the generation of M(D) by standard monomials, consider a non-standard monomial F in M(D). Since q(1)l is the largest monomial of a given degree l, we may suppose F to be: F = r(A1 , B1 ) · · · r(Ai , Bi )s(I1 ) · · · s(Ik )t (J1 ) · · · t (Jl ) In view of Theorem 11.3.1.1, (1), we may suppose that F = r(A1 , B1 ) · · · r(Ai , Bi )s(I1 ) · · · s(Ik ) or r(A1 , B1 ) · · · r(Ai , Bi )t (J1 ) · · · t (Jl ), say, F = r(A1 , B1 ) · · · r(Ai , Bi )s(I1 ) · · · s(Ik ). Then F = θD (H ), where H = x(A1 , B1 ) · · · x(Ai , Bi )y(I1 ) · · · y(Ik ). The required result follows from Proposition 11.3.4.2. Proposition 11.3.5.2 Standard monomials in c(A, B), d(I ), e(J ) give a basis for the K-vector space S(D). The proof is completely analogous to that of Proposition 11.3.5.1 (in view of the fact that S(D) = M(D)(q(1)) ). Theorem 11.3.5.3 Standard monomials in p(A, B), u(I ), ξ(J ) form a basis for the K-vector space S. Proof. We already have established the linear independence of standard monomials (cf. Proposition 11.3.2.2). The generation by standard monomials follows by considering the surjective map ϕD : S(D) → S and using the generation of S(D) by standard monomials (cf. Proposition 11.3.5.2). Theorem 11.3.5.4 The map ϕD : S(D) → S is an isomorphism of K-algebras. Proof. Under ϕD , the standard monomials in S(D) are mapped bijectively onto the standard monomials in S. The result follows from Proposition 11.3.5.2 and Theorem 11.3.5.3.

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Theorem 11.3.5.5 A Presentation for S: (i) The K-algebra S is generated by {p(A, B), u(I ), ξ(J ), (A, B) ∈ Hp , I ∈ Hu , J ∈ Hξ }. (ii) The ideal of relations among the generators {p(A, B), u(I ), ξ(J ), (A, B) ∈ Hp , I ∈ Hu , J ∈ Hξ } is generated by the six type of relations as given by Theorem 11.3.1.1. Proof. The result follows from Theorem 11.3.5.4.

11.4 Normality and Cohen-Macaulayness of the K-algebra S In this section, we present a proof (cf. [72]) of the normality and Cohen-Macaulayness of Spec S by relating it to a toric variety. From §11.2, we have • {u(I ), ξ(J ), p(A, B), I ∈ Hu , J ∈ Hξ , (A, B) ∈ Hp } generates S as a K-algebra. • Standard monomials in {u(I ), ξ(J ), p(A, B), I ∈ Hu , J ∈ Hξ , (A, B) ∈ Hp } form a K-basis for S. • Considering S as a quotient of the polynomial algebra K[X(A, B), Y (I ), Z(J ), (A, B) ∈ Hp , I ∈ Hu , J ∈ Hξ ] the ideal a of relations is generated by the six kinds of quadratic relations as given in Theorem 11.3.1.1. 11.4.1 The algebra associated to a distributive lattice Definition 11.4.1.1 A lattice is a partially ordered set (L, ≤) such that, for every pair of elements x, y ∈ L, there exist elements x ∨ y, x ∧ y, called the join, respectively the meet of x and y, satisfying: x ∨ y ≥ x, x ∨ y ≥ y, and if z ≥ x and z ≥ y, then z ≥ x ∨ y, x ∧ y ≤ x, x ∧ y ≤ y, and if z ≤ x and z ≤ y, then z ≤ x ∧ y. Definition 11.4.1.2 A lattice is called distributive if the following identities hold: x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) Definition 11.4.1.3 Given a finite lattice L, the ideal associated to L, denoted by I (L), is the ideal of the polynomial algebra K[L](= K[xα , α ∈ L]) generated by the set of binomials GL = {xy − (x ∧ y)(x ∨ y) | x, y ∈ L non-comparable}. Set A(L) = K[L]/I (L), the algebra associated to L.

11.4 Normality and Cohen-Macaulayness of the K-algebra S

151

The chain lattice C(n1 , . . . , nd ): Given an integer n ≥ 1, let C(n) denote the chain {1 < · · · < n}, and for n1 , . . . , nd > 1, let C(n1 , . . . , nd ) denote the chain product lattice C(n1 ) × · · · × C(nd ) consisting of all d-tuples (i1 , . . . , id ), with 1 ≤ i1 ≤ n1 , . . . , 1 ≤ id ≤ nd . For (i1 , . . . , id ), (j1 , . . . , jd ) in C(n1 , . . . , nd ), we define (i1 , . . . , id ) ≤ (j1 , . . . , jd ) ⇐⇒ i1 ≤ j1 , . . . , id ≤ jd . We have (i1 , . . . , id ) ∨ (j1 , . . . , jd ) = (max{i1 , j1 }, . . . , max{id , jd }) (i1 , . . . , id ) ∧ (j1 , . . . , jd ) = (min{i1 , j1 }, . . . , min{id , jd }). Clearly, C(n1 , . . . , nd ) is a finite distributive lattice. 11.4.2 Flat degenerations of certain K-algebras Let L be a finite lattice, and R(L) a K-algebra with generators {pα | α ∈ L}. Definition 11.4.2.1 A monomial pα1 . . . pαr is said to be standard if α1 ≥ · · · ≥ αr . Suppose that the standard monomials form a K-basis for R(L). Given any nonstandard monomial F , the expression  ci ∈ K ∗ F = ci Fi , for F as a sum of standard monomials will be referred to as a straightening relation. Consider the surjective map π : K[L] → R(L),

xα → pα .

Let us denote ker π by I . For α, β ∈ H with α > β, we set ]β, α[= {γ ∈ L | α > γ > β}. Recall the following theorem (cf. Gonciulea-Lakshmibai [33], Theorem 5.2) Theorem 11.4.2.2 Let L, R(L), I be as above. Suppose that there exists a lattice embedding L → C, where C = C(n1 , . . . , nd ) for some n1 , . . . , nd ≥ 1, such that the entries of the d-tuple (θ1 , . . . , θd ) representing an element θ of L form a non≤ · · · ≤ θd . Suppose that I is generated as an ideal by decreasing sequence, i.e. θ1 elements of the form xτ xϕ − cαβ xα xβ (where τ, ϕ are non-comparable, and α ≥ β). Further suppose that in the straightening relation  pτ pφ = cαβ pα pβ , (*) the following hold: (a) pτ ∨φ pτ ∧φ occurs on the right-hand side of (*) with coefficient 1.

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(b) τ, φ ∈]β, α[, for every pair (α, β) appearing on the right-hand side of (*). ˙ = α ∪β, ˙ for every (α, β) on the (c) Under the embedding L → C, we have τ ∪φ right-hand side of (*). Then there exists a flat family over Spec K[t] whose special fiber (t = 0) is Spec A(L) and general fiber (t invertible) is Spec R(L). Corollary 11.4.2.3 Spec R(L) flatly degenerates to a (normal) toric variety. In particular, Spec R(L) is normal and Cohen-Macaulay. Proof. We have (cf. [39]) that A(L) is a normal domain. Hence we obtain that I (L) is a binomial prime ideal. On the other hand, we have (cf. [28]) that a binomial prime ideal is a toric ideal (in the sense of [114]). It follows that Spec A(L) is a (normal) toric variety and we obtain the first assertion. The first assertion together with Theorem 11.4.2.2 and the fact that a toric variety is normal and Cohen-Macaulay implies that Spec R(L) is normal and Cohen-Macaulay. 11.4.3 The distributive lattice D Consider the partially ordered set D = H ∪ {1} ∪ {0} defined in §11.3.3. We equip D with the structure of a distributive lattice by embedding it inside the chain lattice C(m, q) := C(m, m · · · , m, q, q · · · , q ), as follows:       n+1 times

n+1 times

To each element of D, we associate a 2n + 2-tuple: For A = (a1 , · · · , ar ) ∈ I (r, m), B = (b1 , · · · , br ) ∈ I (r, q), let A, B denote the n + 1-tuples: A := (a1 , · · · , ar , m, m, · · · , m, 1), B := (b1 , · · · , br , q, q, · · · , q, 1) (i) Let (A, B) ∈ Hp , say, A ∈ I (r, m), B ∈ I (r, q), for some r, 1 ≤ r ≤ n. We let (A, B) be the (2n + 2)-tuple: (A, B) = (A, B). (ii) Let I ∈ Hu , say, I = (i1 , · · · , in ) (∈ I (n, m)) . We let I˜ be the (2n + 2)-tuple: I˜ = (i1 , · · · , in , 1, 1, · · · , 1)    n+1 times

(iii) Let ξ ∈ Hξ , say, J = (j1 , · · · , jn ) (∈ I (n, m))), we let J˜ be the (2n + 2)-tuple: J˜ = (1, · · · , 1, j1 , · · · , jn , 1).    n+1 times

˜ 0˜ be the (2n + 2)-tuples: (iv) Corresponding to 1, 0, we let 1, 1˜ = (m, m · · · , m, q, q · · · , q ), 0˜ = (1, · · · , 1 )          n+1 times

n+1 times

2n+2 times

This induces a canonical embedding of D inside the chain lattice C(m, m · · · , m, q, q · · · , q ).       n+1 times

n+1 times

11.4 Normality and Cohen-Macaulayness of the K-algebra S

153

Lemma 11.4.3.1 Let τ1 , τ2 ∈ C(m, q). Suppose τ1 , τ2 ∈ D. Then τ1 ∨ τ2 , τ1 ∧ τ2 are also in D. Thus D acquires the structure of a distributive lattice. Proof. Clearly the Lemma requires a proof only when τ1 , τ2 are non-comparable. We consider the following cases. For two s-tuples E = {e1 , · · · , es }, F = {f1 , · · · , fs }, we shall denote E ∨ F : = (max{e1 , f1 }, . . . , max{es , fs }) E ∧ F : = (min{e1 , f1 }, . . . , min{es , fs }). Case 1: τ1 , τ2 ∈ Hp , say τ1 = (A1 , B1 ), τ2 = (A2 , B2 ). We have τ1 ∨ τ2 = (A1 ∨ A2 , B1 ∨ B2 ), τ1 ∧ τ2 = (A1 ∧ A2 , B1 ∧ B2 ) Clearly τ1 ∨ τ2 , τ1 ∧ τ2 are in Hp , and hence in D. Case 2: τ1 ∈ Hp , τ2 ∈ Hu , say τ1 = (A, B), τ2 = I˜ (for some I ∈ Hu ). Let I be the n + 1-tuple (I, 1) (entries of I followed by 1). We have τ1 ∨ τ2 = (A ∨ I , B), τ1 ∧ τ2 = (A ∧ I , (1, · · · , 1))    n+1 times

Clearly τ1 ∨ τ2 ∈ Hp , τ1 ∧ τ2 ∈ Hu . Case 3: τ1 ∈ Hp , τ2 ∈ Hξ , say τ1 = (A, B), τ2 = J˜ (for some J ∈ Hξ ). Let J be the n + 1-tuple (J, 1) (entries of I followed by 1). We have τ1 ∨ τ2 = (A, B ∨ J ), τ1 ∧ τ2 = (1, · · · , 1, B ∧ J )    n+1 times

Clearly τ1 ∨ τ2 ∈ Hp , τ1 ∧ τ2 ∈ Hξ . Case 4: τ1 , τ2 ∈ Hu , say τ1 = I˜1 , τ2 = I˜2 (for some I1 , I2 ∈ Hu ). We have τ1 ∨ τ2 = I 1 ∨ I2 , τ1 ∧ τ2 = I 1 ∧ I2 Clearly τ1 ∨ τ2 , τ1 ∧ τ2 are in Hu . Case 5: τ1 , τ2 ∈ Hξ . This case is similar to Case 4. Case 6: τ1 ∈ Hu , τ2 ∈ Hξ , say τ1 = I˜, τ2 = J˜ (for some I, J in Hu , Hξ respectively). We have τ1 ∨ τ2 = (I , J ), τ1 ∧ τ2 = 0˜ Clearly τ1 ∨ τ2 ∈ Hp , τ1 ∧ τ2 ∈ D. Lemma 11.4.3.2 We have rank (D) = (m + q)n − n2 + 2 (= d + 2, where d = (m + q)n − n2 ). In particular, dim A(D) = d + 3 This is immediate from Lemma 11.2.1.2.

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11.4.4 Flat degeneration of Spec R(D) to the toric variety Spec A(D) In this subsection, we show that Spec R(D) flatly degenerates to the toric variety Spec A(D), where A(D) is the algebra associated as in 11.4.1 to the distributive lattice D, by showing that R(D) satisfies the hypotheses of Theorem 11.4.2.2. We first prove some preparatory Lemmas. Lemma 11.4.4.1 Let τ, φ be two non-comparable elements of H . Then in the straightening relation for pτ pφ as given by Theorem 11.3.1.1, pτ ∨φ pτ ∧φ occurs with coefficient 1 (here for an element ϕ of H, pϕ stands for p(A, B), u(I ) or ξ(J ) according as ϕ = (A, B) ∈ Hp , I ∈ Hu or J ∈ Hξ ). Proof. The assertion is clear if the relation is of the type (1) of Theorem 11.3.1.1. If the relation is of the type (4) of Theorem 11.3.1.1, then the result follows from Proposition 11.1.0.2,(3). Similarly, if the relation is of the type (2) (resp. (3)) of Theorem 11.3.1.1, then the result follows as above (in view of Proposition 11.1.0.2,(3)) Let then the relation be of the type (5) or (6) of Theorem 11.3.1.1, say of type (5) (the proof is similar if it is of type (6)):  ct p(At , Bt )u(It ) (∗) p(A, B)u(I ) = t

where I ∈ I (n, m), (A, B) ∈ Hp , and A ≥ I . As in the proof of Theorem 11.3.1.1, (5), we multiply through out by ξ(In ) to arrive at  ai p(Ci1 , Ci2 )p(Di1 , Di2 ), ai ∈ K ∗ (∗∗) p(A, B)p(I, In ) = where (Ci1 , Ci2 ), (Di1 , Di2 ) belong to Hp . As above, using Proposition 11.1.0.2,(3), we obtain that p((A, B) ∨ (I, In ))p((A, B) ∧ (I, In )) occurs in (**) with coefficient 1. We have (in view of Lemma 11.4.3.1, rather its proof), p((A, B)∨(I, In ))p((A, B)∧(I, In )) = p(A∨I , B)p(A∧I , In ) = p(A∨I , B)u(A∧I )ξ(In )

Also from the proof of Theorem 11.3.1.1, (5), we have, for every i, Di2 = In (in (**)). Hence writing p(Di1 , Di2 ) = u(Di1 )ξ(In ), cancelling out ξ(In ) (note that L.H.S. of (**)= p(A, B)u(I )ξ(In )), we obtain that p(A ∨ I , B)u(A ∧ I ) occurs in (*) with coefficient 1 (note that by Case 2 in the proof of Lemma 11.4.3.1, we have (A, B) ∨ I = (A ∨ I , B), (A, B) ∧ I = (A ∧ I , (1, · · · , 1))).    n+1 times

Thus the result follows if the relation is of the type (5) (or (6)) of Theorem 11.3.1.1. Lemma 11.4.4.2 Let τ, φ be two non-comparable elements of D. Then for every (α, β) on the right-hand side of the straightening relation (in R(D), as given by Theorem 11.3.1.1), we have (i) τ, φ ∈]β, α[,

11.5 The ring of invariants K[X]SLn (K)

155

˙ = α ∪β ˙ (ii) τ ∪φ (here, ∪˙ denotes a disjoint union). Proof. The assertions follow from Theorem 11.3.1.1 (and the identification of D as a sublattice of C(m, q)). Theorem 11.4.4.3 There exists a flat family over A1 , with Spec R(D) as the generic fiber and Spec A(D) as the special fiber. In particular, R(D) is a normal CohenMacaulay ring of dimension d + 3 (where d = (m + q)n − n2 ). Proof. In view of Theorem 11.4.2.2, and Corollary 11.4.2.3, it suffices to show that (a)- (c) of Theorem 11.4.2.2 hold for R(D). (a) follows from Lemma 11.4.4.1; (b) and (c) follow from Lemma 11.4.4.2. Clearly R(D) has dim d + 3 (since dim A(D) = d + 3 (cf. Lemma 11.4.3.2)). Theorem 11.4.4.4 The K-algebra S is normal, Cohen-Macaulay of dimension (m + q)n − n2 + 1. Proof. The algebra M(D)(= R(D)(x(0)) ) being a homogeneous localization of the normal, Cohen-Macaulay ring R(D), is a normal, Cohen-Macaulay ring of dim d +2. Considering M(D) as a graded ring (cf. §11.3.3), we have S(D) = M(D)(x(1)) . Hence S(D) being a homogeneous localization of the normal, Cohen-Macaulay ring M(D), is a normal, Cohen-Macaulay ring of dimension d + 1. This together with Theorem 11.3.5.4 implies that S is a normal, Cohen-Macaulay ring of dimension d + 1 (note that d = (m + q)n − n2 ).

11.5 The ring of invariants K[X]SLn (K) We preserve the notation of §11.2. In this section, we shall show that the inclusion S ⊆ R SLn (K) is in fact an equality, i.e., S = R SLn (K) . We now apply Chapter 10, Lemma 10.1.0.4 to our situation. Let G = SLn (K). Consider m

q

∗ X=V · · ⊕ V ⊕ V · · ⊕ V ∗ = Spec R, AN = Mm,q (K) × K ( n ) × K (n)  ⊕ ·  ⊕ · m copies

q copies

Let {ui , ξj ), 1 ≤ i ≤ m, 1 ≤ j ≤ q, u(I ), ξ(J ), I ∈ Hu , J ∈ Hξ } be denoted by {f1 , · · · , fN } (note that f1 , · · · , fN are G-invariant elements in R). Let x = (u, ξ ) ∈ X. Let ψ : X → AN be the map, ψ(x) = (f1 (x), · · · , fN (x)). Clearly ψ(X) = Spec S. Let us denote Y = Spec S. Proposition 11.5.0.1 With X, AN , ψ, Y as above, the hypotheses of Chapter 10, Lemma 10.1.0.4 are satisfied.

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Proof. (i) Let x ∈ X ss . We need to show that ψ(x) = 0. If possible, let us assume that ψ(x) = 0. Let x = (u, ξ ). Let Wu (resp. Wξ ) be the span of {u1 , · · · , um } (resp.{ξ1 , · · · , ξq }). Further, let dim Wu = r, dim Wξ = s. The assumption that ψ(x) = 0 implies in particular that u(I )(x) = 0, ∀I ∈ I (n, m), ξ(J )(x) = 0, ∀J ∈ I (n, q). Hence, Wu (resp. Wξ ) is not equal to V (resp.V ∗ ). Therefore, we get r < n, s < n. Also at least one of {r, s} is non-zero; otherwise, r = 0 = s would imply ui = 0, ∀i, ξj = 0, ∀j , i.e., x = 0 which is not possible, since x ∈ X ss . Let us suppose that r = 0. (The proof is similar if s = 0.) The assumption that ψ(x) = 0 implies in particular that ui , ξj  = 0, for all i, j ; hence, Wξ ⊆ Wu⊥ . Therefore, s ≤ n − r. Hence we can choose a basis {e1 , · · · , en } of V such that Wu = the K∗ , · · · , e∗ }. Writing each vector span of {e1 , · · · , er }, and Wξ ⊆ the K-span of {er+1 n ui as a row vector (with respect to this basis), we may represent the u’s by the m × n matrix U given by ⎛ ⎞ u11 u12 . . . u1r 0 . . . 0 ⎜ u21 u22 . . . u2r 0 . . . 0⎟ ⎜ ⎟ U := ⎜ . .. .. .. .. .. ⎟ ⎝ .. . . . . .⎠ um1 um2 . . . umr 0 . . . 0 Similarly, writing each vector ξj as a column vector (with respect to the above basis), we may represent ξ ’s by the n × q matrix Λ given by ⎛ ⎞ 0 0 ... 0 ⎜ .. .. .. ⎟ ⎜ . . . ⎟ ⎜ ⎟ ⎜ 0 0 ... 0 ⎟ ⎟ Λ := ⎜ ⎜ξr+1 1 ξr+1 2 . . . ξr+1 q ⎟ ⎜ ⎟ ⎜ .. .. .. ⎟ ⎝ . . . ⎠ ξn 1

ξ n 2 . . . ξn q

Choose integers a1 , · · · , ar , br+1 , · · · , bn , all of them > 0 so that ai = bj . a a −b Let gt be the diagonal matrix in G(= SLn (K)), gt = diag(t 1 , · · · t r , t r+1 , · · · , t −bn ). We have, gt x = g · (U, Λ) = (Ugt , gt−1 Λ) = (Ut , Λt ), where ⎛ a ⎞ t 1 u11 t a2 u12 . . . t ar u1r 0 . . . 0 ⎜ t a1 u21 t a2 u22 . . . t ar u2r 0 . . . 0⎟ ⎜ ⎟ Ut = ⎜ . .. .. .. .. .. ⎟ ⎝ .. . . . . .⎠ a a a r 1 2 t um1 t um2 . . . t umr 0 . . . 0 and

⎛

0 .. .

0 .. .

...

0 .. .

⎞

⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 0 ... 0 ⎟ Λt = ⎜ ⎜t br+1 ξr+1 1 t br+1 ξr+1 2 . . . t br+1 ξr+1 q ⎟ ⎟ ⎜ ⎟ ⎜ .. .. .. ⎠ ⎝ . . . t bn ξn 2 . . . t bn ξn q t bn ξn 1

11.5 The ring of invariants K[X]SLn (K)

157

Hence gt x → 0 as t → 0, and this implies that 0 ∈ G · x which is a contradiction to the hypothesis that x is semi-stable. Therefore our assumption that ψ(x) = 0 is wrong and (i) of Lemma 10.1.0.4, Chapter 10 is satisfied. (ii) Let U = {(u, ξ ) ∈ X | {u1 , . . . , un }, {ξ1 , . . . , ξn } are linearly independent} Clearly, U is a G-stable open subset of X. Claim: G operates freely on U , U → U mod G is a G-principal fiber space, and F induces an immersion U/G → AN . Proof of Claim: Let H = GLn (K). We have a G-equivariant identification (∗)

U∼ · · × V × V ∗ × · · · × V ∗ = E × F, say = H × H × V × · (m−n) copies

(q−n) copies

· · × V ∗ . From this it is clear that where E = H × H, F = V × · · · × V × V ∗ × · (m−n) copies

(q−n) copies

G operates freely on U . Further, we see that U mod G may be identified with the fiber space with base (H × H ) mod G (G acting on H × H as g · (h1 , h2 ) = · · × V × V ∗ × · · · × V ∗ associ(h1 g, g −1 h2 ), g ∈ G, h1 , h2 ∈ H ), and fiber V × · (m−n) copies

(q−n) copies

ated to the principal fiber space H ×H → (H ×H ) / G. It remains to show that ψ induces an immersion U/G → AN , i.e., to show that the map ψ : U/G → AN and its differential dψ are both injective. We first prove the injectivity of ψ : U/G → AN . Let x, x  in U/G be such that ψ(x) = ψ(x  ). Let η, η ∈ U be lifts for x, x  respectively. Using the identification (*) above, we may write η = (A, un+1 , · · · , um , B, ξn+1 , · · · , ξq ), A, B ∈ H  η = (A , un+1 , · · · , um , B  , ξn+1 , · · · , ξq  ), A , B  ∈ H (here, ui , 1 ≤ i ≤ n are given by the rows of A, while ξi , 1 ≤ i ≤ n are given by the columns of B; similar remarks on ui , ξi ). The hypothesis that ψ(x) = ψ(x  ) implies in particular that ui , ξj  = ui , ξj , 1 ≤ i, j ≤ n which may be written as AB = A B  . This implies that (∗∗)

A = A · g,

where g = BB  −1 (∈ H ). Further, the hypothesis that u(I )(x) = u(I )(x  ), ∀I , implies in particular that u(In )(x) = u(In )(x  ) (where In = (1, 2, · · · , n)). Hence we obtain (∗ ∗ ∗)

detA = detA

Now (**) and (***) imply that g in fact belongs to G(= SLn (K)). Hence on U/G, we may suppose that

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x = (u1 , · · · , un , un+1 , · · · , um , ξ1 , · · · , ξq ) x  = (u1 , · · · , un , un+1 , · · · , um , ξ1 , · · · , ξq ) where {u1 , · · · , un } is linearly independent. For a given j , we have, ui , ξj  = ui , ξj , 1 ≤ i ≤ n, implies, ξj = ξj (since, {u1 , · · · , un } is linearly independent). Thus we obtain ξj = ξj , for all j

(†)

On the other hand, we have (by definition of U ) that {ξ1 , . . . , ξn } is linearly independent. Hence fixing an i, n + 1 ≤ i ≤ m, we get ui , ξj  = ui , ξj (= ui , ξj ), 1 ≤ j ≤ n implies, ui = ui . Thus we obtain (††)

ui = ui , for all i

The injectivity of ψ : U/G → AN follows from (†),(††). To prove that the differential dψ is injective, we merely note that the above argument remains valid for the points over K[], the algebra of dual numbers (= K ⊕K, the K-algebra with one generator , and one relation  2 = 0), i.e., it remains valid if we replace K by K[], or in fact by any K-algebra. Thus condition (ii) of Lemma 10.1.0.4, Chapter 10 holds. (iii) The above Claim implies in particular that dim U/G = dim U − dim G = (m + q)n − (n2 − 1) = dim Spec S (cf. Theorem 11.4.4.4). Thus condition (iii) of Lemma 10.1.0.4, Chapter 10 holds. The condition (iv) of Lemma 10.1.0.4, Chapter 10 follows from Theorem 11.4.4.4. ∗ · · ⊕ V × V · · ⊕ V ∗ , where m, q > Theorem 11.5.0.2 Let V = K n , X = V  ⊕ ·  ⊕ · m copies

n. Then for the diagonal action of G := SLn (K), we have

q copies

(i) First Fundamental Theorem for SLn (K)-invariants: K[X]G is generated by {p(A, B), u(I ), ξ(J ), (A, B) ∈ Hp , I ∈ Hu , J ∈ Hξ }. (ii) Second Fundamental Theorem for SLn (K)-invariants: The ideal of relations among the generators {p(A, B), u(I ), ξ(J ), (A, B) ∈ Hp , I ∈ Hu , J ∈ Hξ } is generated by the six type of relations as given by Theorem 11.3.1.1. (iii) A Standard Monomial Basis for SLn (K)-invariants: Standard monomials in {p(A, B), u(I ), ξ(J ), (A, B) ∈ Hp , I ∈ Hu , J ∈ Hξ } form a K-basis for K[X]G . (iv) K[X]G is Cohen-Macaulay. Proof. Chapter 10, Lemma 10.1.0.4 implies (in view of Proposition 11.5.0.1) that Spec S is the categorical quotient of X by G and ψ : X → Spec S is the canonical quotient map. Assertions (i) and (ii) follow from Theorem 11.3.5.5. Assertion (iii) follows from Theorem 11.3.5.3. Assertion (iv) follows from Theorem 11.4.4.4.

12 SOn (K)-action

In this chapter, we consider the SOn (K)-action on X = V · · ⊕ V, V = K n .  ⊕ · m copies

We present a proof (cf. [63]) of the Cohen-Macaulayness of K[X]SOn (K) . While Spec K[X]On (K) being a certain determinantal variety inside Sym Mm , the space of symmetric m × m matrices (cf. chapter 10)) gets identified with the opposite cell in a certain Schubert variety, Spec K[X]SOn (K) does not have a direct relationship to Schubert varieties. Nevertheless, as in Chapter 11, certain “Schubert variety connections” are used in a crucial way (cf. §12.1,§12.2,§12.3 ). As in Chapter 11, we adopt the “deformation technique” for this purpose. To be more precise, the proof of the Cohen-Macaulayness of K[X]SOn (K) is accomplished in the following steps: • One first constructs a K-subalgebra S of K[X]SOn (K) by prescribing a set of algebra generators {fα , α ∈ D}, D being a doset associated to a finite partially ordered set P (here, “doset” is as defined in DeConcini-Lakshmibai [21]; see also Definition 12.4.0.1). • One then constructs a “standard monomial” basis for S by (i) defining “standard monomials” in the fα ’s (cf. Definition 12.2.1.1) (ii) writing down the straightening relation for a non-standard (degree 2) monomial fα fβ (cf. Proposition 12.3.1.1) (iii) proving linear independence of standard monomials (by relating the generators of S to certain determinantal varieties inside the space of symmetric matrices) (cf. Proposition 12.2.2.2) (iv) proving the generation of S (as a vector space) by standard monomials (using (ii)). In fact, to prove the generation for S, one first proves generation for a “graded version” R(D) of S, where D as above is a doset associated to a finite partially ordered set P . One then deduces the generation for S. In fact, one constructs a “standard monomial” basis for R(D). While the generation by standard monomials for S is deduced from the generation by standard monomials for R(D), the linear independence of standard monomials in R(D) is deduced from the linear independence of standard monomials in S (cf. (iii) above).

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• As a consequence, one obtains a presentation for S as a K-algebra (cf. Theorem 12.3.3.4). • The Cohen-Macaulayness (cf. Proposition 12.4.1.2, Corollary 12.4.1.3) of R(D) is proved by realizing it as a “doset algebra with straightening law” (cf. DeConcini-Lakshmibai [21]), and using the results of [21]. • One then deduces the Cohen-Macaulayness of S from that of R(D) (cf. Theorem 12.4.1.4). • One proves (cf. Proposition 12.5.0.5) that Spec S is regular in codimension 1 by using the fact that Spec K[X]SOn (K) → Spec K[X]On (K) is a 2-sheeted cover. • One deduces (cf. Proposition 12.5.0.6) the normality of Spec S (using Serre’s criterion for normality - Spec A is normal if and only if A has S2 and R1 ). • One then shows that the inclusion S ⊆ K[X]SOn (K) is in fact an equality by showing that the morphism Spec K[X]SOn (K) → Spec S (induced by the inclusion S ⊆ K[X]SOn (K) ) satisfies the hypotheses in Zariski Main Theorem (and hence is an isomorphism (cf. Theorem 12.5.0.7)). One then deduces the CohenMacaulayness of K[X]SOn (K) (cf. Corollary 12.5.0.9) As the first set of main consequences, we obtain (cf. Theorem 12.5.0.8) • First Fundamental Theorem for SOn (K)-invariants, i.e., describing algebra generators for K[X]SOn (K) . • Second Fundamental Theorem for SOn (K)-invariants, i.e., describing generators for the ideal of relations among these algebra generators for K[X]SOn (K) . • A Standard Monomial Basis for K[X]SOn (K) . As the second main consequence, we obtain (cf. §12.6) a characteristic-free proof of the Cohen-Macaulayness of the moduli space M of equivalence classes of semistable, rank 2 vector bundles on a smooth projective curve of genus > 2 by relating it to K[X]SO3 (K) . In Mehta-Ramadas [80], the Cohen-Macaulayness for M is deduced by proving the Frobenius-split properties for M. As the third main consequence, we obtain (cf. Theorem 12.7.0.8) a characteristicfree basis for the ring of invariants for the (diagonal) adjoint action of SL2 (K) on sl (K) ⊕ · · · ⊕ sl2 (K).   2 m copies

Our main goal in this chapter is to prove the Cohen-Macaulayness of K[X]SOn (K) ; as mentioned above, this is accomplished by first constructing a “standard monomial” basis for the subalgebra S of K[X]SOn (K) , deducing Cohen-Macaulayness of S, and then proving that S in fact equals K[X]SOn (K) . Thus one does not use the results of DeConcini-Procesi [22] (especially, Theorem 5.6 of [22]), we rather give a different proof of Theorem 5.6 of [22].

12.1 Preliminaries For the convenience of the discussion of the results in this chapter, we recollect below the results from chapter 6, specifically the standard monomial basis for the co-

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ordinate rings of symmetric determinantal varieties. Since the results of §12.2.1 rely on an explicit description of the straightening relations (of a degree 2 non-standard monomial), in this section we derive such straightening relations (cf. Proposition 12.1.7.3) by relating symmetric determinantal varieties to Schubert varieties in the Lagrangian Grassmannian. 12.1.1 The Lagrangian Grassmannian variety Let V = K 2m , (K being the base field which we suppose to be algebraically closed of characteristic = 2) together with a non-degenerate, skew-symmetric bilinear form , . Let G = Sp(V ) (the group of linear automorphisms of V preserving , ). If J is the matrix of the form, then G may be identified with the fixed point set of the involution σ : SL(V ) → SL(V ), σ (A) = J −1 (t A)−1 J . Taking J to be   0 Jm J = −Jm 0 where Jm is the m × m matrix with 1’s along the anti-diagonal, and 0’s off the anti-diagonal, B (resp. T ), the set of upper triangular (resp. diagonal) matrices in G is a Borel sub group (resp. a maximal torus) in G (cf. Steinberg [113]). Let Lm = {maximal totally isotropic sub spaces of V }, the symplectic (or also Lagrangian) Grassmannian Variety. We have a canonical inclusion Lm → Gm,2m , where Gm,2m is the Grassmannian variety of m-dimensional subspaces of K 2m . 12.1.2 Schubert varieties in Lm Let I (m, 2m) = {i = (i1 , . . . , im )|1 ≤ i1 < · · · < im ≤ 2m}. For j, 1 ≤ j ≤ 2m, let j  = 2m + 1 − j . Let IG (m, 2m) = {i ∈ I (m, 2m)| for every j, 1 ≤ j ≤ 2m, precisely one of {j, j  } occurs in i}

Recall (cf. Chapter 6) that the Schubert varieties in Lm are indexed by IG (m, 2m); further, the partial order on the set of Schubert varieties in Lm (given by inclusion) induces a partial order on IG (m, 2m): i ≥ j ⇔ it ≥ jt , ∀t. Let w ∈ IG (m, 2m), and let X(w) be the associated Schubert variety. For the projective embedding fm : Lm → P(∧m V ) (induced by the Plücker embedding Gm,2m → P(∧m V )), let R(w) denote the homogeneous co-ordinate ring of X(w). A Standard Monomial Basis for R(w): We have (cf. Chapter 8) a basis {pτ,ϕ } for R(w)1 indexed by admissible pairs on X(w), i.e., admissible pairs (τ, ϕ) such that w ≥ τ . This basis includes the extremal weight vectors pτ , τ ∈ IG (m, 2m) corresponding to the admissible pair (τ, τ ). Thus denoting by A the set of all admissible pairs, we have that A includes the diagonal of IG (m, 2m) × IG (m, 2m). Recall the following (cf. Chapter 8): Definition 12.1.2.1 A monomial of the form pτ1 ,ϕ1 pτ2 ,ϕ2 · · · pτr ,ϕr , τ1 ≥ ϕ1 ≥ τ2 ≥ · · · ≥ ϕr

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is called a standard monomial (of degree r). Such a monomial is said to be standard on X(w), if in addition w ≥ τ1 . Theorem 12.1.2.2 Standard monomials on X(w) of degree r form a basis of R(w)r . As a consequence, we have a qualitative description of a typical quadratic relation on a Schubert variety X(w) as given by the following Proposition. Proposition 12.1.2.3 (cf. [21, 62]) Let (τ1 , φ1 ), (τ2 , φ2 ) be two admissible pairs on X(w) such that pτ1 ,ϕ1 pτ2 ,ϕ2 is non-standard so that φ1 ≥ τ2 , φ2 ≥ τ1 . Let  pτ1 ,ϕ1 pτ2 ,ϕ2 = ci pαi ,βi pγi ,δi , ci ∈ K ∗ (*) i

be the expression for pτ1 ,ϕ1 pτ2 ,ϕ2 as a sum of standard monomials on X(w). (i) For every i, we have, αi ≥ both τ1 and τ2 . Further, for some i, if αi = τ1 (resp. τ2 ), then βi > ϕ1 (resp. ϕ2 ). (ii) For every i, we have, δi ≤ both ϕ1 and ϕ2 . Further, for some i, if δi = ϕ1 (resp. ϕ2 ), then γi < τ1 (resp. τ2 ). (iii) Suppose there exists a permutation σ of {τ1 , ϕ1 , τ2 , ϕ2 } such that σ (τ1 ) ≥ σ (ϕ1 ) ≥ σ (τ2 ) ≥ σ (ϕ2 ), then (σ (τ1 ), σ (ϕ1 )), (σ (τ2 ), σ (ϕ2 )) are both admissible pairs, and pσ (τ1 ),σ (ϕ1 ) pσ (τ2 ),σ (ϕ2 ) occurs with coefficient ±1 in (*). We shall refer to a relation as in (*) as a straightening relation. A presentation for R(w): For w ∈ IG (m, 2m), let Aw = {(τ, ϕ) ∈ A | w ≥ τ }. Consider the polynomial algebra K[xτ,ϕ , (τ, ϕ) ∈ Aw ]. For two admissible pairs in Aw such that pτ1 ,ϕ1 pτ2 ,ϕ2 is non-standard, denote F(τ1 ,φ1 ),(τ2 ,φ2 ) = (τ1 , φ1 ), (τ2 , φ2 ) xτ1 ,φ1 , xτ2 ,φ2 − i ci xαi ,βi xγi ,δi , αi , βi , γi , δi , ci being as in Proposition 12.1.2.3. Let Iw be the ideal in K[xτ,ϕ , (τ, ϕ) ∈ Aw ] generated by such F(τ1 ,φ1 ),(τ2 ,φ2 ) ’s. Consider the surjective map fw : K[xτ,ϕ , (τ, ϕ) ∈ Aw ] → R(w), xτ,ϕ → pτ,ϕ . We have from Chapter 8 the following: Proposition 12.1.2.4 fw induces an isomorphism K[xτ,ϕ , (τ, ϕ) ∈ Aw ]/Iw ∼ = R(w). 12.1.3 The opposite big cell in Lm Recall the following from Chapter 6: Facts: 1. We have a natural embedding Lm → Gm,2m Gm,2m being the Grassmannian variety of m-dimensional sub spaces of K 2m . 2. Indexing the simple roots of G as in Bourbaki [8], we have an identification G/P ∼ = Lm

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P being the maximal parabolic sub group of G corresponding to “omitting” the simple root αm (the right-end root in the Dynkin diagram of G). 3. The opposite big cell O − in Gm,2m may be identified as    Im , Y ∈ Mm,m (K) (∗) O− = Y where Im is the identity m × m matrix, and Mm,m K) is the space of m × m matrices − in (with entries in K). This gives rise to an identification of the opposite big cell OG Lm as    Im − , Z ∈ Sym Mm (∗∗) OG = Z where Sym Mm is the space of symmetric m × m matrices (with entries in K). − 12.1.4 The functions fτ,ϕ on OG

Let j ∈ I (m, 2m), pj the corresponding Plücker co-ordinate on Gm,2m . Denote by fj , the restriction of pj to O − . As seen in Chapter 6, we have (under the identification (*)) that if z ∈ O − corresponds to the matrix A, then fj (z) is simply the following minor of A: let j = (j1 , . . . , jm ), and let jr be the largest entry ≤ m. Let {k1 , . . . , km−r } be the complement of {j1 , . . . , jr } in {1, . . . , m}. Then fj (z) is the (m − r)-minor of A with column indices k1 , . . . km−r , and row indices jr+1 , . . . , jm (here the rows of A are indexed as m + 1, . . . , 2m). Conversely, given a minor of A, say, with column indices b1 , . . . , bs , and row indices jm−s+1 , . . . , jm (again, the rows of A are indexed as m + 1, . . . , 2m), it is fj (z), where j = (j1 , . . . , jm ) is given as follows: {j1 , . . . , jm−s } is the complement of {b1 , . . . , bs } in {1, . . . , m}, and jd−s+1 , . . . , jm are simply the row indices. The Partial Order ≥: Given A = (a1 , · · · , as ), A = (a1 , · · · , as  ), for some s, s  ≥ 1, recall the partial order: A ≥ A if s ≤ s  , aj ≥ aj , 1 ≤ j ≤ s. As seen in Chapter 6, we have that on G/P , any pτ,ϕ ((τ, ϕ) being an admissible pair) is the restriction of some Plücker co-ordinate on Gm,2m . Let us denote − − . Given z ∈ OG , let M be the correspondby fτ,ϕ the restriction of pτ,ϕ to OG ing matrix in Sym Mm (under the identification (**)); then as above, fτ,ϕ (z) is a certain minor of A, say with row (resp. column) indices A := {a1 , · · · , as } (resp. B := {b1 , · · · , bs }); we have, A ≥ B. Denote this minor by p(A, B). Conversely, such a minor corresponds to a unique fτ,ϕ . Thus we have a bijection bij

{admissible pairs} → {minors p(A, B) of X, A ≥ B} X being a symmetric m × m matrix of indeterminates. Convention. If τ = ϕ = (1, . . . , m), then fτ,ϕ evaluated at z is 1; we shall make it correspond to the minor of X with row indices (and column indices) given by the empty set.

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12.1.5 The opposite cell in X(w) − For a Schubert variety X(w) in Lm , let us denote OG ∩ X(w) by Y (w). We consider − Y (w) as a closed subvariety of OG . In view of Proposition 12.1.2.4, we obtain that − is generated by the ideal defining Y (w) in OG

{fτ,ϕ |, w ≥ τ }. 12.1.6 Symmetric determinantal varieties Let Z = Sym Mm , the space of all symmetric m × m matrices with entries in K. We shall identify Z with AN , where N = 12 m(m + 1). We have K[Z] = K[zi,j , 1 ≤ i ≤ j ≤ m]. The variety Dt (Sym Mm ). Let X = (xij ), 1 ≤ i, j ≤ m, xij = xj i be a m × m symmetric matrix of indeterminates. Let A, B, A ⊂ {1, · · · , m}, B ⊂ {1, · · · , m}, #A = #B = s, where s ≤ m. Denote by p(A, B) the s-minor of X with row indices given by A, and column indices given by B. For n, 1 ≤ n ≤ m, let It be the ideal in K[Z] generated by {p(A, B), A ⊂ {1, · · · , m}, B ⊂ {1, · · · , m}, #A = #B = t}. Let Dt (Sym Mm ) be the symmetric determinantal variety (a closed subvariety of Z), with It as the defining ideal. In the discussion below, we also allow t = m + 1 in which case Dt (Sym Mm ) = Z. Identification of Dt (Sym Mm ) with Y (φ). From Chapter 6, we have an identification   Im − OG = X where X is a symmetric m × m matrix. As mentioned above, we have a bijection: bij

{fτ,ϕ , (τ, ϕ) ∈ A} → { minors p(A, B), A, B ∈ I (r, m), A ≥ B, 1 ≤ r ≤ m of X}

(here, A is the set of all admissible pairs, and I (r, m) = {i = (i1 , . . . , ir )|1 ≤ i1 < · · · < ir ≤ m}. Let φ be the m-tuple, φ = (t, t + 1, · · · , m, 2m + 2 − t, 2m + 3 − t, · · · , 2m)(= (t, t + 1, · · · , m, (t − 1) , (t − 2) , · · · , 1 )) (note that φ consists of the two blocks [t, m], [2m + 2 − t, 2m] of consecutive integers - here, for i < j , [i, j ] denotes the set {i, i + 1, · · · , j }, and for 1 ≤ s ≤ 2m, s  = 2m + 1 − s). If t = m + 1, then − ∼ (= Sym Mm ). From we set φ = (m , (m − 1) , · · · , 1 ); note then that Y (φ) = OG Chapter 6, we have Theorem 12.1.6.1 Dt (Sym Mm ) ∼ = Y (φ); further, dim Dt (Sym Mm ) = 1)(2m + 2 − t) = 12 (t − 1)(t − 1) .

1 2 (t

−

Corollary 12.1.6.2 K[Dt (Sym Mm )] ∼ = R(φ)(pid ) , the homogeneous localization of R(φ) at pid (id being the m-tuple (1, · · · , m)).

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Singular Locus of Dt (Sym Mm ): For our discussion in §12.5 (especially, in the proof of Lemma 12.5.0.5), we will be required to know Sing Dt (Sym Mm ), the singular locus of Dt (Sym Mm ). Let φ = ([t, m], [2m+2−t, 2m]) as above. From [57], we have Sing X(φ) = X(φ  ) where φ  = ([t − 1, m], [2m + 3 − t, 2m]). This together with Theorem 12.1.6.1 implies the following theorem: Theorem 12.1.6.3 Sing Dt (Sym Mm ) = Dt−1 (Sym Mm ) 12.1.7 The set Hm Let Hm = {(A, B) ∈

∪ I (s, m) × I (s, m) | A ≥ B}

0≤s≤m

where our convention is that (∅, ∅) is the element of Hm corresponding to s = 0. We define  on Hm as follows: • We declare (∅, ∅) as the largest element of Hm . • For (A, B), (A , B  ) in Hm , say, A = (a1 , · · · , as ), B = (b1 , · · · , bs ), A =  (a1 , · · · , as  ), B  = (b1 , · · · , bs  ) for some s, s  ≥ 1, we define (A, B)  (A , B  ) (or also p(A, B)  p(A , B  )) if B ≥ A (here, the partial order ≥ is as in §12.1.4). The Bijection θ : Let X = (xij ) be a generic symmetric m × m matrix. The above partial order  induces a partial order  on the set of all minors of X, namely, p(A, B)  p(A , B  ), if (A, B)  (A , B  ). Let (τ, ϕ) ∈ A, i.e., (τ, ϕ) is an admissible pair. As elements of IG (m, 2m), let τ = (a1 , · · · , ar , b1 , · · · , bs ), ϕ = (c1 , · · · , cr , d1 , · · · , ds ) where r + s = m, ai , bj , ci , dj are ≤ n, and (recall that) for 1 ≤ q ≤ 2m, q  = 2m + 1 − q. We would like to remark that the fact that (τ, ϕ) is an admissible pair implies that number of entries ≤ m in τ and ϕ are the same (see [67], [57] for details). Denote i := (i1 , · · · , im ) := (a1 , · · · , ar , d1 , · · · , ds ) (∈ I (m, 2m)) (here, I (m, 2m) is as in §12.1.2). Set Ai = {2m + 1 − im , 2m + 1 − im−1 , · · · , 2m + 1 − ir+1 }, Bi = the complement of {i1 , i2 · · · , ir } in {1, 2, · · · , m}. Define θ : A → {all minors of X} by setting θ (i) = p(Ai , Bi ) (here, the constant function 1 is considered as the minor of X with row indices (and column indices) given by the empty set). Then θ is a bijection. Note that θ reverses the respective (partial) orders, i.e., given i, i  ∈ I (m, 2m), corresponding to admissible pairs (τ, ϕ), (τ  , ϕ  ) we have, i ≤ i  ⇐⇒ θ (i)  θ (i  ). Using , we define standard monomials in p(A, B)’s for (A, B) ∈ Hm :

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Definition 12.1.7.1 A monomial p(A1 , B1 ) · · · p(As , Bs ), s ∈ N is standard if p(A1 , B1 )  · · ·  p(As , Bs ). Let X = V · · ⊕ V, V = K n . From Chapter 10, §10.4, we have  ⊕ · m copies

Theorem 12.1.7.2 (i) Spec K[X]O(V ) , the categorical quotient X//O(V ) gets identified with Dn+1 (Sym Mm ). (ii) Let Hn = {(A, B) ∈ Hm | #A ≤ n}. Standard monomials in {p(A, B), (A, B) ∈ Hn } form a basis for K[Dn+1 (Sym Mm )], the algebra of regular functions on Dn+1 (Sym Mm ) (⊂ Sym Mm ). As a direct consequence of Proposition 12.1.2.3, we obtain Proposition 12.1.7.3 Let p(A1 , A2 ), p(B1 , B2 ) (in K[Dt (Sym Mm )]) be non standard. Let  p(A1 , A2 )p(B1 , B2 ) = ai p(Ci1 , Ci2 )p(Di1 , Di2 ), ai ∈ K ∗ (∗) be the straightening relation, i.e., R.H.S. is a sum of standard monomials. Then for every i, Ci1 , Ci2 , Di1 , Di2 have cardinalities ≤ t − 1; further, (i) Ci1 ≥ both A1 and B1 ; further, if for some i, Ci1 equals A1 (resp. B1 ), then Ci2 > A2 (resp. > B2 ). (ii) Di2 ≤ both A2 and B2 ; further, if for some i, Di2 equals A2 (resp. B2 ), then Di1 < A1 (resp. < B1 ). (iii) Suppose there exists a σ ∈ S4 (the symmetric group on 4 letters) such that σ (A1 ) ≥ σ (A2 ) ≥ σ (B1 ) ≥ σ (B2 ), then (σ (A1 ), σ (A2 )), (σ (B1 ), σ (B2 )) are in Ht−1 , and p(σ (A1 ), σ (A2 ))p(σ (B1 ), σ (B2 )) occurs with coefficient ±1 in (*). Remark 12.1.7.4. On the R.H.S. of (*), Ci1 , Ci2 could both be the empty set (in which case p(Ci1 , Ci2 ) is understood as 1). For example, with X being a 2 × 2 symmetric matrix of indeterminates, we have 2 p2,1 = p2,2 p1,1 − p∅,∅ p12,12 .

Remark 12.1.7.5. In the sequel, while writing a straightening relation as in Proposition 12.1.7.3, if for some i, Ci1 , Ci2 are both the empty set, we keep the corresponding p(Ci1 , Ci2 ) on the right hand side of the straightening relation (even though its value is 1) in order to have homogeneity in the relation. Taking t = m + 1 (in which case Dt (Sym Mm ) = Z = Sym Mm ) in Theorem 12.1.7.2 and Proposition 12.1.7.3, we obtain Theorem 12.1.7.6 (i) Standard monomials in {p(A, B), (A, B) ∈ Hm }’s form a basis for K[Z](∼ = K[xij , 1 ≤ i ≤ j ≤ m]). (ii) Relations similar to those in Proposition 12.1.7.3 hold on Z.

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167

Remark 12.1.7.7. Note that Theorem 12.1.7.2 recovers Theorem 5.1 of DeConciniProcesi [22]. But we had taken the above approach of deducing Theorem12.1.7.2 from Theorems 12.1.2.2, 12.1.6.1 in order to derive the straightening relations as given by Proposition 12.1.7.3 (which are crucial for the discussion in §12.2.1). A presentation for K[Dt (Sym Mm )]. Consider the polynomial algebra K[x(A, B), (A, B) ∈ Ht−1 ]. For two non-comparable pairs (A1 , A2 ), (B1 , B2 ) in Ht−1 , denote  F ((A1 , A2 ); (B1 , B2 )) = x(A1 , A2 )(B1 , B2 ) − ai x(Ci1 , Ci2 )x(Di1 , Di2 ) where Ci1 , Ci2 , Di1 , Di2 , ai are as in Proposition 12.1.7.3. Let Jt−1 be the ideal generated by {F ((A1 , A2 ); (B1 , B2 )), (A1 , A2 ), (B1 , B2 ) non-comparable} Consider the surjective map ft−1 : K[x(A, B), (A, B) ∈ Ht−1 ] → K[Dt (Sym Mm )], x(A, B) → p(A, B). Then in view of Proposition 12.1.2.4 and Theorem 12.1.6.1, we obtain Proposition 12.1.7.8 ft−1 induces an isomorphism K[x(A, B), (A, B) ∈ Ht−1 ]/Jt−1 ∼ = K[Dt (Sym Mm )]

12.2 The algebra S Let V = K n together with a non-degenerate bilinear form , . Let X = V ⊕m (= V ⊕ · · · ⊕ V (m copies)), and G = SOn (K). Denote R = K[X]. We present the proof of Cohen-Macaulayness of R G (cf. [63]). This is accomplished by proving the Cohen-Macaulayness of a certain subalgebra S of R G , and showing S in fact equals R G . For u = (u1 , u2 , ..., um ) ∈ X, writing ui = (ui1 , · · · , uin ) (with respect to the standard basis for K n ), we shall identify u with the m × n matrix U := (uij )m×n . The Functions p(A, B): Let u := (u1 , · · · , um ) ∈ X. For A, B in I (r, m), where 1 ≤ r ≤ n, and A ≥ B, let p(A, B) denote the regular on X,  function  p(A, B)((u)) = the r-minor of the symmetric m × m matrix ui , uj  with row indices given by the entries of A, and column indices given by the entries of B. The Functions u(I ): For I ∈ I (n, m), let u(I ) be the function u(I ) : X → K, u(I )(u) := the n-minor of U with the row indices given by the entries of I (here, U = (uij ) is as above). We have, g · u(I ) = (det g)u(I ), g ∈ O(V ). Hence u(I ) is in K[X]SO(V ) . Lemma 12.2.0.1 For I, J ∈ I (n, m), we have u(I )u(J ) = p(I, J ).

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Proof. Let M, N denote the n × n submatrices of the m × n matrix U = (uij ) with row indices given by I, J respectively. We have u(I )u(J ) = (det M) (det N ) = (det M) (det t N ) = det ( M t N ) = p(I, J ) (note that M t N is the submatrix of (ui , uj ) with row indices given by the entries of I , and column indices given by the entries of J ). The Algebra S: Let S be the subalgebra of R G generated by p(A, B), A, B ∈ I (r, m), r ≤ n − 1, A ≥ B, u(I ), I ∈ I (n, m). Remark 12.2.0.2. The K-algebra S could have been simply defined as the Ksubalgebra of R G generated by {ui , uj } (i.e., by {p(A, B), #A = #B = 1}) ∪ {u(I ), I ∈ I (n, m)}. But we have a purpose in defining it as above, namely, the standard monomials (in S) will be built out of the p(A, B)’s with #A ≤ n − 1, and the u(I )’s (cf. Definition 12.2.1.1 below). 12.2.1 Standard monomials and their linear independence In this subsection, we define standard monomial in S, and prove their linear independence. The Set H : Let Hu := I (n, m) (note that Hu indexes the u(I )’s). Let Hp = {(A, B) ∈ I (r, m) × I (r, m), 1 ≤ r ≤ n − 1, A ≥ B} (note that Hp indexes the p(A, B)’s). Let H = Hp ∪ Hu We define a partial order ≥ on H as follows: (i) For elements Hp , it is just the partial order on Hp . (ii) For elements Hu , it is the partial order ≥ as defined in §12.1.4. (iii) No element of Hu is greater than any element of Hp . (iv) For (A, B) ∈ Hp , I ∈ Hu , (A, B) ≥ I , if B ≥ I (again, ≥ being as in §12.1.4). Thus S has a set of algebra generators {p(A, B), (A, B) ∈ Hp , u(I ), I ∈ Hu } indexed by the partially-ordered set H . Definition 12.2.1.1 A monomial F = p(A1 , B1 ) · · · p(Ar , Br )u(I1 ) · · · u(Is ) in {p(A, B), u(I ), (A, B) ∈ Hp , I ∈ Hu } is said to be standard if (A1 , B1 ) ≥ · · · ≥ (A1 , B1 ) ≥ I1 ≥ · · · ≥ Is i.e, A1 ≥ B1 ≥ A2 ≥ · · · ≥ Br ≥ I1 ≥ · · · ≥ Is .

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12.2.2 Linear independence of standard monomials In this subsection, we prove the linear independence of standard monomials. Lemma 12.2.2.1 Let (A, B) ∈ Hp , I ∈ Hu . (i) The set of standard monomials in the p(A, B)’s is linearly independent. (ii) The set of standard monomials in the u(I )’s is linearly independent. Proof. The subalgebra generated by {p(A, B), A, B ∈ Hn } being R O(V ) , gets identified with K[Dn+1 (Sym( Mm )], and hence (i) follows from Theorem 12.1.7.2. (ii) follows from [72], Theorem 1.6.6,(1) applied to K[uij , 1 ≤ i ≤ m, 1 ≤ j ≤ n]. Proposition 12.2.2.2 Standard monomials (in S) are linearly independent. Proof. Let F = G + H = 0,

(∗)

be a relation among standard monomials, where G = ci Gi , H = dj Hj and for each p(A1 , B1 ) · · · p(Ar , Br )u(I1 ) · · · u(Is ) in G (resp. H ), s is even - including 0 - (resp. odd). Consider a g in On (K), with det g = −1. Then noting that u(I )u(J ) = p(I, J ), and using the g · u(I ) = facts that g · p(A, B) = p(A, B), dj Hj = (det g)u(I ), we have, F − gF = 2dj Hj = 0. Hence, if we show that 0, then it would follow (in view of Theorem 12.1.7.2) that (*) is the trivial sum. Thus we may suppose  (∗∗) F = dj Hj where each Hj is a standard monomial of the form: Hj = p(R1 , S1 ) · · · p(Rl , Sl )u(I ) Now multiplying (∗∗) by u(In ), In being (1, · · · , n) (and noting as above that u(I )u(In ) = p(I, In )), we obtain  di Pi = 0 where each Pi is a standard monomial of the form p(U1 , Q1 ) · · · p(Um , Qm ) (note that for each standard monomial Hj appearing in (∗∗), Hj u(In ) is again standard). Now the required result follows from the linear independence of p(A, B)’s in K[X]O(V ) (cf. Theorem 12.1.7.2).

12.3 The algebra S(D) Let S be as in the previous section.

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12.3.1 Quadratic relations In this subsection, we describe certain straightening relations (i.e., expressions for non-standard monomials as linear sums of standard monomials) to be used while proving the generation (of S as a K-vector space) by standard monomials. Proposition 12.3.1.1

(i) Let I, I  ∈ Hu be not comparable. We have,  br u(Ir )u(Ir ), br ∈ K ∗ u(I )u(I  ) = r

I, I  ,

where for all r, Ir ≥ both and Ir ≤ both I, I  ; in fact, Ir > both I, I  (for,   if Ir = I or I , then I, I would be comparable). (ii) Let (A1 , A2 ), (B1 , B2 ) ∈ Hp be not comparable. Then we have  p(A1 , A2 )p(B1 , B2 ) = ai p(Ci1 , Ci2 )p(Di1 , Di2 ), ai ∈ K ∗ , where (Ci1 , Ci2 ), (Di1 , Di2 ) belong to Hp , and Ci2 ≥ Di1 ; further, for every i, we have a) Ci1 ≥ both A1 and B1 ; if Ci1 = A1 (resp. B1 ), then Ci2 > A2 (resp.B2 ). b) Di2 ≤ both A2 and B2 ; if Di2 = A2 (resp. B2 ), then Di1 < A1 (resp.B1 ). (iii) Let I ∈ Hu , (A, B) ∈ Hp be such that B ≥ I . We have,  p(A, B)u(I ) = dt p(At , Bt )u(It ), dt ∈ K ∗ t

where for every t, we have, (At , Bt ) ∈ Hp , Bt ≥ It . Further, At ≥ both A and I ; if At = A, then Bt > B. Proof. In the course of the proof, we will be repeatedly using the fact that the sub algebra generated by {p(A, B), A, B ∈ Hn } (where recall that Hn = {(A, B) ∈ Hm , #A ≤ n}, Hm being as in §12.1.7) being R O(V ) (cf. Chapter 10, Theorem 10.4.0.3), the relations given by Proposition 12.1.7.3 hold in K[X]O(V ) . Assertion (i) follows from [72], Theorem 4.1.1,(2). Assertion (ii) follows from Proposition 12.1.7.3. (iii). We have, p(A, B)u(I )u(In ) = p(A, B)p(I, In ) (In being (1, 2, · · · , n)). The hypothesis that B ≥ I implies that p(A, B)p(I, In ) is not standard. Hence Proposition 12.1.7.3 implies that in K[Dn+1 (Sym( Mm )]  ai p(Ci1 , Ci2 )p(Di1 , Di2 ), ai ∈ K ∗ p(A, B)p(I, In ) = where R.H.S. is a standard sum, i.e., Ci1 ≥ Ci2 ≥ Di1 ≥ Di2 , ∀i. Further, for every i, Ci1 ≥ both A and I ; if Ci1 = A, then Ci2 > B; Ci2 ≥ both B and In ; Di2 ≤ both B and In which forces Di2 = In (note that in view of Proposition 12.1.7.3, all minors in the above relation have size ≤ n); and hence #Di1 = n, for all i. Hence p(Di1 , Di2 ) = u(Di1 )u(In ), for all i. Hence cancelling u(In ), we obtain  p(A, B)u(I ) = ai p(Ci1 , Ci2 )u(Di1 ), where Ci1 ≥ both A and I ; if Ci1 = A, then Ci2 > B. This proves (iii).

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To prove the generation of S (as a K-vector space) by standard monomials, we define a K-algebra S(D), construct a standard monomial basis for S(D) and deduce the results for S (in fact, it will turn out that S(D) ∼ = S). We first define the K-algebra R(D) as follows: Let D = H ∪ {1} H being as in §12.2.1. Extend the partial order on H to D by declaring {1} as the largest element. Let P (D) be the polynomial algebra P (D) := K[X(A, B), Y (I ), X(1), (A, B) ∈ Hp , I ∈ Hu ] Let a(D) be the homogeneous ideal in the polynomial algebra P (D) generated by the relations (i)-(iii) of Proposition 12.3.1.1 (X(A, B), Y (I ) replacing p(A, B), u(I ) respectively), with relation (ii) homogenized as  ai X(Ci1 , Ci2 )X(Di1 , Di2 ) X(A1 , A2 )X(B1 , B2 ) = where X(Ci1 , Ci2 ) is to be understood as X(1) if both Ci1 , Ci2 equal the empty set (cf. Remark 12.1.7.5). Let R(D) = P (D)/a(D) We shall denote the classes of X(A, B), Y (I ), X(1) in R(D) by x(A, B), y(I ), x(1) respectively. The Algebra S(D): Set S(D) = R(D)(x(1)) , the homogeneous localization of R(D) y(I ) at x(1). We shall denote x(A,B) x(1) , x(1) (in S(D)) by c(A, B), d(I ) respectively. Let ϕD : S(D) → S be the map ϕD (c(A, B)) = p(A, B), ϕD (d(I )) = u(I ). Let θD : R(D) → S(D) be the canonical map. Denote γD : R(D) → S as the composite γD = ϕD ◦ θD . 12.3.2 A standard monomial basis for R(D) We define a monomial in x(A, B), y(I ), x(1)) (in R(D)) to be standard in exactly the same way as in Definition 12.2.1.1. Proposition 12.3.2.1 The standard monomials in the x(A, B), y(I ), x(1) are linearly independent. Proof. The result follows by considering γD : R(D) → S, and using the linear independence of standard monomials in S (cf. Proposition 12.2.2.2). Generation of R(D) by Standard Monomials: We shall now show that any nonstandard monomial F in R(D) is a linear sum of standard monomials. Observe that if M is a standard monomial, then x(1)l M is again standard; hence we may suppose F to be:

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F = x(A1 , B1 ) · · · x(Ar , Br )y(I1 ) · · · y(Is ) Fix an integer N sufficiently large. To each element A ∈ ∪nr=1 I (r, m), we associate an (n + 1)-tuple as follows: Let A ∈ I (r, m), for some r, A = (a1 , · · · , ar ). To A, we associate the n + 1-tuple A = (a1 , · · · , ar , m, m, · · · , m, 1). To F , we associate the integer nF (and call it the weight of F ) which has the entries of A1 , B1 , A2 , B2 , · · · , Ar , Br , I1 , · · · , Is as digits (in the N-ary presentation). The hypothesis that F is non-standard implies that either x(Ai , Bi )x(Ai+1 , Bi+1 ) is non-standard for some i ≤ r − 1, or x(Ar , Br )y(I1 ) is non-standard, or u(It )u(It+1 ) is non-standard for some t ≤ s − 1 is non-standard. Straightening these using Proposition 12.3.1.1, we obtain that F = ai Fi where nFi > nF , for all i, and the result follows by decreasing induction on nF (note that while straightening a degree 2 relation using Proposition 12.3.1.1, if x(1) occurs in a monomial G, then the digits in nG corresponding to x(1) are taken to be m, m, · · · , m (2n+2-times)). Also note that the largest F of degree r in x(A, B)’s and degree s in the y(I )’s is x({m}, {m})r u(I0 )s (where I0 is the n-tuple (m + 1 − n, m + 2 − n, · · · , m)) which is clearly standard. Hence we obtain Proposition 12.3.2.2 Standard monomials in x(A, B), y(I ), x(1)) generate R(D) as a K-vector space. Combining Propositions 12.3.2.1, 12.3.2.2, we obtain Theorem 12.3.2.3 Standard monomials in x(A, B), y(I ), x(1) give a basis for the K-vector space R(D). 12.3.3 Standard monomial bases for S(D) Standard monomials in c(A, B), d(I ) in S(D) S(D)) are defined in exactly the same way as in Definition 12.2.1.1. Theorem 12.3.3.1 Standard monomials in c(A, B), d(I ) give a basis for the Kvector space S(D). Proof. The linear independence of standard monomials follows as in the proof of Proposition 12.3.2.1 by considering ϕD : S(D) → S, and using the linear independence of standard monomials in S (cf. Proposition 12.2.2.2). To see the generation of S(D) by standard monomials, consider a non-standard monomial F in S(D), say, F = c(A1 , B1 ) · · · c(Ai , Bi )d(I1 ) · · · d(Ik ) Then F = θD (G), where G = x(A1 , B1 ) · · · x(Ai , Bi )y(I1 ) · · · y(Ik ). The required result follows from Proposition 12.3.2.2. Theorem 12.3.3.2 Standard monomials in p(A, B), u(I ) form a basis for the Kvector space S.

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173

Proof. We already have established the linear independence of standard monomials (cf. Proposition 12.2.2.2). The generation by standard monomials follows by considering the surjective map ϕD : S(D) → S and using the generation of S(D) by standard monomials (cf. Theorem 12.3.3.1). Theorem 12.3.3.3 The map ϕD : S(D) → S is an isomorphism of K-algebras. Proof. Under ϕD , the standard monomials in S(D) are mapped bijectively onto the standard monomials in S. The result follows from Theorems 12.3.3.1 and 12.3.3.2. Theorem 12.3.3.4 A Presentation for S: (i) The K-algebra S is generated by {p(A, B), u(I ), (A, B) ∈ Hp , I ∈ Hu }. (ii) The ideal of relations among the generators {p(A, B), u(I )} is generated by relations (i)-(iii) of Proposition 12.3.1.1. (iii) (Standard Monomial Basis) Standard monomials in {p(A, B), u(I ), (A, B) ∈ Hp , I ∈ Hu } form a basis for S. Proof. The result follows from Theorem 12.3.3.3 (and the definition of S(D)) Remark 12.3.3.5. We shall see (cf. Theorem 12.5.0.7 below) that the inclusion S → R G is in fact an equality.

12.4 Cohen-Macaulayness of S In this section, we prove the Cohen-Macaulayness of S in the following steps: • (i) We prove that S is a doset algebra with straightening law over a doset D contained in P × P , for a certain partially ordered set P . • (ii) P is a wonderful poset. • (iii) Conclude the Cohen-Macaulayness of S using [21]. Let P be a partially ordered set. Recall Definition 12.4.0.1 (cf. [21]) A doset of P is a subset D of P × P such that (i) The diagonal Δ(P ) ⊂ D. (ii) If (a, b) ∈ D, then a ≥ b. (iii) Let a ≥ b ≥ c in P . (a) Let (a, b), (b, c) be in D. Then (a, c) ∈ D. (b) Let (a, c) ∈ D. Then (a, b), (b, c) are in D. Remark 12.4.0.2. (i) We shall refer to an element (α, β) of D as an admissible pair; (α, α) will be called a trivial admissible pair. (ii) As a consequence of (iii) above, we have the following: If a ≥ b ≥ c ≥ d in P are such that (a, d) is in D, then (b, c) ∈ D. Definition 12.4.0.3 (cf. [21]) A doset algebra with straightening law over the doset D is a K-algebra E with a set of algebra generators {x(A, B), (A, B) ∈ D} such that

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(i) E is graded with E0 = K, and Er equals the K-span of monomials of degree r in x(A, B)’s. (ii) “Standard monomials” x(A1 , B1 ) · · · x(Ar , Br ) (i.e., A1 ≥ B1 ≥ A2 ≥ B2 · · · Ar ≥ Br ) is a K-basis for Er . (iii) Given a non-standard monomial F := x(A1 , B1 ) · · · x(Ar , Br ), in the straightening relation  (∗) F = ai Fi expressing F as a sum of standard monomials, writing Fi = x(Ai1 , Bi1 ) · · · x(Air , Bir ) (a standard monomial) we have the following: (a) For every permutation σ of {A1 , B1 , A2 , B2 , · · · , Ar , Br }, we have, {Ai1 , Bi1 , Ai2 , Bi2 , · · · , Air , Bir } is lexicographically greater than or equal to {σ (A1 ), σ (B1 ), σ (A2 ), σ (B2 ), · · · , σ (Ar ), σ (Br )} . (b) If there exists a permutation τ of {A1 , B1 , A2 , B2 , · · · , Ar , Br } such that τ (A1 ) ≥ τ (B1 ) ≥ τ (A2 ) ≥ τ (B2 ) ≥ · · · ≥ τ (Ar ) ≥ τ (Br ), then the monomial x(τ (A1 ), τ (B1 )) · · · x(τ (Ar ), τ (Br )) occurs on the R.H.S. of (*) with coefficient ±1. (We shall refer to this situation as “{A1 , B1 , A2 , B2 , · · · , Ar , Br } is totally ordered up to a reshuffle”) (Note that in (iii)(b), we have (in view of (iii) in Definition 12.4.0.1)) that (τ (Ai ), τ (B i ))’s are admissible pairs. The Discrete Doset Algebra K{D}: The discrete doset algebra K{D} is the doset algebra with straightening relations given as follows: Let F := x(A1 , B1 ) · · · x(Ar , Br ) be a non-standard monomial. Then  x(τ (A1 ), τ (B1 )) · · · x(τ (Ar ), τ (Br )), if τ (A1 ) ≥ τ (B1 ) ≥ · · · ≥ τ (Ar ) ≥ τ (Br ) F = 0, otherwise for some τ as in (iii),(b) above (if such a τ exists). Remark 12.4.0.4. The conditions in (iii) are equivalent to the corresponding conditions for r = 2. Let K{P } be the discrete algebra over P , namely, the Stanley-Reisner algebra, defined as the quotient of the polynomial algebra K[xα , α ∈ P ] by the ideal generated by {xα xβ , α, β ∈ P non-comparable}. Recall Proposition 12.4.0.5 (cf. [21], Theorem 3.5) Let E be a doset algebra with straightening law over a doset D inside P × P . (i) There exists a sequence E = B1 , B2 , · · · , Br = K{D} of doset algebras with straightening law over the doset D such that there exists a flat family Bj , 1 ≤ j ≤ r − 1 (over Spec K[t]) with generic fiber Spec Bj , and special fiber Spec Bj +1 . (ii) K{D} is Cohen-Macaulay if K{P } is.

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175

12.4.1 A doset algebra structure for R(D) We first define The Partially Ordered Set P : Let P := ∪nr=1 I (r, m) ∪ {1} Extend the partial order on ∪nr=1 I (r, m) to P by declaring 1 to be the largest element. Let D be as in §12.3, namely, D = Hp ∪ Hu ∪ {1} Lemma 12.4.1.1 D is a doset inside P × P . Proof. Clearly, D ⊂ P × P , and contains Δ(P ). Note that Hu (as a subset of P × P ) is identified with the diagonal Δ(Hu ) (inside P × P ); similarly, 1 is identified with (1, 1). Also note that the non-trivial admissible pairs in D are among the (A, B)’s ((A, B) ∈ Hp ). The remaining conditions in Definition 12.4.0.1 hold in view of the results in §12.1.6 and the results of [62, 69]. Proposition 12.4.1.2 R(D) is a doset algebra with straightening laws over the doset D. Proof. Condition (1) in Definition 12.4.0.3 follows from the definition of the Kalgebra R(D); note that R(D) has algebra generators given by {x(A, B), (A, B) ∈ Hp }, y(I ), I ∈ Hu , x(1), indexed by D. Note that the generators {y(I ), I ∈ Hu , x(1)} are indexed by trivial admissible pairs, and the generators indexed by non-trivial admissible pairs are among {x(A, B), (A, B) ∈ Hp }. Condition (2) in Definition 12.4.0.3 follows from Theorem 12.3.2.3. Verification of Condition (3): As in the proof of Theorem 4.1 of [21], in view of Proposition 12.3.1.1, it suffices to verify condition (3) in Definition 12.4.0.3 for a degree two non-standard monomial F (cf. Remark 12.4.0.4). We divide the verification into the following cases: Case 1: Let F = y(I )y(J ). In this case, the situation of 3(b) (in Definition 12.4.0.3) does not exist. The condition 3(a) follows from Proposition 12.3.1.1,(1). Case 2: Let F = x(A, B)y(I ). In this case again, the situation of 3(b) (in Definition 12.4.0.3) does not exist (since B ≥ I , and I can not be > A or B - note that no element of Hu is greater than any element of Hp ). Condition 3(a) follows from Proposition 12.3.1.1,(3). Case 3: Let F = x(A1 , A2 )x(B1 , B2 ). Condition 3(a) follows from Proposition 12.3.1.1,(2). Condition 3(b) follows from Theorem 12.1.6.1, and DeConciniLakshmibai [21], Theorem 4.1 (especially, its proof); here, we should remark that one first concludes such relations for Y (φ) (Y (φ) as in Theorem 12.1.6.1), then for S, and hence for R(D) (note that S being R(D)(x(1)) such relations in S imply similar relations in R(D), since x(1)l is the largest monomial in any given degree l). Corollary 12.4.1.3 R(D) is Cohen-Macaulay.

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Proof. This follows from Propositions 12.4.0.5, 12.4.1.2. Note that P is a wonderful poset (in the sense of DeConcini-Eisenbud-Procesi [20]), in fact, a distributive lattice, and hence the discrete algebra K{P } is Cohen-Macaulay (cf. [20], Theorem 8.1). The above Corollary together with the fact that S is a homogeneous localization of R(D) implies Theorem 12.4.1.4 S is Cohen-Macaulay.

12.5 The equality R SOn (K) = S We preserve the notation from the previous sections. In this section, we shall first prove that the morphism q : Spec R SOn (K) −→ Spec S induced by the inclusion S ⊆ R SOn (K) , is finite, surjective, and birational. Then, we shall prove that Spec S is normal, and deduce (using Zariski’s Main Theorem) that q is an isomorphism thus proving that the inclusion S ⊆ R SOn (K) is an equality, i.e., R SOn (K) = S. As a consequence, we will obtain (in view of Theorem 12.4.1.4) that R SOn (K) is CohenMacaulay. Notation: In this section, for an integral domain A, κ(A) will denote the quotient field of A. / be a finitely generated K-algebra (K being an algebraically Lemma 12.5.0.1 Let A / be a domain. Let γ be closed field of characteristic different from 2). Further, let A / Let A = A /Γ where Γ ∼ an involutive K-algebra automorphism of A. = Z/2Z is the group generated by γ . We have, / −→ Spec A induced by the inclusion A ⊂ A / is (i) The canonical map p : Spec A a finite, surjective morphism. / is a quadratic extension of κ(A). (ii) κ(A) / can be Proof. (i) It is easy to see that a finite set B of A-algebra generators of A chosen so that they are all eigenvectors of γ corresponding to the eigenvalue −1. / one has f = 1/2(f + γ (f )) + 1/2(f − γ (f )).) (This is because for any f ∈ A, / each generator satisfies the equation Since a := f.γ (f ) ∈ A for any f ∈ A, x 2 + a = 0 (over A). Also, since product of any two generators is an eigenvector / is generated as an A-module corresponding to the eigenvalue 1, it follows that A / is a finite module by the set of algebra generators B together with 1. Therefore A over A. Hence p is a finite morphism; surjectivity of p follows from the fact (cf. Mumford [86], Ch I.7, Proposition 3) that a finite morphism f : Spec B −→ Spec A of affine varieties is surjective if and only if f ∗ : A −→ B is injective. Assertion (1) follows. / has algebra generators, (ii) From the discussion in (i), we obtain that the A-algebra A say, {f1 , · · · , fr } such that • fi2 ∈ A, 1 ≤ i ≤ r • fi fj ∈ A, ∀i, j

12.5 The equality R SOn (K) = S

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/ α 2 ∈ A. This implies that every s ∈ κ(A) / Hence we obtain that for every α ∈ A, 2 / satisfies a quadratic equation x + a over κ(A); further, κ(A) is a (finite) separable extension of κ(A) (since char K = 2). Assertion (2) follows from this. / = R SOn (K) . We have, Corollary 12.5.0.2 Let A = R On (K) , A / −→ Spec A induced by the inclusion A ⊂ A / is (i) The canonical map p : Spec A a finite, surjective morphism. / is a quadratic extension of κ(A). (ii) κ(A) Proof. Taking γ ∈ On (K) to be an order 2 element which projects onto the generator of On (K))/SOn (K) =: Γ (= Z/2Z), we have, γ defines an involutive K-algebra / with A /Γ = A. The result follows from Lemma 12.5.0.1. automorphism of A, Proposition 12.5.0.3 The morphism q : Spec R SOn (K) −→ Spec S induced by the inclusion S ⊆ R SOn (K) is surjective, finite, and birational. / = Spec(R SOn (K) ), Y = Spec(R On (K) ), Z = Spec S. Consider Proof. Denote Y the inclusions (∗)

R On (K) ⊂ S ⊆ R SOn (K)

Note that the first inclusion is a strict inclusion, since, S = R On (K) [u(I ), I ∈ I (n, m)] (and u(I ) ∈ R On (K) for I ∈ I (n, m)). This induces the following commutative diagram / Y   p q      Y Z The finiteness of q follows from the finiteness of p (cf. Corollary 12.5.0.2, (1)); this together with the inclusion q ∗ : S → R SOn (K) implies the surjectivity of q (cf. Mumford [86], ch I.7, Proposition 3). Now the inclusions given by (*) give the following inclusions of the respective quotient fields: κ(R On (K) ) ⊂ κ(S) ⊆ κ(R SOn (K) ) (with the first inclusion being a strict inclusion). This together with the fact that κ(R SOn (K) ) is a quadratic extension of κ(R On (K) ) (cf. Corollary 12.5.0.2, (2)) implies that κ(S) is a quadratic extension of κ(R On (K) ) as well. Hence we obtain that κ(S) = κ(R SOn (K) ) proving the birationality of q. Finally, to verify the hypotheses in Zariski’s Main Theorem for the morphism q, it remains to show that S is a normal domain. Again in view of Theorem 12.4.1.4 and Serre’s criterion for normality (namely, Spec A is normal if and only if A has S2 and R1 ), to prove the normality of S, it suffices to show that Spec S is regular in codimension 1 (i.e., singular locus of Spec S has codimension at least 2). Towards

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proving this, we first obtain a criterion for the branch locus of a finite morphism to have codimension at least 2. Let π : Z −→ Y be a finite morphism where Y is a reduced and irreducible affine scheme over an algebraically closed field K of arbitrary characteristic. Then, there exists an open subscheme Y1 ⊂ Y (namely, the set of unramified points for π) such that res(π) : π −1 (Y1 ) → Y1 is an étale morphism (see Mumford [86], Ch.III.10; here, res(π) denotes the restriction of π.). Let Y, Z be affine, say Y = Spec A, Z = Spec B, where A, B are finitely generated K-algebras. Further, let A be a domain, and B an integral extension of A which is finitely generated as an A module (so that π : Z −→ Y is a finite morphism). Suppose that κ(B) is a finite separable extension of κ(A). Then, there exists a s ∈ B such that κ(B) = κ(A)[s]. Indeed if bb is a primitive element for the extension κ(B) of κ(A) with b, b ∈ B, then there exists a λ ∈ κ(A) such that κ(B) = κ(A)[b + λb ]. To see this, observe that since there are only finitely many intermediate fields between κ(A) and κ(B) (while κ(A) is infinite), we have that not all κ(A)[b + λb ], λ ∈ κ(A) can be distinct. Hence, for some λ, μ ∈ κ(A), λ = μ, b + μb is in κ(A)[b + λb ]. From this it follows that bb is in κ(A)[b + λb ]. Lemma 12.5.0.4 With the above notation, let Y = Spec A and let Z = Spec B. Consider the finite morphism π : Z −→ Y induced by the inclusion A ⊂ B. As above, let b be a primitive element such that κ(B) = κ(A)[b]. Let a ∈ A be such that A[1/a][b] = B[1/a]. Further, let the discriminant of b be invertible in A[1/a]. Then res(π) : Spec B[1/a] → Spec A[1/a] is étale. (Here, res(π) denotes the restriction of π.) Proof. Let U = Spec (A[1/a]), V = π −1 (U ) = Spec (B[1/a]). Since B[1/a] = A[1/a][b] (by Hypothesis), we obtain that B[1/a] is a free A[1/a]-module with basis {1, b, · · · , bN −1 } (here, N = [κ(B) : κ(A)]). Further, by Hypothesis, discriminant (b) is invertible in A[1/a]. Hence we obtain that res(π) : Spec (B[1/a]) → Spec (A[1/a]) is étale. Proposition 12.5.0.5 The variety Spec S is regular in codimension 1. Proof. Taking A = R On (K) , B = S, as seen in the proof of Proposition 12.5.0.3, we have that κ(B) is a quadratic extension of κ(A). Further, the generators uI (of the A-algebra B = A[u(I ), I ∈ I (n, m)]) satisfy the relation u(I )2 − p(I, I ) = 0 over A (note that A = K[p(I, J ), I, J ∈ I (r, m), r ≤ n] (cf. Theorem 12.1.7.2)). Hence, κ(B) is also separable over κ(A), since char(K) = 2. Now we take Y = Spec A, Z = Spec B, and π : Z → Y the morphism induced by the inclusion R On (K) ⊆ S, in Lemma 12.5.0.4. Fixing a particular I ∈ I (n, m), we have that the relation u(I )u(J ) = p(I, J ) implies that u(J ) = p(I, J )u(I )/p(I, I ) and hence u(J ) ∈ S[1/p(I, I )] for all J . In particular S[1/p(I, I )] is a free R On (K) [1/p(I, I )]-module with basis {1, u(I )}. Also the discriminant δ(uI ) is seen to be 4p(I, I ) which is invertible in R On (K) [1/p(I, I )]. Hence, taking a = p(I, I ), b = u(I ), the hypothesis of Lemma 12.5.0.4 are satisfied. Hence res(π) :

∪

I ∈I (n,m)

S[1/p(I, I )] →

∪

I ∈I (n,m)

R On (K) [1/p(I, I )]

12.5 The equality R SOn (K) = S

179

is étale. Let a be the ideal generated by {p(I, I ), I ∈ I (n, m)}. Note that in view of the relations u(I )u(J ) = p(I, J ), I, J ∈ I (n, m) (cf. Lemma 12.2.0.1), we have p(I, J )2 = p(I, I )p(J, J ) √ √ Thus we have, p(I, J ) ∈ a, ∀I, J ∈ I (n, m). Let Y0 = V ( a). Then Y0 is simply Dn (Sym Mm ). Also, Y being Dn+1 (Sym Mm ) (cf. Theorem 12.1.7.2), we have Y0 is the singular locus of Y (cf. Theorem 12.1.6.3). Now the branch locus Yb of π : Z −→ Y is contained in Y0 . Hence the fact that codimY Y0 ≥ 2 (since Y is normal) implies that codimY Yb ≥ 2

(∗) Denote

Ye := {unramified points for π} We have, res(π) : π −1 (Ye ) → Ye is étale, and Yb = Y \ Ye . Denote Ze := π −1 (Ye ), Zb := π −1 (Yb ) Denoting by Zsing the singular locus of Z, we have (cf. (*)) codimZ (Zsing ∩ Zb ) (≥ codimZ Zb ) ≥ 2

(∗∗)

On the other hand, res(π) : Ze → Ye being étale, we have, π −1 (Ysing ∩ Ye ) = Zsing ∩ Ze Hence we obtain, (∗ ∗ ∗)

codimZ (Zsing ∩ Ze ) = codimY (Ysing ∩ Ye ) ≥ codimY (Ysing ) ≥ 2

(**) and (***) imply that codimZ (Zsing ) ≥ 2, and the result follows. The above Proposition together with Theorem 12.4.1.4 implies the following Proposition 12.5.0.6 Spec S is normal. The result follows from Serre’s criterion for normality: Spec A is normal if and only if A has S2 and R1 . Theorem 12.5.0.7 The inclusion S ⊆ R SOn (K) is an equality, i.e., R SOn (K) = S. Proof. Propositions 12.5.0.3, 12.5.0.6 imply (in view of Zariski Main Theorem (cf. Mumford [86], ch III.9)) that the morphism q : Spec R SOn (K) −→ Spec S is in fact an isomorphism. The result follows from this. Combining the above Theorem with Theorems 12.3.3.4, 12.4.1.4 we obtain the following theorems.

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Theorem 12.5.0.8 A presentation for R G : (i) (First fundamental Theorem) The K-algebra R G (= S) is generated by {p(A, B), u(I ), (A, B) ∈ Hp , I ∈ Hu }. (ii) (Second fundamental Theorem) The ideal of relations among the generators {p(A, B), u(I )} is generated by relations (1)–(3) of Proposition 12.3.1.1. (iii) (Standard Monomial Basis) Standard monomials in {p(A, B), u(I ), (A, B) ∈ Hp , I ∈ Hu } form a basis for R G . Theorem 12.5.0.9 R SOn (K) is Cohen-Macaulay. The result follows from the above Theorem and Theorem 12.4.1.4.

12.6 Application to moduli problem In this section, using Theorem 12.5.0.9, we give a characteristic-free proof of the Cohen-Macaulayness of the moduli space M2 of equivalence classes of semi-stable rank 2, degree 0 vector bundles on a smooth projective curve of genus > 2 by relating it to K[X]SO3 (K) . It is known (cf. [95],§7, Theorem 3) that M2 is smooth when the genus is 2. Assume for the moment that the characteristic of the field K is zero. Consider the moduli space Mn of equivalence classes of semi-stable, rank n, degree 0 vector bundles on a smooth projective curve C of genus m > 2. Let V be the trivial vector bundle on C. The automorphism group of V (∼ = H 0 (C, Aut V )) can be identified with GLn (K). The tangent space at V of the versal deformation space (cf. [108]) of V is H 1 (C, End V ) and hence it can be identified with m copies of the space Mn (K) of n×n matrices, V being identified with the “origin”. The canonical action of Aut V on this tangent space gets identified with the diagonal adjoint action of GLn (K) on m copies of Mn (K). Now the moduli space Mn is a GIT quotient Z//H , for a suitable Z and H a projective linear group of suitable rank. The versal deformation space of V gets embedded in Z and by “Luna slice” type of arguments (cf. [87], Appendix to chapter 1, D), the analytic local ring of Mn at V gets identified with the analytic local ring of H 1 (C, End V )//P GLn (K) at the “origin”. Suppose now that n = 2. If V is a stable vector bundle, then the point in M2 that it defines is smooth. If V is not stable, then the point in M2 that it defines can be represented by L1 ⊕L2 , where L1 and L2 are line bundles of degree zero. If L1 ∼ = L2 , then End (V ) ∼ = M2 (K), and the considerations are the same as for the case when V is trivial. If L1 is not isomorphic to L2 , then Aut V is the torus T := Gm × Gm , and the analytic local ring of M2 at V is isomorphic to the analytic local ring of H 1 (C, End V )//T at the origin (in fact the action of T is trivial). This is certainly Cohen-Macaulay. Although we are only interested in the rank 2 case, let us remark that considerations of the previous paragraph hold when the rank of V is arbitrary, and the Zi , G = Gi , analytical local ring of Mn is isomorphic to Z//G, where Z = where Zi is g copies of Mni (K) and Gi = P GLni (K).

12.7 Results for the adjoint action of SL2 (K)

181

Although we have assumed that the characteristic of the field to be zero, the above considerations remain valid in positive characteristic but with certain restrictions; for example, if n = 2, then the characteristic should not be 2 or 3. Thus, in order to prove that M2 is Cohen-Macaulay, it suffices to show that the point corresponding to the trivial bundle is so. Further, since the analytic local ring at the point corresponding to the trivial bundle gets identified with the completion of the local ring at the origin of M2 (K)⊕m //P GL2 (K), it suffices to prove that M2 (K)⊕m //P GL2 (K) is Cohen-Macaulay. Let then n = 2, M2 := M2,2 (K), M20 := sl2 (K). Let Z = M2 ⊕ · · · ⊕ M2 (g copies), Z0 = M20 ⊕ · · · ⊕ M20 (g copies) Let A = K[Z], A0 = K[Z0 ]. Let G = SL2 (K). Consider the diagonal action of G on Z, Z0 induced by the action of G on M2 , M20 by conjugation. From the above discussion, we have that the completion of the local ring at the point in M corresponding to the trivial rank 2 vector bundle is isomorphic to the completion of AG at the point which is the image of the origin (in Z(= A4g )) under Z → SpecAG . On the other hand, we have, AG = AG 0 [x1 , · · · , xg ], xi ’s being 0 ∼ indeterminates (since, M2 = M2 ⊕ K); further, we have that the adjoint action of SL2 (K) on sl2 (k) is isomorphic to the natural representation of SO3 (K) on K 3 (note that the Lie algebras sl2 (K) and so3 (K) are isomorphic). Hence the ring AG 0 gets identified with K[X]SO3 (K) , X being V ⊕ · · · ⊕ V (g copies), V = K 3 . Hence we obtain Theorem 12.6.0.1 (i) The GIT quotients Z//G, Z0 //G are Cohen-Macaulay. (ii) The moduli space M is Cohen-Macaulay. (iii) We have standard monomial bases for the co-ordinate rings of these spaces. (iv) We have first & second fundamental theorems for these spaces, i.e., algebra generators, and generators for the ideal of relations among the generators. Remark 12.6.0.2. The results in Theorem 12.6.0.1 being characteristic-free, we may deduce from Theorem 12.6.0.1 that the moduli space M behaves well under specializations; for instance, if the curve C is defined over Z, and if MZ is the corresponding moduli space, then for any algebraically closed field K of characteristic = 2, 3, the base change of MZ by K gives the moduli space MK over K. Remark 12.6.0.3. This section is motivated by Mehta-Ramadas [80]. In loc.cit, the Cohen-Macaulayness for Z//G, Z0 //G, M are deduced by proving the Frobeniussplit properties for these spaces.

12.7 Results for the adjoint action of SL2 (K) Consider G = SL2 (K), ch K = 2, 3. Let Z = sl2 (K) ⊕ · · · ⊕ sl2 (K) = Spec R, say    m copies

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Using the results of §12.4, §12.5, we shall describe a “standard monomial basis” for R G ; the elements of this basis will be certain monomials in tr (Ai Aj )’s, and tr (Ai Aj Ak )’s. Identification of sl2 (K) and so3 (K): Let       01 1 0 00 X= ,H = ,Y = 00 0 −1 10 be the Chevalley basis of sl2 (K). Let ,  be a symmetric non-degenerate bilinear form on V = K 3 . Taking the matrix of the form ,  to be J = anti-diagonal(1, 1, 1), we have, SO3 (K) = {A ∈ SL3 (K) | J −1 (t A)−1 J = A} so3 (K) = {A ∈ sl3 (K) | J −1 (t A)J = −A} The Chevalley basis of so3 (K) is given by ⎛ √ ⎛ ⎞ ⎛ ⎞ ⎞ 0 0 0 0 2 √ 20 0 0 √ 0 0⎠ X  = ⎝0 0 − 2⎠ , H  = ⎝0 0 0 ⎠ , Y  = ⎝ 2 √ 0 0 −2 0 0 0 0 − 20 The map X → X  , H → H  , Y → Y  gives an isomorphism sl2 (K) ∼ = so3 (K) of Lie algebras. Further, the map     b c a b 3 θ : sl2 (K) → K ,  → √ , −a, √ c −a 2 2 identifies the adjoint action of SL2 (K) on sl2 (K) with the natural action of SO3 (K) on K 3 ; and the induced map θm : Z −→ V · · ⊕ V, V = K 3  ⊕ · m copies

identifies the diagonal (adjoint) action of SL2 (K) on Z with the diagonal (adjoint) · · ⊕ V. action of SO3 (K) on V  ⊕ · Lemma 12.7.0.1 Let z ∈ Z, say z = (A1 , · · · , Am ). Let θm (z) = (u1 , · · · , um ). We have (i) tr (Ai Aj ) = 2ui , uj . (ii) Let I ∈ I (3, m), say, I = (i, j, k). Then tr (Ai Aj Ak ) = 2u(I ). (here, u(I ) is as in §12.2) The proof is an easy verification. Notation: In the sequel we shall denote U (i, j ) := tr (Ai Aj ), i ≥ j, U (i, j, k) := tr (Ai Aj Ak ), (i, j, k) ∈ I (3, m)

12.7 Results for the adjoint action of SL2 (K)

183

Remark 12.7.0.2. Note that if i, j, k are not distinct, then tr (Ai Aj Ak ) = 0; for, say, i = j , then Ai being a 2 × 2 traceless matrix, we have (in view of CayleyHamilton theorem), A2i = −|Ai |, and hence tr (A2i Ak ) = −|Ai |tr( Ak ) = 0 (since Ak ∈ sl2 (K)). In view of the identification given by θm , we obtain (cf. Theorems 12.5.0.7, 12.3.3.2) a “standard monomial basis” for S := R SL2 (K) . By Theorem 12.3.3.2, monomials p(α)p(A, B)u(I ) where p(α) := p(α1 , α2 ) · · · p(α2r−1 , α2r ), for some r, αi ∈ [1, m] p(A, B) := p(A1 , B1 ) · · · p(As , Bs ), for some s, Ai , Bi ∈ I (2, m) u(I ) := u(I1 ) · · · u(It ), for some t, I ∈ I (3, m) α1 ≥ · · · ≥ α2r ≥ A1 ≥ B1 ≥ A2 ≥ · · · ≥ Bs ≥ I1 ≥ · · · ≥ I give a basis for R SO3 (K) (here, [1, m] denotes the set {1, · · · , m}). We shall refer to p(α)p(A, B)u(I ) as a standard monomial of multi-degree (r, s, t). Denote Mr,s,t := {standard monomials of multi-degree (r, s, t)} Standard Monomials of Type I,II,III: Let notation be as above. We shall refer to p(α1 , α2 ) · · · p(α2r−1 , α2r ), α1 ≥ · · · ≥ α2r p(A1 , B1 ) · · · p(As , Bs ), A1 ≥ B1 ≥ A2 ≥ · · · ≥ Bs u(I1 ) · · · u(It ), I1 ≥ · · · ≥ I as standard monomials of Type I,II,III (and of degree r, s, t) respectively. We now define three types of standard monomials in U (i, j )(= tr (Ai Aj )), i ≥ j, U (i, j, k)(= tr (Ai Aj Ak )), (i, j, k) ∈ I (3, m) analogous to the above three types. Type I and Type III are direct extensions to traces: Definition 12.7.0.3 A monomial of the form U (α1 , α2 ) · · · U (α2r−1 , α2r ), α1 ≥ · · · ≥ α2r will be called a Type I standard monomial. Definition 12.7.0.4 A monomial of the form U (I1 ) · · · U (It ), I1 ≥ · · · ≥ I will be called a Type III standard monomial.

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12 SOn (K)-action

For defining Type II standard monomials, we first define a bijection between {p(A, B), A, B ∈ I (2, m), A ≥ B} and {U (j, i)U (l, k), j ≥ i, l ≥ k, i ≥ l}. Note that given U (j, i)U (l, k), j ≥ i, l ≥ k, we may suppose that j is the greatest among {i, j, k, l} (if l is the greatest, then we may write U (j, i)U (l, k) as U (l, k)U (j, i)); then the latter set is simply {all non-standard Type I, degree 2 monomials in the U (a, b)’s}

Let us denote the two sets by A, B respectively. Let p(A, B) ∈ A, say, A = (a, b), b > a, B = (c, d), d > c. Define ω : A → B as follows:  U (b, c)U (d, a), if d ≥ a ω(p(A, B)) = U (b, d)U (a, c), if d < a Given a standard monomial p(A1 , B1 ) · · · p(As , Bs ), Ai , Bi ∈ I (2, m), we shall define ω(p(A1 , B1 ) · · · p(As , Bs )) := ω(p(A1 , B1 )) · · · ω(p(As , Bs )) Definition 12.7.0.5 A monomial U (j1 , i1 )U (l1 , k1 ) · · · U (js , is )U (ls , ks ), jt > it , lt > kt , 1 ≤ t ≤ s is called a Type II standard monomial if it equals ω(p(A1 , B1 ) · · · p(As , Bs )) for some (Type II) standard monomial p(A1 , B1 ) · · · p(As , Bs ). Remark 12.7.0.6. Note in particular that for s = 1, the Type II standard monomials (in the U (j, i)’s) are precisely the non-standard Type I, degree 2 monomials in the U (a, b)’s. Let us extend ω to Mr,s,t , the definition of ω(F), for F a Type I or III standard monomial being obvious, namely, ω(p(α1 , α2 ) · · · p(α2r−1 , α2r )) = U (α1 , α2 ) · · · U (α2r−1 , α2r ) u(I1 ) · · · u(It ) = U (I1 ) · · · U (It ) Define ω(p(α)p(A, B)u(I )) = ω(p(α))ω(p(A, B))ω(u(I )), p(α)p(A, B)u(I ) ∈ Mr,s,t Definition 12.7.0.7 A monomial in the U (j, i)’s, j ≥ i and U (I )’s, I ∈ I (3, m) of the form ω(p(α)p(A, B)u(I )), p(α)p(A, B)u(I ) ∈ Mr,s,t will be called a standard monomial of type (r, s, t). Notation: We shall denote Nr,s,t = ω(Mr,s,t )

12.7 Results for the adjoint action of SL2 (K)

185

Theorem 12.7.0.8 Let Z = sl2 (K) ⊕ · · · ⊕ sl2 (K) = Spec R, say, where, m > 3.    m copies

Standard monomials (in the traces) of type (r, s, t) where r, s, t are positive integers form a basis for R SL2 (K) for the adjoint action of SL2 (K) on Z Proof. Denote S := R SL2 (K) . Write S = ⊕ Sr,s,t (r,s,t)

where r, s, t are positive integers, and Sr,s,t is the K-span of Mr,s,t . In fact, we have (in view of linear independence of standard monomials (cf. Theorem 12.3.3.2)) that Mr,s,t is a basis for Sr,s,t . Using the bijection ω, we shall show that Nr,s,t is also a basis for Sr,s,t . Clearly this requires a proof only in the case s = 0. Let Nr,s,t = #Mr,s,t . The relations (cf. Lemma 12.7.0.1; note that p(i, j ) = ui , uj ) p(i, j ) = p(A, B) =

1 U (i, j ) 2

1 (U (a, c)U (b, d) − U (b, c)U (a, d)) 4 1 u(I ) = U (I ) 2

give raise to the transition matrix, say, M. Then it is easy to see that for a suitable indexing of the elements of Mr,s,t , and Nr,s,t , the matrix M takes the upper triangular form with the diagonal entries being non-zero. To be very precise, to p(α)p(A, B)u(I ) ∈ Mr,s,t , we associate a (4s + 2r + 3t)-tuple n(α, A, B, I ) as follows. Let p(α) = p(α1 , α2 ) · · · p(α2r−1 , α2r ), α1 ≥ · · · ≥ α2r p(A, B) = p(A1 , B1 ) · · · p(As , Bs ), A1 ≥ B1 ≥ A2 ≥ · · · ≥ Bs u(I ) = u(I1 ) · · · u(It ), I1 ≥ · · · ≥ I Ai = (ai1 , ai2 ), ai1 < ai2 , Bi = (bi1 , bi2 ), bi1 < bi2 , Ai ≥ Bi 1 ≤ i ≤ s Set n(α, A, B, I ) = (a12 , a11 , b12 , b11 , a22 , a21 · · · , bs2 , bs1 , α, I ) where α = (α1 , · · · , α2r ), and I = (I1 , · · · , It ). We take an indexing on Mr,s,t induced by the lexicographic order on the n(α, A, B, I )’s, and take the induced indexing on Nr,s,t (via the bijection ω). With respect to these indexings on Mr,s,t , Nr,s,t , it is easily seen that the transition matrix M is upper triangular with the diagonal entries being non-zero. It follows that Nr,s,t is a basis for Sr,s,t . Remark 12.7.0.9. Note that the standard monomial basis (in the traces) as given by Theorem 12.7.0.8 is characteristic-free. Also, note that Theorem 12.7.0.8 recovers the result of Procesi [100], Theorem 3.4(a), for the case of SL2 (K), namely, tr (Ai Aj ), tr (Ai Aj Ak ), i, j, k ∈ [1, m] generate R G as a K-algebra (in a characteristic-free way).

13 Applications of standard monomial theory

In this chapter, we present some applications of SMT. One of the most important applications of SMT is the determination of the singular loci of Schubert varieties. We present results for Schubert varieties in the flag variety SLn (K)/B. Using the standard monomial basis for Schubert varieties, Lakshmibai and Seshadri determined ([68]) the singular loci of Schubert varieties in the flag variety. It might be said that this work is the beginning of the study of singularities of Schubert varieties. After this work, over the past two decades, a lot of work has been done (by several authors) on the singularities of Schubert varieties. For a complete account, see [4]. After presenting the results on singularities of Schubert varieties in the flag variety, we discuss some important algebraic varieties - ladder determinantal varieties, quiver varieties, varieties of complexes - for which geometric properties such as normality, CohenMacaulayness etc., are concluded by relating them to Schubert varieties. We also present (as yet another application of SMT), the degeneration of Schubert varieties (in the Grassmannian) to toric varieties (cf. [33]). Here again, though many more results have been proved in recent times (cf. [12, 17]), we have presented the results for Schubert varieties in the Grassmannian because of the simplicity of the proof in this case!

13.1 Tangent space and smoothness 13.1.1 The Zariski tangent space Let x be a point on a variety X. Let mx be the maximal ideal of the local ring OX, x with residue field K(x)(= OX, x /mx ). Note that K(x) = K (since K is algebraically closed). The Zariski tangent space to X at x is defined as Tx (X) = DerK (OX, x , K(x)) = {D : OX, x → K(x), K-linear such that D(ab) = D(a)b + aD(b)} (here K(x) is regarded as an OX, x -module). It can be seen easily that Tx (X) is canonically isomorphic to HomK-mod (mx /m2x , K).

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13 Applications of standard monomial theory

13.1.2 Smooth and non-smooth points A point x on a variety X is said to be a simple or smooth or nonsingular point of X if OX, x is a regular local ring. A point x which is not simple is called a multiple or non-smooth or singular point of X. The set Sing X = {x ∈ X | x is a singular point} is called the singular locus of X. A variety X is said to be smooth if SingX = ∅. Theorem 13.1.2.1 Let x ∈ X. Then dimK Tx (X) ≥ dim OX, x (dim OX, x is also denoted dimx X) with equality if and only if x is a simple point of X. Proof. We have dimK Tx (X) = dimK (mx /m2x ) ≥ dim OX, x , with equality if and only if OX, x is regular. The result now follows. 13.1.3 The space T (w, τ ) Let G be a semisimple algebraic group. Let T , B, W, R, S etc., be as in Chapter 3. For α ∈ R, let Xα be the element of the Chevalley basis for g(= LieG), corresponding to α. We follow [8] for denoting elements of R, R + , S etc. For τ ≤ w, let T (w, τ ) be the Zariski tangent space to X(w) at eτ . Let w0 be the element of largest length in W . Now the tangent space to G at eid is g, and hence the tangent space to G/B at eid is ⊕ β∈R + g−β . For τ ∈ W , identifying G/B with τ G/τ B (where τ B = τ Bτ −1 ) via the map gB → (nτ gn−1 τ ) B, nτ being a fixed lift of τ in NG (T ), we have, the tangent space to G/B at eτ is ⊕ β∈τ (R + ) g−β , i.e., T (w0 , τ ) = ⊕ β∈τ (R + ) g−β . Set

N (w, τ ) = {β ∈ τ (R + ) | X−β ∈ T (w, τ )}.

Since T (w, τ ) is a T -stable subspace of T (w0 , τ ), we have T (w, τ ) = the span of {X−β , β ∈ N (w, τ )}. 13.1.4 A canonical affine neighborhood of a T -fixed point Let τ ∈ W . Let Uτ− be the unipotent subgroup of G generated by the root subgroups U−β , β ∈ τ (R + ) (note that Uτ− is the unipotent part of the Borel sub group τ B − , opposite to τ B (= τ Bτ −1 )). We have U−β ∼ = Ga , Uτ− ∼ = Πβ∈τ (R + ) U−β . Now, Uτ− eτ gets identified with AN , where N = #R + via the above identification and we shall denote the induced coordinate system on Uτ− eτ by {x−β , β ∈ τ (R + )}. In the sequel, we shall denote Uτ− eτ by Oτ− . Thus we obtain that Oτ− is an affine neighborhood of eτ in G/B.

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189

13.1.5 The affine variety Y (w, τ ) For w ∈ W, w ≥ τ , let us denote Y (w, τ ) := Oτ− ∩ X(w). It is a nonempty affine open subvariety of X(w), and a closed subvariety of the affine space Oτ− . 13.1.6 Equations defining Y (w, τ ) in Oτ− Let w ∈ W . For 1 ≤ i ≤ l(= rankG), fix a basis Di for the kernel of the surjective map H 0 (G/B, Li ) → H 0 (X(w), Li ) (given by restriction), Li being the ample generator of Pic(G/Pi ). We have (Chapter 3, §3.3.3) that the ideal sheaf of X(w) in G/B is generated by {Di , 1 ≤ i ≤ l}. Let τ ∈ W be such that τ ≤ w. Let Y (w, τ ) be as in §13.1.5, I (w, τ ) the ideal defining Y (w, τ ) as a closed subvariety of Oτ− . We have I (w, τ ) is generated by {f |Oτ− , f ∈ Di , 1 ≤ i ≤ l}. For example, take G = SL(4). The ideal of Y ((3412), id) in OI−d is generated by {p4 , p234 }, and the ideal of Y ((2413), id) is generated by {p3 , p4 , p34 , p134 , p234 }. 13.1.7 Jacobian criterion for smoothness Let Y be an affine variety in An , and let I (Y ) be the ideal defining Y in An . Let I (Y ) ∂fi ). Then we have, be generated by {f1 , f2 , . . . , fr }. Let J be the Jacobian matrix ( ∂x j rank JP ≤ codimAn Y with equality if and only if P is a smooth point of Y (here JP denotes J evaluated at P ). Example 1: Consider the Schubert variety X(3412) in SL4 /B. As stated in §13.1.6, − are {p4 , p234 }, and the equations definthe equations defining Y ((3412), id) in Oid − with the group of ing Y ((2413), I d) are {p3 , p4 , p34 , p134 , p234 }. Identifying Oid unipotent lower triangular matrices in SL4 , we have p3 = x31 p4 = x41 p34 = x42 x31 − x41 x32 p134 = x43 x32 − x42 p234 = x21 (x43 x32 − x42 ) − (x43 x31 − x41 ) Hence, the equations defining Y ((3412), I d) are {x41 , x21 (x43 x32 − x42 ) − (x43 x31 − x41 )}. The Jacobian matrix is 0 1 0 0 0 1 0 0 . (x43 x32 − x42 ) −x43 x21 x43 1 −x21 −x31 Here, the columns are indexed by x21 , x31 , x32 , x41 , x42 , x43 . The rank of this matrix at P =id (i.e. all xij = 0) is 1 < 2 = 6 − l(3412) = codimY ((3412), I d). Therefore, X(3412) is singular at eid .

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13 Applications of standard monomial theory

Example 2: Consider the Schubert variety X(2413) in SL4 /B; the equations defining Y ((2413), I d) are {x31 , x41 , x31 x42 − x41 x32 , x42 − x43 x32 , x21 (x43 x32 − x42 ) − (x43 x31 − x41 )}. The Jacobian matrix is ⎡ ⎤ 0 1 0 0 0 0 ⎢ ⎥ 0 0 0 1 0 0 ⎢ ⎥ ⎢ ⎥. 0 x −x −x x 0 42 41 32 31 ⎢ ⎥ ⎣ ⎦ 1 −x32 0 0 −x43 0 (x43 x32 − x42 ) −x43 x21 x43 1 −x21 (x32 x21 − x31 ) This matrix is easily seen to have rank 3 when all variables are set to zero; 3 = 6 − l(2413) =codimY ((2413), I d) which implies X(2413) smooth at eid . 13.1.8 T -stable curves For a root α > 0, let Zα denote the SL(2)-copy in G corresponding to α; note that Zα is simply the subgroup of G generated by Uα and U−α . Given x ∈ W , precisely one of {Uα , U−α } fixes the point ex . Thus Zα · ex is a T -stable curve in G/B (note that Zα · ex ∼ = P1 ), and conversely any T -stable curve in G/B is of this form (cf. [13]). Corresponding to each α > 0, the number of T -stable curves in G/B is 12 (#W ), and thus the number of T -stable curves in G/B is 12 (#W )l(w0 ), w0 being the element of largest length in W . Now a T -stable curve Zα · ex is contained in a Schubert variety X(w) if and only if ex , esα x are both in X(w). Given y ≤ w, let R(y, w) = {α > 0 | sα y ≤ w}. Then there are precisely r(y, w) T -stable curves in X(w) passing through ey , where r(y, w) = #R(y, w). Deodhar’s Inequality: (see [13] for example) For x ≤ y ≤ w, #{α ∈ R + | x ≤ sα y ≤ w} ≥ l(w) − l(x). In view of Deodhar’s Inequality, we have r(y, w) ≥ l(w) (taking x = id in Deodhar’s Inequality). Thus there are at least l(w) T -stable curves in X(w) passing through ey . Lemma 13.1.8.1 Let w, τ ∈ W, w ≥ τ . Let β ∈ τ (R + ). If w ≥ sβ τ , then X−β ∈ T (w, τ ). Proof. The hypothesis that w ≥ sβ τ implies that the curve Zβ · eτ is contained in X(w). Now the tangent space to Zβ · eτ at eτ is the one-dimensional span of X−β . The required result now follows. We shall show in the following section that w, τ being as above, T (w, τ ) is precisely the span of {X−β , β ∈ τ (R + ) | w ≥ sβ τ }, G being SLn (K).

13.2 Singularities of Schubert varieties in the flag variety In this section, we shall describe the singular locus of a Schubert variety in the flag variety. Let G = SLn (K), T the maximal torus in G consisting of diagonal matrices, and B the Borel subgroup consisting of upper triangular matrices.

13.2 Singularities of Schubert varieties in the flag variety

191

13.2.1 Ideal of Y (w, τ ) Let w, τ ∈ W, w ≥ τ . Let Y (w, τ ) be as in §13.1.5, and I (w, τ ) the ideal defining Y (w, τ ) as a closed subvariety of Oτ− . For 1 ≤ d ≤ l, we have H 0 (G/B, L(ωd )) = (∧d K n )∗ , the linear dual of ∧d K n . Further, the Plücker coordinates pθ , θ ∈ Id,n form a basis of H 0 (G/B, L(ωd )). The kernel of H 0 (G/B, L(ωd )) → H 0 (X(w), L(ωd )) has a basis consisting of {pθ , θ ∈ Id,n |w (d) ≥ θ }, where w (d) is the element of Id,n (= W Pd , Pd being as in Chapter 4, §4.1.6) given by πd (X(w)) = XPd (w (d) ), πd being the canonical projection G/B → G/Pd . Hence we obtain the following Theorem: Theorem 13.2.1.1 The ideal I (w, τ ) is generated by {pθ | Oτ− , θ ∈ Id,n , 1 ≤ d ≤ n − 1 | w (d) ≥ θ }. Let w, τ be as above. We shall now give a criterion for X(w) to be smooth at eτ . We first remark that (1) For θ, μ ∈ W d (= W Pd ), 1 ≤ d ≤ n − 1, pθ (eμ ) = 0 ⇐⇒ θ = μ, where, recall that for θ = (i1 · · · id ) ∈ W d (= Id,n ), eθ denotes the vector ei1 ∧ · · · ∧ eid in ∧d K n , and pθ denotes the Plücker coordinate associated to θ . (2) Let Xα be the element of the Chevalley basis of g, corresponding to α ∈ R. If Xα pμ = 0, μ ∈ W d , then Xα pμ = ±psα μ , where sα is the reflection corresponding to the root α. (3) For α = β, if Xα pμ , Xβ pμ are non-zero, then Xα pμ = Xβ pμ . The first remark is obvious, since {pθ | θ ∈ W d } is the basis of (∧d k n )∗ (= H 0 (G/Pd , L(ωd ))), dual to the basis {eφ , φ ∈ Id,n } of ∧d k n . The second remark is a consequence of SL2 theory, using the following facts    = 0 or 1, χ being the weight of pμ . (a) |χ, α ∗ | =  2(χ,α)  (α,α) (b) pμ is the lowest weight vector for the Borel subgroup μ B = μBμ−1 . The third remark follows from weight considerations (note that if Xα pμ = 0, then Xα pμ is a weight vector (for the T -action) of weight χ + α, χ being the weight of pμ ). Theorem 13.2.1.2 (cf. [68]) Let τ, w ∈ W, τ ≤ w. Then dim T (w, τ ) = #{γ ∈ τ (R + ) | w ≥ sγ τ }. Proof. By Theorem 13.2.1.1, we have, I (w, τ ) is generated by {pθ | Oτ− , θ ∈ Id,n , 1 ≤ d ≤ n − 1 | w (d) ≥ θ }. Denoting the affine coordinates on Oτ− by xβ , β ∈ ∂pθ and Xβ pθ at eτ coincide. Let Jw denote τ (R − ), we have that the evaluations of ∂x β the Jacobian matrix of Y (w, τ ) (considered as a subvariety of the affine space Oτ− ). We shall index the rows of Jw by {pθ | Oτ− , θ ∈ Id,n , 1 ≤ d ≤ n − 1 | w (d) ≥ θ }

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13 Applications of standard monomial theory

and the columns by xβ , β ∈ τ (R − ). Let Jw (τ ) denote Jw evaluated at τ . Now in view of (1) & (2) above, the (pθ , xβ )-th entry in Jw (τ ) is non-zero if and only if Xβ pθ = ±pτ . Hence in view of (3) above, we obtain that in each row of Jw (τ ), there is at most one non-zero entry. Hence rankJw (τ ) = the number of non-zero columns of Jw (τ ). Now, Xβ pθ = ±pτ if and only if θ ≡ sβ τ ( mod WPd ). Thus the column of Jw (τ ) indexed by xβ is non-zero if and only if w ≥ sβ τ . Hence rankJw (τ ) = #{γ ∈ τ (R + ) | w ≥ sγ } and thus we obtain dim T (w, τ ) = #{γ ∈ τ (R + ) | w ≥ sγ τ }. Theorem 13.2.1.3 Let w, τ be as in Theorem 13.2.1.2. Then {Xβ , β ∈ τ (R − ) | w ≥ sβ τ } is a basis for T (w, τ ). Proof. Let β ∈ τ (R − ) be such that w ≥ sβ τ . We have (by Lemma 13.1.8.1), Xβ ∈ T (w, τ ), and hence dim T (w, τ ) ≥ #{γ ∈ τ (R + ) | w ≥ sγ τ }. On the other hand, by Theorem 13.2.1.2, dim T (w, τ ) = #{γ ∈ τ (R + ) | w ≥ sγ τ }. The result follows from this. Corollary 13.2.1.4 X(w) is smooth at eτ if and only if l(w) = #{α ∈ R + | w ≥ τ sα }. Proof. We have, X(w) is smooth at eτ if and only if dim T (w, τ ) = dim X(w), and the result follows in view of Theorem 13.2.1.2 (note that if β = τ (α), then sβ τ = τ sα ). Corollary 13.2.1.5 X(w) is smooth if and only if l(w) = #{α ∈ R + | w ≥ sα }. Proof. If SingX(w) (the singular locus of X(w)) is not empty, then it is a closed B-stable subset of X(w), and hence is a union of Schubert subvarieties of X(w). In particular, we obtain that SingX(w) is non-empty if and only if eid ∈ SingX(w). Thus X(τ ) is smooth if and only if it is smooth at eid , and the result follows from Theorem 13.2.1.2. Remark 13.2.1.6. The proof of Theorem 13.2.1.3 uses the fact (and just this fact alone) that the extremal weight vectors in H 0 (G/Pd , L(ωd )) form a basis for H 0 (G/Pd , L(ωd )). We make an abstraction of this fact and deduce the following Theorem 13.2.1.7 Let ω be a minuscule fundamental weight with P as the associated maximal parabolic subgroup. Let τ, w ∈ W P , τ ≤ w. Then the tangent space T (w, τ ) has a basis given by {Xβ , β ∈ τ (R − \ RP− ) | w ≥ sβ τ } The proof follows from the fact that ω being minuscule, the extremal weight vectors in H 0 (G/P , L(ω)) form a basis for H 0 (G/P , L(ω)) (cf. Chapter 8). Application to orthogonal Grassmannian: Let H = SL2n (K), G = SO2n (K). Let Qn , Pn be as in Chapter 7, §7.2. Let τ, w ∈ W Pn , τ ≤ w. Let TH (w, τ ) (resp. TG (w, τ )) denote the tangent space to XQn (w) (resp. XPn (w)) at τ Qn (resp. τ Pn ), notation being as in Chapter 7, §7.2.5. Let σ be as in Chapter 7.

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193

Theorem 13.2.1.8 TH (w, τ ) is σ -stable. Further TG (w, τ ) = TH (w, τ )σ The proof is immediate from Theorem 13.2.1.7 and the description of RG (Chapter 7, §7.1) (note that the fundamental weight ωn (for G) is minuscule). Remark 13.2.1.9. One has similar “root-system” description as in Theorem 13.2.1.3 for the singular locus of a Schubert variety for other classical groups (see [4, 57, 58, 65] for details). We also would like to remark that even though one has now a “type-free” description of singular points on a Schubert variety (thanks to [9, 14, 54]), SMT-theoretic approach seems to be the only approach which gives an explicit determination of the tangent space at a singular point on a Schubert variety! Application to symplectic Grassmannian: Using the “root-system” description for the tangent space T (w, τ ) for the symplectic group case, we have results similar to Theorem 13.2.1.8. In fact more is true as given by the following Theorem 13.2.1.10 (cf. [57]) Let H = SL2n (K), G = Sp2n (K). Let BH , BG be the Borel subgroups of H, G respectively as in Chapter 6. Let τ, w ∈ WG , τ ≤ w. Let TH (w, τ ) (resp. TG (w, τ )) denote the tangent space to XBH (w) (resp. XBG (w)) at τ BH (resp. τ BG ). Let σ be as in Chapter 6. We have TH (w, τ ) is σ -stable. Further TG (w, τ ) = TH (w, τ )σ . 13.2.2 A criterion for smoothness In this sub section, we give a criterion for smoothness of Schubert varieties in SLn (K)/B, in terms of “pattern avoidance” (cf. [66]). We first observe that for n = 4, X(3412), X(4231) are the only two singular Schubert varieties. In [66], it is shown that the situation for a general SL(n) is quite analogous to that of SL(4) as given by the following Theorem: Theorem 13.2.2.1 (cf. [66]) Let η = (a1 . . . an ) ∈ Sn . Then X(η) is singular if and only if there exist i, j, k, l, 1 ≤ i < j < k < l ≤ n such that either (1) or (2) below holds. (1) ak < al < ai < aj (2) al < aj < ak < ai . 13.2.3 Components of the singular locus In this subsection, we give an explicit description of the irreducible components of the singular locus of a Schubert variety in the flag variety. In G = SL4 (K), we have that Sing X(3412) = X(1324) and Sing X(4231) = X(2143). It turns out that for a general n, the situation is “nothing more than this”.

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13 Applications of standard monomial theory

Let then X(w) ⊂ G/B be singular (G being SL(n)). Let w = (a1 · · · an ). Since X(w) is singular, there exist i, j, k, l, 1 ≤ i < j < k < l ≤ n such that either ak < al < ai < aj or al < aj < ak < ai .

(13.1)

It is shown in [66] that in the former case, if w  is obtained from w by replacing ai , aj , ak , al respectively by ak , ai , al , aj , then ew ∈ SingX(w); and in the latter case, if w  is obtained from w by replacing ai , aj , ak , al respectively by aj , al , ai , ak , then ew ∈ SingX(w). For w ∈ W , let Pw (resp. Qw ) be the maximal element of the set of parabolic subgroups which leave BwB (in G) stable under multiplication on the left (resp. right). Definition 13.2.3.1 Given parabolic subgroups P , Q, we say that BwB is P -Q stable if P ⊂ Pw and Q ⊂ Qw . Theorem 13.2.3.2 Let w = (a1 . . . an ). Then SQw = {i − i+1 |ai > ai+1 } SPw = {i − i+1 |i + 1 appears before i in {a1 , a2 , . . . , an } }. Proof. Let α = i − i+1 . Then α ∈ Pw (resp. Qw ) if and only if w −1 (α) (resp. w(α)) is < 0. The result now follows from this (note that for 1 ≤ t ≤ n, w(t) = at , w −1 (t) = m, where m is given by am = t). The set Fη Let η = (a1 . . . an ) ∈ Sn (the symmetric group on n letters). Let Eη be the set of all τ  ≤ η such that either 1) or 2) below holds. 1) There exist i, j, k, l, 1 ≤ i < j < k < l ≤ n, such that (a) ak < al < ai < aj (b) if τ  = (b1 . . . bn ), then there exist i  , j  , k  , l  , 1 ≤ i  < j  < k  < l  ≤ n such that bi  = ak , bj  = ai , bk  = al , bl  = aj (c) if τ (resp. η ) is the element obtained from η (resp. τ  ) by replacing ai , aj , ak , al resp. by ak , ai , al , aj (respectively bi  , bj  , bk  , bl  resp. by bj  , bl  , bi  , bk  ), then τ  ≥ τ and η ≤ η. 2) There exist i, j, k, l, 1 ≤ i < j < k < l ≤ n, such that (a) al < aj < ak < ai (b) if τ  = (b1 . . . bn ), then there exist i  , j  , k  , l  , 1 ≤ i  < j  < k  < l  ≤ n such that bi  = aj , bj  = al , bk  = ai , bl  = ak (c) if τ (resp. η ) is the element obtained from η (resp. τ  ) by replacing ai , aj , ak , al resp. by aj , al , ai , ak (respectively bi  , bj  , bk  , bl  resp. by bk  , bi  , bl  , bj  ), then τ  ≥ τ and η ≤ η. Let Fη = {τ ∈ Eη | Bτ B is Pη -Qη stable}. Theorem 13.2.3.3 For η ∈ Sn , the singular locus of X(η) is equal to ∪λ X(λ), where λ runs over the maximal (under the Bruhat order) elements of Fη . This was first conjectured by Lakshmibai-Sandhya (cf. [66]); later it was proved by several authors (cf. [5, 48, 78]).

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195

13.3 Singular loci of Schubert varieties in the Grassmannian In this section, we describe the components of Sing X, the singular locus of a Schubert variety X in Gd,n . We also describe a recursive formula for the multiplicity at a singular point on X. As seen in chapter 4, the Schubert varieties in Gd,n are indexed by W Pd = {(a1 , . . . , ad ) | 1 ≤ a1 < · · · < ad ≤ n}, Pd being as in Chapter 4, §4.1.6. For our discussion, it is more convenient to use the language of partitions (or Young diagrams). To (a1 , . . . , ad ) ∈ W Pd we associate the Young diagram a := (ad ≥ · · · ≥ a1 ), where ai = ad−i+1 − (d − i + 1). For any partition a = (ad ≥ · · · ≥ a1 ), we will denote by Xa the Schubert variety corresponding to (a1 , . . . , ad ). The dimension of Xa in this setup equals |a| := a1 + . . . ad . It is clear that ai cannot be larger than n − d. Now we describe SingXa , the singular locus of Xa . To describe Sing Xa , we use the “rectangle” notation. Let a be a partition. we write q

q

a = (p1 1 , . . . , pr r ) = (p1 , . . . , p1 , p2 , . . . , p2 , · · · pr , . . . , pr ),          q1

q2

qr

where pi denotes the Young diagram consisting of pi boxes arranged in one row. With this notation, we say that a consists of r rectangles: p1 × q1 , . . . , pr × qr . Theorem 13.3.0.1 (cf. [73]) Let a consist of r rectangles. Then (1) Sing Xa has r − 1 components Xa1 , . . . , Xar−1 , where q

q

q −1

i−1 ai = (p1 1 , . . . , pi−1 , pi i

i+2 , (pi+1 − 1)qi+1 +1 , pi+2 , . . . , pr r ).

q

q

(Note that a/ai , 1 ≤ i ≤ r − 1 are simply the hooks in the Young diagram a. For example, if a = (7, 54 , 32 , 13 ), then a2 = (7, 53 , 23 , 13 )) (2) For a rectangular partition a, Xa is nonsingular. Remark 13.3.0.2. Note that the Young diagram of a component of Sing Xa is obtained by deleting a hook from the Young diagram of Xa . In particular, we have the following Theorem 13.3.0.3 Sing Dt = Dt−1 . (Here, Dt is the determinantal variety in Mr,d (cf. Chapter 5)) Proof. Set n = r + d. We have (cf. Chapter 5, Theorem 5.2.2.7) Dt ∼ = Yφ where φ = (t, t + 1, · · · , d, n + 2 − t, n + 3 − t, · · · , n), and Yφ is the opposite cell in X(φ) (note that φ consists of the two blocks [t, d], [n + 2 − t, n] of consecutive integers; here, for a, b ∈ N, a < b, [a, b] denotes the set {a, a + 1, · · · , b}). The partition determined by φ is given by a = (n − d, · · · , n − d , t − 1, · · · , t − 1).       t−1 times

Now by Theorem 13.3.0.1,

d+1−t times

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13 Applications of standard monomial theory

Sing Xa = Xb , where b is the partition b = (n − d, · · · , n − d , t − 2, · · · , t − 2).       t−2 times

d+2−t times

Now the d-tuple τ associated to b is given by τ = (t − 1, t, · · · , d, n + 3 − t, n + 4 − t, · · · , n) = ([t − 1, d], [n + 3 − t, n]). Hence SingYφ = Yτ . Hence we obtain SingDt = Dt−1 . 13.3.1 Multiplicity at a singular point In this subsection, we give a recursive formula for the multiplicity at a singular point on a Schubert variety X in the Grassmannian variety. Multiplicity of a local ring Let A be a finitely generated K-algebra. Further let A be local with m as the unique maximal ideal. For l ≥ 0, let ψA (l) = length(A/ml ) (= dimk (A/ml )). ψA is called the Hilbert-Samuel function of A. Recall (see [27] for example) the following: Theorem 13.3.1.1 There exists a polynomial PA (x) ∈ Q[x], called the HilbertSamuel polynomial of A such that (i) ψA (l) = PA (l), l # 0. (ii) degPA (x) = dimA. (iii) The leading coefficient of PA (x) is of the form eA /n!, where eA ∈ Z+ and n = dimA. Definition 13.3.1.2 With notations as in Theorem 13.3.1.1, the number eA is called the multiplicity of A. Definition 13.3.1.3 For a point P on an algebraic variety X, the multiplicity of X at P is defined to be the number eA , where A = OX,P , the stalk at P , and is denoted multP X. Proposition 13.3.1.4 Let P be a point on an algebraic variety X. Then P is a smooth point of X if and only if multP X = 1. For a proof, see [86]. Recall (see [27] for example) the following: Theorem 13.3.1.5 Let gr(A, m) = ⊕l≥0 ml /ml+1 . There exists a polynomial QA (x) ∈ Q[x], such that (i) QA (l) = dimk ml /ml+1 , l # 0. (ii) deg QA (x) = dimA − 1 (= n − 1). (iii) The leading coefficient of QA (x) is of the form

fA (n−1)! ,

where fA ∈ Z+ .

13.3 Singular loci of Schubert varieties in the Grassmannian

197

Remark 13.3.1.6. With notations as above, by considering the exact sequence 0 → ml /ml+1 → A/ml+1 → A/ml → 0, we have for l # 0, QA (l) = PA (l + 1) − PA (l) eA eA = [ (l + 1)n + ()(l + 1)n−1 + · · · ] − [ (l)n + ()(l)n−1 + · · · ] n! n! eA n−1 = + ··· . (l) (n − 1)! Hence we obtain eA = f A . Example. Let R be the polynomial algebra K[x1 , · · · xn ], a the maximal ideal generated by {x1 , · · · xn }, and A = Ra (the localization at a). Let m = aRa . We   have, . ml /ml+1 is the span of monomials of total degree l and has dimension l+n−1 n−1 l+n−1 Hence QA (l) = n−1 and is a polynomial in l of degree n − 1; further, the leading 1 . Therefore eA = 1. coefficient of QA (l) is equal to (n−1)! Multiplicity of a graded affine K-algebra For this subsection we refer the reader to [37], [27]. Let B be a graded, finitely + generated K-algebra. The function fB (n) = dimk Bn , n ∈ Z is called the Hilbert function of B, and n∈Z+ fB (n)t n is called the Hilbert series of B. Theorem 13.3.1.7 [37] There exists a polynomial PB (x) ∈ Q[x] of degree dim(X), where X = Proj(B) such that fB (n) = PB (n), for n # 0. Further, the leading coefficient of PB (n) is of the form cB /r!, where cB ∈ N and r = degPB (x). A formula for the multiplicity In this subsection, we give a recursive formula for the multiplicity at a singular point on a Schubert variety X in the Grassmannian. Let X = X(w) be a Schubert variety in the Grassmannian Gd,n (∼ = SLn (K)/Pd , Pd being as in Chapter 4, §4.1.6). Let us denote Pd by just P . Given x ∈ X, we shall denote multx X by just mx X. For x ∈ B · eτ , we have mx (X) = meτ (X). Thus it suffices to determine meτ (X), τ ≤ w. For eτ ∈ X, let us denote meτ (X) by just mτ (w). Consider a τ ∈ W P such that τ ≤ w. Evaluation of Plücker coordinates on Uτ− eτ I. Let us first consider the case τ = id. We identify U − eid with ⎧⎛ ⎫ ⎞ Idd×d ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨⎜xd+1 1 . . . xd+1 d ⎟ ⎬ ⎜ ⎟ ⎜ .. .. ⎟ , xij ∈ k, d + 1 ≤ i ≤ n, 1 ≤ j ≤ d . ⎪ ⎪ ⎝ . . ⎠ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ x n 1 . . . xn d

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13 Applications of standard monomial theory

Let A be the affine algebra of U − eid . Let us identify A with the polynomial algebra K[x−β , β ∈ R + \ RP+ ]. To be very precise, we have R + \ RP+ = {j − i , 1 ≤ j ≤ d, d + 1 ≤ i ≤ n}; given β ∈ R + \ RP+ , say β = j − i , we identify x−β with xij . Hence we obtain that the expression for pθ |U − eid in the local coordinates x−β ’s is homogeneous. Example 13.3.1.8 Consider G2,4 . Then ⎧⎛ 1 ⎪ ⎪ ⎨⎜ 0 − ⎜ U eid = ⎝ x31 ⎪ ⎪ ⎩ x41

⎞ 0 1 ⎟ ⎟, x32 ⎠ x42

⎫ ⎪ ⎪ ⎬

xij ∈ k . ⎪ ⎪ ⎭

On U − eid , we have p12 = 1, p13 = x32 , p14 = x42 , p23 = x31 , p24 = x41 , p34 = x31 x42 − x41 x32 . Thus a Plücker coordinate is homogeneous in the local coordinates xij , d + 1 ≤ i ≤ n, 1 ≤ j ≤ d. This phenomenon does not hold for G/P , if P is not maximal as seen in the following example. Example 13.3.1.9 Consider SL(4)/B. We have ⎧⎛ ⎞ 1 0 0 0 ⎪ ⎪ ⎨⎜ x21 1 0 0⎟ ⎟, U − eid = ⎜ ⎝ x31 x32 1 0⎠ ⎪ ⎪ ⎩ x41 x42 x43 1

⎫ ⎪ ⎪ ⎬

xij ∈ k . ⎪ ⎪ ⎭

Then on U − eid , p23 = −x31 + x32 x21 is non-homogeneous! II. Returning to Gd,n , let now τ be any other element in W P, say  τ = (a1 , . . . , an ). Id (with notations Then Uτ− eτ consists of {Nd,n }, where Nd,n is obtained from X n×d as above) by permuting the rows by τ −1 . (Note that Uτ− eτ = τ U − eid .) Example 13.3.1.10 Consider G2,4 , and let τ = (2314). Then τ −1 = (3124), and ⎧⎛ ⎫ ⎞ x31 x32 ⎪ ⎪ ⎪ ⎪ ⎨⎜ ⎬ ⎟ 1 0 − ⎜ ⎟ . , x ∈ k Uτ eτ = ⎝ ij 0 1 ⎠ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ x41 x42 We have on Uτ− eτ , p12 = −x32 , p13 = x31 , p14 = x31 x42 − x41 x32 , p23 = 1, p24 = x42 , p34 = −x41 . As in the case τ = id, we find that for θ ∈ W P , pθ |Uτ− eτ is homogeneous in local coordinates. In fact we have

13.3 Singular loci of Schubert varieties in the Grassmannian

199

Proposition 13.3.1.11 Let θ ∈ W P . We have a natural isomorphism K[x−β , β ∈ R + \ RP+ ] ∼ = k[x−τ (β) , β ∈ R + \ RP+ ], given by pθ → pτ θ (here we have denoted pθ |U − id , pτ θ |Uτ− eτ by just pθ , pτ θ respectively). As a consequence, we have Corollary 13.3.1.12 Let θ ∈ W P . Then the polynomial expression for pθ |Uτ− eτ in the local coordinates {x−τ (β) , β ∈ R + \ RP+ } is homogeneous. The integer dτ (θ ) Let θ ∈ W P . Let fτ (θ ) be the polynomial expressing pθ |Uτ− eτ in terms of the local coordinates {x−β , β ∈ τ (R + \ RP+ )}. We define dτ (θ ) (also denoted by degτ (θ )) by dτ (θ ) := deg fτ (θ ) (note that fτ (θ ) is homogeneous (cf. Corollary 13.3.1.12)). In fact, we have an explicit expression for dτ (θ ), as follows: Proposition 13.3.1.13 Let θ ∈ W P . Let τ = (a1 , . . . , an ), θ = (b1 , . . . , bn ), τ, θ being considered as permutations. Let r = #{a1 , . . . , ad } ∩ {b1 , . . . , bd }. Then dτ (θ ) = d − r. Proof. Let s = d − r, and {c1 , . . . , cs } = {a1 , . . . , ad } \ {a1 , . . . , ad } ∩ {b1 , . . . , bd } {f1 , . . . , fs } = {b1 , . . . , bd } \ {a1 , . . . , ad } ∩ {b1 , . . . , bd }. For 1 ≤ t ≤ s, let βt = τ (i − j )(= ai − aj ), where i and j are given by ai = ct , aj = ft (note that i ≤ d < j , and hence βt ∈ τ (R + \ RP+ )). For σ = (σ1 , . . . , σn ) ∈ W P , let qσ = eσ1 ∧ . . . ∧ eσd , the extremal weight vector in Vk (ωd ) of weight σ (ωd ). For β ∈ R, say β = i − j , we have  ±qsβ τ , if j ∈ {a1 , . . . , ad }, i ∈ {a1 , . . . , ad } X β qτ = 0, otherwise (note that in the case when j ∈ {a1 , . . . , ad } and i ∈ {a1 , . . . , ad }, (sβ τ )(d) is the dtuple obtained from (a1 , . . . , ad ) by replacing j by i). Setting D  = X−β1 . . . X−βs , we have D  qτ = ±qθ . This implies X−β1 . . . X−βs qτ , pθ  = ±1, where ,  is the canonical g-invariant pairing on ∧d V × (∧d V )∗ , V = K n (in view of the fact that the bases {pθ , θ ∈ W P }, {qθ , θ ∈ W P } are dual to each other).

200

13 Applications of standard monomial theory

Now using the g-invariance of , , we obtain qτ , X−βs . . . X−β1 pθ  = ±1. Thus we obtain X−βs . . . X−β1 pθ = ±pτ , and hence the monomial x−βs . . . x−β1 appears in fτ (θ ) (the polynomial expression for pθ |Uτ− eτ in terms of x−β ’s, β ∈ τ (R + \ RP+ )). The required result now follows. Theorem 13.3.1.14 (cf. [73]) A recursive formula for mτ (w): With notations as above, we have  mτ (w  ), mτ (w) degτ w = where on the right we sum over all divisors X(w  ) in X(w) such that e(τ ) ∈ X(w  ). For other formulas for the multiplicity at a singular point, we refer the reader to [4].

13.4 Results for Schubert varieties in a minuscule G/P Let G be semisimple and P a maximal parabolic subgroup of minuscule type, i.e., the associated fundamental weight ω satisfies (ω, β ∗ ) ≤ 1, for all β ∈ R + ; we refer to such a G/P as a minuscule G/P . For example, any maximal parabolic in SLn (K) is minuscule, the corresponding G/P being a Grassmannian. Other examples of maximal parabolics which are minuscule are G = Sp2n (K), P = P1 , G = SO2n (K), P = Pn−1 , Pn , G = SO2n+1 (K), P = Pn (note that for G = Sp2n (K), the corresponding G/P is a projective space, and for G = SO2n (K), G = SO2n+1 (K), the corresponding G/P ’s are orthogonal Grassmannians consisting of maximal totally isotropic subspaces in K N , N = 2n, 2n + 1 equipped with a non-degenerate symmetric bilinear form). In addition, for G = E6 , P1 , P6 are minuscule maximal parabolic subgroups, while for G = E7 , P1 is the only minuscule maximal parabolic subgroup. The above list in fact exhausts all minuscule maximal parabolic subgroups for all simple G (note that there are no minuscule maximal parabolic subgroups for G of type E8 , F4 , G2 ). See the table in Figure A.1 in the Appendix. For θ ∈ W , and P = Pd (the maximal parabolic subgroup with S \ {αd } as the associated set of simple roots), we shall denote by θ (d) , the element in W Pd representing θ WPd . For the rest of this section, we shall suppose that P is minuscule. The geometry of a minuscule G/P is very similar to that of the Grassmannian; for example, similar to the Grassmannian case, for a minuscule G/P , we have that the extremal weight vectors (namely, the Weyl group translates of a highest weight vector) in H 0 (G/P , L(ω)) form a K-basis for H 0 (G/P , L(ω)). We have results for Sing X (X being a Schubert variety in G/P ) similar to those for a Schubert variety in the Grassmannian. Below we describe briefly these results. For details, we refer the readers to [73]. Let P = Pd and τ, w ∈ W P , w ≥ τ . Set RP (w, τ ) = {β ∈ τ (R + \ RP+ ) | w ≥ (sβ τ )(d) }. Theorem 13.4.0.1 T (w, τ ), the tangent space to XP (w) at eτ is spanned by {X−β , β ∈ RP (w, τ )}.

13.4 Results for Schubert varieties in a minuscule G/P

201

The proof is similar to that of Theorem 13.2.1.3. Let Uτ− be the unipotent subgroup of G generated by the root subgroups U−β , β ∈ τ (R + \ RP+ ). We have U−β ) Ga , Uτ− ) Πβ∈τ (R + \ R + ) U−β . P

We shall denote the coordinate system on Uτ− eτ induced by the above identification by {x−β , β ∈ τ (R + \ RP+ )}. We shall denote by Aτ the polynomial algebra K[x−β , β ∈ τ (R + \ RP+ )]. Let Aτ,w = Aτ /IP (w), where IP (w) is the ideal of elements of Aτ that vanish on XP (w) ∩ Uτ− eτ . Then Aτ,w is generated as an algebra by {x−β , β ∈ RP (w, τ )} (in view of Theorem 13.4.0.1). 13.4.1 Homogeneity of IP (w) Let ω be the fundamental weight associated to P . We shall denote the extremal weight vectors in H 0 (G/P , Lω ) by {pτ , τ ∈ W P }. Recall (cf. Chapter 8, [111]) that {pτ , τ ∈ W P } is a basis for H 0 (G/P , Lω ). Given y ∈ g, we identify y with the corresponding right invariant vector field Dy on G. Considering H 0 (G/P , Lω ) as a g-module, we have Dy f = yf, f ∈ H 0 (G/P , Lω ). Further, the evaluations of ∂x∂f−β and X−β f at eτ coincide for β ∈ τ (R + −RP+ ) (recall that for a root γ , Xγ denotes the element in the Chevalley basis of g associated to γ ). Take an indexing of τ (R + − RP+ ). Given f ∈ H 0 (G/P , Lω ), let Df = {D = X−βs · · · X−β1 | Df = aD pτ , aD ∈ k ∗ } (here, X−βs · · · X−β1 is considered as an element of U (g), the universal enveloping algebra of g). If Df = ∅, then f is a weight vector for the action of T , and in Aτ , f gets identified with D∈Df aD xD , where if D = X−βs · · · X−β1 , then xD is the monomial x−βs · · · x−β1 . Lemma 13.4.1.1 (cf. [73]) Let τ, θ ∈ W P . Suppose D1 = X−βs · · · X−β1 , D2 = X−βt · · · X−β1 are such that Di pθ = aDi pτ , aDi ∈ k ∗ , i = 1, 2. Then s = t, i.e., ord D1 = ord D2 . Hence we obtain that for f = pθ , if Df = ∅, then for D1 , D2 ∈ Df , ord D1 = ord D2 ; thus in Aτ , pθ gets identified with a homogeneous polynomial. Now IP (X(w)), the ideal of XP (w) in G/P is generated by {pθ | w ≥ θ } (cf. [111]). Hence we obtain that IP (w) (= the ideal of elements of Aτ that vanish on XP (w) ∩ Uτ− eτ ) is homogeneous. Thus we get gr(Aτ,w , Mτ,w ) = Aτ,w where Mτ,w is the maximal ideal in Aτ,w that corresponds to the point eτ .

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13 Applications of standard monomial theory

13.4.2 A basis for (Mτ,w )r /(Mτ,w )r+1 Let us fix τ, w ∈ W P , w ≥ τ . Let θ ∈ W P , θ ≤ w be such that θ and τ are comparable i.e. either τ ≥ θ or θ ≥ τ . Lemma 13.4.2.1 (cf. [73]) Let w, τ, θ be as above. Then there exists a D = X−βs · · · X−β1 , βi ∈ RP (w, τ ), 1 ≤ i ≤ s such that Dpθ = aD pτ , aD ∈ k ∗ . Set NP (θ ) = {D = X−βs · · · X−β1 , βi ∈ RP (w, τ ), 1 ≤ i ≤ s | Dpθ = aD pτ , aD ∈ k ∗ }. Note that NP (θ ) = ∅ (in view of the above Lemma). Definition 13.4.2.2 With notations as above, define dθ := ord D, where D ∈ NP (θ ) (note that the right hand side is independent of D (cf. Lemma 13.4.1.1), so the definition makes sense). Remark 13.4.2.3. If θ = w, then we refer to dw as the degree of XP (w) at eτ and denote it by degτ w. Similar to Theorem 13.3.1.14, we have Theorem 13.4.2.4 (cf. [73]) Denoting multeτ X by mτ (w), we have  mτ (w) degτ w = mτ (w  ) where the sum on the right hand side runs over all Schubert divisors XP (w  ) in XP (w) such that eτ ∈ XP (w  ).

13.5 Applications to other varieties In this section, we introduce two classes of affine varieties - ladder determinantal varieties (cf. §13.5.1) and quiver varieties (cf. §13.5.3 ) - and we conclude (cf. [34], [60] ) that these varieties are normal and Cohen-Macaulay by identifying them with YQ (w) for suitable Schubert varieties XQ (w) in suitable SL(n)/Q (note that YQ (w) is normal and Cohen-Macaulay, since XQ (w) is). 13.5.1 Ladder determinantal varieties Let X = (xba ), 1 ≤ b ≤ m, 1 ≤ a ≤ n be a m × n matrix of indeterminates. Given 1 ≤ b1 < · · · < bh < m, 1 < a1 < · · · < ah ≤ n, we consider the subset of X, defined by L = {xba | there exists 1 ≤ i ≤ h such that bi ≤ b ≤ m, 1 ≤ a ≤ ai }. We call L an one-sided ladder in X, defined by the outside corners ωi = xbi ai , 1 ≤ i ≤ h. For simplicity of notation, we identify the variable xba with just (b, a).

13.5 Applications to other varieties

203

Let s = (s1 , s2 . . . , sl ) ∈ Zl+ , t = (t1 , t2 . . . , tl ) ∈ Zl+ such that b1 = s1 < s2 < · · · < sl ≤ m, t1 ≥ t2 ≥ · · · ≥ tl , 1 ≤ ti ≤ min{m − si + 1, ai ∗ } for 1 ≤ i ≤ l, and

(L1)

si − si−1 > ti−1 − ti for 1 < i ≤ l, where for 1 ≤ i ≤ l, we let i ∗ be the largest integer such that bi ∗ ≤ si . For 1 ≤ i ≤ l, let Li = {xba ∈ L | si ≤ b ≤ m}. Let K[L] denote the polynomial ring K[xba | xba ∈ L], and let A(L) = A|L| be the associated affine space. Let Is,t (L) be the ideal in k[L] generated by all the ti -minors contained in Li , 1 ≤ i ≤ l, and Ds,t (L) ⊂ A(L) the variety defined by the ideal Is,t (L). We call Ds,t (L) a ladder determinantal variety (associated to an one-sided ladder). Let Ω = {ω1 , . . . , ωh }. For each 1 < j ≤ l, let Ωj = {ωi | 1 ≤ i ≤ h such that sj −1 < bi < sj and sj − bi ≤ tj −1 − tj }. Let

  l Ωj {(sj , aj ∗ )}. Ω = Ω \ j =2

Ωj =∅

L

Let be the one-sided ladder in X defined by the set of outside corners Ω  . Then it is easily seen that Ds,t (L) ) Ds,t (L ) × Ad , where d = |L| − |L |. Let ωk = (bk , ak ) ∈ Ω  , for some k, 1 ≤ k ≤ h , where h = |Ω  |. If bk ∈ {s1 , . . . , sl }, then bk = bi for some i, 1 ≤ i ≤ h, and we define sj − = bi , tj − = tj −1 , sj + = sj , tj + = tj , where j is the unique integer such that sj < bi < sj +1 . Let s (resp. t ) be the sequence obtained from s (resp. t) by replacing sj (resp. tj ) with sj − and sj + (resp. tj − and tj + ) for all k such that bk ∈ {s1 , . . . , sl }, j being the unique integer such that sj −1 < bi < sj , and i being given by bk = bi . Let l  = |s |. Then s and t satisfy (L1), and in addition we have {b1 , . . . , bh  } ⊂ {s1 , . . . , sl }. It is easily seen that Ds,t (L ) = Ds ,t (L ), and hence Ds,t (L) ) Ds ,t (L ) × Ad . Therefore it is enough to study Ds,t (L) with s, t ∈ Zl+ such that {s1 , . . . , sl } ⊃ {b1 , . . . , bh }.

(L2)

Without loss of generality, we can also assume that tl ≥ 2, and ti−1 > ti if si ∈ {b1 , . . . , bh }, 1 < i ≤ l. For 1 ≤ i ≤ l, let L(i) = {xba | si ≤ b ≤ m, 1 ≤ a ≤ ai ∗ }.

(L3)

204

13 Applications of standard monomial theory

Note that the ideal Is,t (L) is generated by the ti -minors of X contained in L(i), 1 ≤ i ≤ l. The ladder determinantal varieties (associated to one-sided ladders) get related to Schubert varieties(cf. [34]). We describe below the main results of [34]. The Varieties Z and XQ (w) Let G = SL(n), Q = Pa1 ∩· · ·∩Pah . Let O− be the opposite big cell in G/Q. Let H be the one-sided ladder defined by the outside corners (ai + 1, ai ), 1 ≤ i ≤ h. Let s, t ∈ Zl+ satisfy (L1), (L2), (L3) above. For each 1 ≤ i ≤ l, let L(i) = {xba | sj ≤ b ≤ n, 1 ≤ a ≤ ai ∗ }. Let Z be the variety in A(H ) ) O− defined by the vanishing of the ti -minors in L(i), 1 ≤ i ≤ l. Note that Z ) Ds,t (L) × A(H \ L) ) Ds,t (L) × Ar , where r = dim SL(n)/Q − |L|. min , such that the variety Z identifies We shall now define an element w ∈ WQ min with the opposite cell in the Schubert variety XQ (w) in G/Q. We define w ∈ WQ by specifying w (ai ) ∈ W ai 1 ≤ i ≤ h, where πi (X(w)) = X(w (ai ) ) under the projection πi : G/Q → G/Pai . Define w (ai ) , 1 ≤ i ≤ h, inductively, as the (unique) maximal element in W ai such that (1) w (ai ) (ai − tj + 1) = sj − 1 for all j ∈ {1, . . . , l} such that sj ≥ bi , and tj = tj −1 if j > 1. (2) if i > 1, then w (ai−1 ) ⊂ w (ai ) . Note that w (ai ) , 1 ≤ i ≤ h, is well defined in W i , and w is well defined as an min . element in WQ Theorem 13.5.1.1 (cf. [34]) The variety Z (= Ds,t (L) × Ar ) identifies with the opposite cell in XQ (w), i.e. Z = XQ (w) ∩ O− (scheme theoretically). As a consequence of the above Theorem, we obtain (cf. [34]) Theorem 13.5.1.2 The variety Ds,t (L) is irreducible, normal, Cohen-Macaulay. 13.5.2 The varieties Vi , 1 ≤ i ≤ l Let Vi , 1 ≤ i ≤ l be the subvariety of Ds,t (L) defined by the vanishing of all (ti −1)minors in L(i), where L(i) is as in §13.5.1. In [34] the singular locus of Ds,t (L) has also been determined, as described below. Theorem 13.5.2.1 Sing Ds,t (L) = ∪li=1 Vi . Remark 13.5.2.2. Ladder determinantal varieties were first introduced by Abhyankar (cf. [2]). Remark 13.5.2.3. A similar identification as in Theorem 13.5.1.1 for the case t1 = · · · = tl has also been obtained by Mulay (cf. [85]).

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205

13.5.3 Quiver varieties Fulton [31] and Buch-Fulton [11] have recently given a theory of “universal degeneracy loci”, characteristic classes associated to maps among vector bundles, in which the role of Schubert varieties is taken by certain degeneracy schemes. The underlying varieties of these schemes arise in the theory of quivers: they are the closures of orbits in the space of representations of the equi-oriented quiver Ah . Many other classical varieties also appear as quiver varieties, such as determinantal varieties and the variety of complexes (see [23], [40], [90]). In [60], the quiver varieties (corresponding to the equi-oriented type A quiver) are shown to be normal and Cohen-Macaulay (in arbitrary characteristic) by identifying them with YQ (w) for suitable Schubert varieties XQ (w) in suitable SLn (K)/Q. Fix an h-tuple of non-negative integers n = (n1 , . . . , nh ) and a list of vector spaces K-vector spaces V1 , . . . , Vh with respective dimensions n1 , . . . , nh . Define Z, the variety of quiver representations (of dimension n, of the equi-oriented quiver of type Ah ) to be the affine space of all (h−1)-tuples of linear maps (f1 , . . . , fh−1 ) : f1

f2

fh−2

fh−1

V1 → V2 → · · · → Vh−1 → Vh . If we endow each Vi with a basis, we get Vi ∼ = K ni and Z∼ = Mn2 ,n1 ×· · · × Mnh ,nh−1 , where Ml,m denotes the affine space of matrices over K with l rows and m columns. The group Gn := GL(n1 ) × · · · × GL(nh ) acts on Z by −1 (g1 , g2 , · · · , gh ) · (f1 , f2 , · · · , fh−1 ) = (g2 f1 g1−1 , g3 f2 g2−1 , · · · , gh fh−1 gh−1 )

Now, let r = (rij )1≤i≤j ≤h be an array of non-negative integers with rii = ni , and define rij = 0 for any indices other than 1 ≤ i ≤ j ≤ h. Define the set Z ◦ (r) = {(f1 , · · · , fh−1 ) ∈ Z | ∀ i < j, rank(fj−1 · · · fi : Vi → Vj ) = rij }. (This set might be empty for a bad choice of r.) Proposition 13.5.3.1 (cf. [24]) The Gn -orbits of Z are exactly the sets Z ◦ (r) for r = (rij ) with rij − ri,j+1 − ri−1,j + ri−1,j+1 ≥ 0,

∀ 1 ≤ i < j ≤ h.

Definition 13.5.3.2 We define the quiver variety as the algebraic set Z(r) = {(f1 , · · · , fh−1 ) ∈ Z | ∀i, j, rank(fj−1 · · · fi : Vi → Vj ) ≤ rij }. Remark 13.5.3.3. The variety Z(r) is simply the Zariski closure of Z ◦ (r) (cf. [1], [60]).

206

13 Applications of standard monomial theory

Given n = (n1 , · · · , nh ), for 1 ≤ i ≤ h, let ai = n 1 + n 2 + · · · + n i ,

a0 = 0,

and

n = n1 + · · · + nh .

For positive integers i ≤ j , we shall frequently use the notations [i, j ] = {i, i + 1, . . . , j },

[i] = [1, i],

[0] = {}.

Let K n ∼ = V1 ⊕ · · · ⊕ Vh have basis e1 , . . . , en compatible with the Vi . Consider its general linear group GLn (K), the subgroup B of upper-triangular matrices, and the parabolic subgroup Q of block upper-triangular matrices Q = {(aij ) ∈ GL(n) | aij = 0 whenever j ≤ ak < i for some k}. In this section, we look at G/Q as the space of partial flags as follows: a partial flag of type (a1 < a2 < · · · < ah = n) (or simply a flag) is a sequence of subspaces U. = (U1 ⊂ U2 ⊂ · · · ⊂ Uh = K n ) with dim Ui = ai . Let Ei = V1 ⊕ · · · ⊕ Vi = e1 , . . . , eai , and Ei = Vi+1 ⊕ · · · ⊕ Vh = eai +1 , . . . , en , so that Ei ⊕ Ei = K n . The flag variety Fl is the set of all flags U. as above. Fl has a transitive GLn (K)action induced from K n , and Q = StabGLn (K) (E.), so we have the identification Fl ∼ = GLn (K)/Q, g·E. ↔ gQ. The Schubert varieties are the closures of B-orbits on Fl. Such orbits are usually indexed by certain permutations of [n], but we prefer to use flags of subsets of [n], of the form τ = (τ1 ⊂ τ2 ⊂ · · · ⊂ τh = [n]),

#τi = ai .

A permutation w : [n] → [n] corresponds to the subset-flag with τi = w[ai ] = {w(1), w(2), . . . , w(ai )}. This gives a one-to-one correspondence between cosets of the symmetric group W = Sn modulo the Young subgroup Wn = Sn1 × · · · × Snh , and subset-flags. Given such τ , let Ei (τ ) = ej | j ∈ τi  be a coordinate subspace of K n , and E.(τ ) = (E1 (τ ) ⊂ E2 (τ ) ⊂ · · · ) ∈ Fl. Then we may define the Schubert cell ◦ (τ ) = B · E(τ ) XQ  = (U1 ⊂ U2 ⊂ · · · ) ∈ Fl

   dim Ui ∩ K j = # τi ∩ [j ]   1 ≤ i ≤ h, 1 ≤ j ≤ n

and the Schubert variety ◦ (τ ) XQ (τ ) = XQ  = (U1 ⊂ U2 ⊂ · · · ) ∈ Fl

where K j = e1 , . . . , ej  ⊂ K n .

   dim Ui ∩ K j ≥ # τi ∩ [j ]   1 ≤ i ≤ h, 1 ≤ j ≤ n

13.5 Applications to other varieties

207

Under the identification of G/Q with Fl, the opposite cell O− in G/Q gets identified with the set of flags in general position with respect to the spaces E1 ⊃ · · · ⊃  : Eh−1 O− = {(U1 ⊂ U2 ⊂ · · · ) ∈ Fl | Ui ∩ Ei = 0}. Let YQ (τ ) = XQ (τ ) ∩ O− , the opposite cell of X(τ ). We define a special subset-flag τ max = (τ1max ⊂ · · · ⊂ τhmax = [n]) corresponding to n = (n1 , . . . , nh ). We want each τimax to contain numbers as large as possible given the constraints [aj−1 ] ⊂ τjmax for all j . Namely, we define τimax recursively by τhmax = [n];

max τimax = [ai−1 ] ∪ {largest ni elements of τi+1 }.

Furthermore, given r = (rij )1≤i≤j ≤h indexing a quiver variety, define a subset-flag τ r to contain numbers as large as possible given the constraints  ai − ri,j +1 for i ≤ j # τir ∩ [aj ] = for i > j aj Namely, τir = { 1 . . . ai−1 . . . . . . . ai . . . . . . . ai+1 . . . . . . . ai+2 . . . . . . . . . . n }                ai−1

rii −ri,i+1 ri,i+1 −ri,i+2 ri,i+2 −ri,i+3

ri,h

where we use the visual notation ·· · · · · a = [a −b+1, a]. b

Recall that aj = aj −1 +nj and 0 ≤ rij −ri,j +1 ≤ nj , so that each τir is an increasing r . Thus, τ r list of integers. Also rij − ri,j +1 ≤ ri+1,j − ri+1,j +1 , so that τir ⊂ τi+1 are indeed subset-flags. Examples We give below four examples. Example 1 A small generic case. Let h = 4, n = (2, 3, 2, 2), 2200 r= 311 22 2 where rij are written in the usual matrix positions. Then we get (a1 , a2 , a3 , a4 ) = (2, 5, 7, 9), n = 9, and τ max = (89 ⊂ 12589 ⊂ 1234589 ⊂ [9]),

τ r = (45 ⊂ 12459 ⊂ 1234589 ⊂ [9]),

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13 Applications of standard monomial theory

which correspond to the cosets in W/Wn w max = 89|125|34|67,

w r = 45|129|38|67.

(The minimal-length representatives of these cosets are the permutations as written; the other elements are obtained by permuting numbers within each block.) The partial flag variety is Fl = {U1 ⊂ U2 ⊂ U3 ⊂ K 9 | dim Ui = ai }, and the Schubert varieties are:     2    K ⊂ U2  U ⊂ k 5 ⊂ U3 , K 2 ⊂ U2 , XQ (τ r ) = U.  1 . XQ (τ max ) = U.  5 K ⊂ U3 dim U2 ∩ K 5 ≥ 4 The opposite cells YQ (τ ) are defined by the extra conditions Ui ∩ Ei = 0. Example 2. Fulton’s universal degeneracy schemes (cf. [31]). Given m > 0, let Z be the affine space associated to the quiver data h = 2m, n = (1, 2, . . . , m, m, . . . , 2, 1). For each w ∈ Sm+1 , Fulton defines a “degeneracy scheme” Ωw = Z(r) as follows. Denote i = 2m+1−i, and define r = r(w) = (rij ) by: rij = rj i = i rij = # [i] ∩ w[j ] for 1 ≤ i, j ≤ m. The associated Schubert varieties XQ (τ r ) are given by τ r = (τ1r ⊂ /=w /1 | · · · |/ w1 ∈ W/Wn · · · ⊂ τ r ) or by cosets w 1

τir = [ai−1 ] ∪ {aw−1 (1) , aw−1 (2) , . . . , aw−1 (i) }, τ r = [ai −1] ∪ {a1 , a2 , . . . , am } i

w /i = [ai−2 +1, ai−1 ] ∪ {aw−1 (i) } w /m = [am−1+1, am−1] ∪ {aw−1 (m+1) } w /j = [aj−2+1, aj−1 ]

for 1 ≤ i ≤ m, 1 ≤ j ≤ m − 1. Furthermore τ max = τ r(w) and w /max = w /r(w) for w = e ∈ Sm+1 , the identity permutation. Example 3. The variety of complexes. For a given h and n, the variety of complexes is defined as the union C = ∪r Z(r) over all r = (rij ) with ri,i+2 = 0 for all i. The subvarieties Z(r) correspond to the multiplicity matrices m = (mij ) with mij = 0 for all i + 2 ≤ j , and mii + mi−1,i + mi,i+1 = ni for all i. In [90], Musili-Seshadri have shown that each component of C, is isomorphic to the opposite cell in a Schubert variety. Example 4. The Classical Determinantal Variety. The classical determinantal variety of k × l matrices of rank ≤ t is D = Z(r) for   

r=

l m l −m m and m = 0 k 0 k−m

τ max = ([k + 1, k + l] ⊂ [n]),

where m = min(t + 1, k, l). Also n = k + l,

τ r = ([m + 1, l] ∪ [k + l − m + 1, k + l] ⊂ [n])

X(τ max ) = Gl,n , X(τ r ) ∼ = {U ∈ Gl,n | U ∩ K l = l − m}, D = Z(r) ∼ = Y (τ r ) = {U ∈ Gl,n | U ∩ K l = l − m, U ∩ E  = 0}, where E  = el+1 , el+2 , . . . , en .

13.6 Variety of complexes

209

Main theorem Denote a generic element of the quiver space Z = M(n2 , n1 ) × · · · × M(nh , nh−1 ) by (A1 , . . . , Ah−1 ), so that the coordinate ring of Z is the polynomial ring in the entries of all the matrices Ai . Let r = (rij ) index the quiver variety Z(r) = {(A1 , . . . , Ah−1 ) | rank Aj −1 · · · Ai ≤ rij }. Let J (r) ⊂ K[Z] be the ideal generated by the determinantal conditions implied by the definition of Z(r):  8 9  j > i, λ ⊂ [nj ], μ ⊂ [ni ] J (r) = det(Aj −1 Aj −2 · · · Ai )λ×μ  . #λ = #μ = rij + 1 Clearly J (r) defines Z(r) set-theoretically. Theorem 13.5.3.4 (cf. [60]) J (r) is a prime ideal and is the vanishing ideal of Z(r) ⊂ Z. There are isomorphisms of reduced schemes Z(r) = Spec(k[Z] / J (r)) ∼ = Spec(k[O− ] / I(τ r )) = YQ (τ r ). That is, the quiver scheme Z(r) defined by J (r) is isomorphic to the (reduced) variety YQ (τ r ), the opposite cell of a Schubert variety. In proving the above theorem again, one uses the standard monomial theory for Schubert varieties.

13.6 Variety of complexes In this section we consider a special case of the quiver varieties called the variety of complexes. Let V be a variety of complexes, namely, the subvariety of Z consisting of {(A1 , A2 , . . . , Ah ) | Ai Ai−1 = 0, 2 ≤ i ≤ h}. We have from the previous section V = Z(r), r

where the union is taken over all r = (rij ) such that rij = 0 for all j = i, i + 1. Let us denote ki = ri,i+1 , k = (k1 , k2 , . . . , kh ), and V (k) = Z(r). Let n = (n1 , n2 , . . . , nh+1 ), and An = {k = (k1 , k2 , . . . , kh ) | ki ≤ min {ni , ni+1 }, 1 ≤ i ≤ h ki−1 + ki ≤ ni , 2 ≤ i ≤ h}. Then V =

k∈An

Let us denote τ r by τk .

V (k).

210

13 Applications of standard monomial theory

13.6.1 A partial order on {(k1 , k2 , . . . , kh )} The partial order on the set {Z(k), k ∈ An } of Gn -orbit closures in V given by inclusion induces a partial order ≥ on An , namely, for k = (k1 , k2 , . . . , kh ), k = (k1 , k2 , . . . , kh ) in An , k ≥ k ⇐⇒ kt ≥ kt , 1 ≤ t ≤ h. Theorem 13.6.1.1 [60, 90] (i) V (k) ∼ = YQ (τk ). (ii) The irreducible components of V are given by {V (k) | k is maximal for the above partial order}. Proposition 13.6.1.2 (cf. [60, 90]) Let k ∈ An . Then dimV (k) = 1≤i≤h+1 (ni − ki ) (ki−1 + ki ), where k0 = kh+1 = 0. Let n = (n1 , . . . , nh+1 ). Fix (k1 , k2 , . . . , kh ) so that ki ≤ min{ni , ni+1 }, 1 ≤ i ≤ h, ki−1 + ki ≤ ni , 2 ≤ i ≤ h, where k0 = kh+1 = 0. Let Vk = V (k1 , k2 , . . . , kh ) := {(f1 , . . . , fh ) | rank(fi ) ≤ ki }, fi ◦ φi−1 = 0, 2 ≤ i ≤ h}, f1 , . . . , fh being as above. For 1 ≤ i ≤ h, let Vi = V (k1 , . . . , ki−1 , ki − 1, ki+1 , . . . , kh ), and for 1 ≤ j ≤ h − 1, let Vj,j +1 = V (k1 , . . . , kj −1 , kj − 1, kj +1 − 1, kj +2 , . . . , kh ). Theorem 13.6.1.3 (cf. [32]) The irreducible components of SingVk are Vi , i ∈ Ω, and Vj,j +1 , j ∈ Ω, where Ω is the set of all 1 ≤ i ≤ h such that ki−1 + ki < ni and ki + ki+1 < ni+1 .

13.7 Degenerations of Schubert varieties to toric varieties In this section, we discuss the toric degenerations of Schubert varieties in the Grassmannian (cf. [33]). As mentioned in the beginning of this chapter, though there are recent results (cf. [12, 17]) on toric degenerations of Schubert varieties in any G/Q, G semi simple, in this section we present the results on the toric degenerations of Schubert varieties in the Grassmannian because of the simplicity of the proof in this case! These results are proved using Standard monomial bases for Schubert varieties in SL(n)/B. 13.7.1 Generalities on distributive lattices Definition 13.7.1.1 A lattice is a partially ordered set (L, ≤) such that, for every pair of elements x, y ∈ L, there exist elements x ∨ y and x ∧ y, called the join, respectively the meet of x and y, defined by:

13.7 Degenerations of Schubert varieties to toric varieties

211

x ∨ y ≥ x, x ∨ y ≥ y, and if z ≥ x and z ≥ y, then z ≥ x ∨ y, x ∧ y ≤ x, x ∧ y ≤ y, and if z ≤ x and z ≤ y, then z ≤ x ∧ y. It is easy to check that the operations ∨ and ∧ are commutative and associative. Definition 13.7.1.2 An element z ∈ L is called the zero of L, denoted by 0, if z ≤ x for all x in L. An element z ∈ L is called the one of L, denoted by 1, if z ≥ x for all x in L. Definition 13.7.1.3 Given a lattice L, a subset L ⊂ L is called a sublattice of L if x, y ∈ L implies x ∧ y ∈ L , x ∨ y ∈ L . Definition 13.7.1.4 Two lattices L1 and L2 are isomorphic if there exists a bijection ϕ : L1 → L2 such that, for all x, y ∈ L1 , ϕ(x ∨ y) = ϕ(x) ∨ ϕ(y) and ϕ(x ∧ y) = ϕ(x) ∧ ϕ(y). Definition 13.7.1.5 A lattice is called distributive if the following identities hold: x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z). Example 13.7.1.6 The lattice of all subsets of the set {1, 2, . . . , n} is denoted by B(n), and called the Boolean algebra of rank n. 13.7.2 An important example Given an integer n ≥ 1, C(n) will denote the chain {1 < · · · < n}, and for n1 , . . . , nd > 1, C(n1 , . . . , nd ) will denote the chain product lattice C(n1 ) × · · · × C(nd ) consisting of all d-tuples (i1 , . . . , id ), with 1 ≤ i1 ≤ n1 , . . . , 1 ≤ id ≤ nd . For (i1 , . . . , id ), (j1 , . . . , jd ) in C(n1 , . . . , nd ), we define (i1 , . . . , id ) ≤ (j1 , . . . , jd ) ⇐⇒ i1 ≤ j1 , . . . , id ≤ jd . We have (i1 , . . . , id ) ∨ (j1 , . . . , jd ) = (max{i1 , j1 }, . . . , max{id , jd }) (i1 , . . . , id ) ∧ (j1 , . . . , jd ) = (min{i1 , j1 }, . . . , min{id , jd }). C(n1 , . . . , nd ) is a finite distributive lattice, and its zero and one are (1, . . . , 1), (n1 , . . . , nd ) respectively. Note that there is a total order . on C(n1 , . . . , nd ) extending <, namely the lexicographic order, defined by (i1 , . . . , id ) . (j1 , . . . , jd ) if and only if there exists l < d such that i1 = j1 , . . . , il = jl , il+1 < jl+1 . Also note that two elements (i1 , . . . , id ) . (j1 , . . . , jd ) are non-comparable with respect to ≤ if and only if there exists 1 < h ≤ d such that ih > jh . Sometimes we denote the elements of C(n1 , n2 , . . . , nd ) by xi1 ...id , with 1 ≤ i1 ≤ n1 , . . . , 1 ≤ id ≤ nd . One has the following (see [3]):

212

13 Applications of standard monomial theory

Theorem 13.7.2.1 Any finite distributive lattice is isomorphic to a sublattice of a Boolean algebra of finite rank, and hence, in particular, to a sublattice of a finite chain product. 13.7.3 Generalities on toric varieties Definition 13.7.3.1 (cf. [30], [49]) An equivariant affine embedding of a torus T is an affine variety X containing T as an open subset and equipped with a T -action T × X → X extending the action T × T → T given by multiplication. If in addition X is normal, then X is called an affine toric variety. For our purpose, we also need the definition of a toric ideal (cf. [114]). Let T = (K ∗ )m be the m-dimensional torus. Let M be the character group (= Homalg.gp. (T , Gm )) of T . Then M can be identified with Zm . Let A = {a1 , . . . , an } be a subset of Zm . Consider the map πA : Zn+ → Zm ,

u = (u1 , . . . , un ) → u1 a1 + · · · + un an .

−1 ]. Let K[x] = K[x1 , . . . , xn ], K[t±1 ] = K[t1 , . . . , tm , t1−1 , . . . , tm The map πA induces a homomorphism of semigroup algebras ±1 π! A : K[x] → K[t ],

xi → tai .

Definition 13.7.3.2 The kernel of  πA is denoted by IA and called the toric ideal associated to A. A variety of the form V(IA ), the affine variety of the zeroes in K n of IA , is called an affine toric variety. Note that a toric ideal is prime. Remark 13.7.3.3. Consider the action of T on K n given by tei = tai ei (here, ei , 1 ≤ i ≤ n are the standard basis vectors of K n ). Then V(IA ) is simply the Zariski closure of the T -orbit through (1, 1, . . . , 1). In particular, V(IA ) is an equivariant affine embedding of T = (K ∗ )m . Remark 13.7.3.4. In the above definition, we do not require V(IA ) to be normal. Using [49], we have that V(IA ) is normal if and only if the semi subgroup S of M generated by A is saturated (here, S is said to be saturated if for a ∈ M, ra ∈ S /⇒ a ∈ S (where r is an integer > 1) (cf. [49])). Recall the following (see [114]). Proposition 13.7.3.5 The toric ideal IA is spanned as a K-vector space by the set of binomials (*) {xu − xv | u, v ∈ Zn+ with πA (u) = πA (v)}. (Here, a binomial is a polynomial with at most two terms.)

13.7 Degenerations of Schubert varieties to toric varieties

213

13.7.4 An example d n , m = d n . Let Let us fix the integers n1 , . . . , nd > 1, and let n = Πi=1 i i=1 i el1 , . . . , elnl be the unit vectors in Znl , for 1 ≤ l ≤ d. For 1 ≤ ξ1 ≤ n1 , . . . , 1 ≤ ξd ≤ nd , define aξ1 ...ξd = e1ξ1 + · · · + edξd ∈ Zn1 ⊕ · · · ⊕ Znd

and let An1 ,...,nd = {aξ1 ...ξd | 1 ≤ ξ1 ≤ n1 , . . . , 1 ≤ ξd ≤ nd }. The corresponding map n ·····nd

πA : Z+1

→ Zn1 +···+nd

is defined as follows: for 1 ≤ l ≤ d and 1 ≤ il ≤ nl fixed, the (n1 +· · ·+nl−1 +il )-th uξ1 ...ξl−1 ξl ξl+1 ...ξd , the sum being taken over the coordinate of πA (u) is given by elements (ξ1 , . . . , ξl−1 , ξl , ξl+1 , . . . , ξd ) of C(n1 , . . . , nd ) with ξl = il . We call this subset the l-th slice of C(n1 , . . . , nd ) defined by il , and denote it by {ξl = il }. The components (or entries) of an element u ∈ Zn1 ·····nd are indexed by the elements (i1 , . . . , id ) of C(n1 , . . . , nd ). If (j1 , . . . , jd ) ∈ {ξl = il }, sometimes we also say that uj1 ...jd itself belongs to the slice {ξl = il }. The map πA induces the map π! A : K[x11...1 , . . . , xξ1 ξ2 ...ξd , . . . , xn1 n2 ...nd ] → K[t11 , . . . , t1n1 , . . . , td1 , . . . , tdnd ] given by xξ1 ...ξd → t1ξ1 . . . tdξd , for 1 ≤ ξ1 ≤ n1 , . . . , 1 ≤ ξd ≤ nd . 13.7.5 The algebra associated to a distributive lattice Definition 13.7.5.1 Given a finite lattice L, the ideal associated to L, denoted by I (L), is the ideal of the polynomial ring K[L] generated by the set of binomials GL = {xy − (x ∧ y)(x ∨ y) | x, y ∈ L non-comparable}. By Theorem 13.7.2.1, a finite distributive lattice L may be identified with a sublattice of a finite chain product lattice. Hence it inherits a total order extending the given partial order. In turn, this total order induces the lexicographic order on the monomials in K[L]. The following theorems show that the ideal associated to a chain product lattice (resp. a finite distributive lattice) is toric. Theorem 13.7.5.2 We have I (C(n1 , . . . , nd )) = IAn1 ,...,nd . Theorem 13.7.5.3 Let L be a finite distributive lattice. Then the ideal I (L) is toric. For a proof of Theorems 13.7.5.2,13.7.5.3, see [33].

214

13 Applications of standard monomial theory

Remark 13.7.5.4. (1) In [39], it is shown that RL = K[L]/I (L) (L being a finite distributive lattice) is a normal domain; on the other hand, one knows (see [28] for example) that a prime binomial ideal is toric, in the sense of Definition 13.7.3.2 (here, by a binomial, we mean a polynomial with at most two terms). Thus the result that I (L) is toric (L being a finite distributive lattice) may also be concluded using [39] and [28]. (2) In fact, given a finite lattice L, I (L) is toric if and only if L is distributive. (3) We have included below a short geometric proof of the result that a prime binomial ideal is toric. 13.7.6 Varieties defined by binomials Let N ≥ 1, and let X be an affine variety in AN , not contained in any of the coordinate hyperplanes {xi = 0}. Further, let X be irreducible, and let its defining prime ideal I (X) be generated by l binomials a

b

a

b

x1 i1 . . . xNiN − λi x1 i1 . . . xNiN ,

1 ≤ i ≤ l.

(∗)

Consider the natural action of the torus TN = (k ∗ )N on AN , (t1 , . . . , tN ) · (a1 , . . . , aN ) = (t1 a1 , . . . , tN aN ). Let X(TN ) = Hom(TN , Gm ) be the character group of TN , and let i ∈ X(TN ) be the character 1 ≤ i ≤ N. i (t1 , . . . , tN ) = ti , For 1 ≤ i ≤ l, let

N  ϕi = (ait − bit )t . t=1

Set T =

∩li=1 ker ϕi ,

and

X◦

= {(x1 , . . . , xN ) ∈ X | xi = 0 for all 1 ≤ i ≤ N}.

Proposition 13.7.6.1 Let notations be as above. (1) There is a canonical action of T on X. (2) X ◦ is T -stable. Further, the action of T on X ◦ is simple and transitive. (3) T is a subtorus of TN , and X is an equivariant affine embedding of T . Proof. (1) We consider the (obvious) action of T on AN . Let (x1 , . . . , xN ) ∈ X, t = (t1 , . . . , tN ) ∈ T , and (y1 , . . . , yN ) = t · (x1 , . . . , xN ) = (t1 x1 , . . . , tN xN ). Using the fact that (x1 , . . . , xN ) satisfies (∗), we obtain a

a

a

a

a

a

b

b

b

b

b

b

y1 i1 . . . yNiN = t1 i1 . . . tNiN x1 i1 . . . xNiN = λi t1 i1 . . . tNiN x1 i1 . . . xNiN = λi y1 i1 . . . yNiN , for all 1 ≤ i ≤ l, i.e. (y1 , . . . , yN ) ∈ X. Hence t · (a1 , . . . , aN ) ∈ X for all t ∈ T . (2) Let x = (x1 , . . . , xN ) ∈ X ◦ , and t = (t1 , . . . , tN ) ∈ T . Then, clearly t · (x1 , . . . , xN ) ∈ X ◦ . Considering x as a point in AN , the isotropy subgroup in

13.7 Degenerations of Schubert varieties to toric varieties

215

TN at x is {id}. Hence the isotropy subgroup in T at x is also {id}. Thus the action of T on X ◦ is simple.  ) ∈ X ◦ . Set t = (t , . . . , t ), where t = x /x  . Let (x1 , . . . , xN ), (x1 , . . . , xN 1 N i i i  ). Hence the action of T on Then, clearly t ∈ T . Thus (x1 , . . . , xN ) = t · (x1 , . . . , xN X ◦ is simple and transitive. (3) Now, fixing a point x ∈ X ◦ , we obtain from (2) that the orbit map t → t · x is in fact an isomorphism of T onto X ◦ . Also, since X is not contained in any of the coordinate hyperplanes, the open set Xi = {(x1 , . . . , xN ) ∈ X | xi = 0} is nonempty for all 1 ≤ i ≤ N. The irreducibility of X implies that the sets Xi , 1 ≤ i ≤ N, are open dense in X, and hence their intersection X◦ =

N 

Xi = {(x1 , . . . , xN ) ∈ X | xi = 0 for any i}

i=1

is an open dense set in X, and thus X ◦ is irreducible. This implies that T is irreducible (and hence connected). Thus T is a subtorus of TN . The assertion that X is an equivariant affine embedding of T follows from (1) and (2). Remark 13.7.6.2. One can see that the ideal I (X) is a toric ideal in the sense of Definition 13.7.3.2; more precisely,  I (X) = IA , with m = dim T , n = N, A±1= ], {ρi , 1 ≤ i ≤ N}, where ρi = i T (here K[T ] is identified with K[t1±1 , . . . , tm and the character group X(T ) with Zm ). 13.7.7 Degenerations of Schubert varieties in the Grassmannian to toric varieties Let G = SL(n). Fix d, 1 ≤ d ≤ n − 1. Let Gd,n , Id,n etc., be as in Chapter 4. Proposition 13.7.7.1 (Id,n , ≥) is a distributive lattice. Proof. Let τ = (i1 , · · · , id ), φ = (j1 , · · · , jd ) be two non-comparable elements in Id,n . Set λ = (k1 , · · · , kd ), μ = (l1 , · · · , ld ) where kt = max {it , jt }, lt = min {it , jt }, 1 ≤ t ≤ d. Note that (kt−1 , kt ) = (it−1 , it ) or (it−1 , jt ) or (jt−1 , it ) or (jt−1 , jt ), from which it follows that kt−1 < kt . Similarly, lt−1 < lt . Thus λ, μ are well-defined elements of Id,n . Further from the definition of λ, μ, it is easily seen that λ (resp. μ) is the unique minimal (resp. maximal) element of Id,n which is greater (resp. less) than both τ and φ. Corollary 13.7.7.2 Let τ, φ, λ, μ ∈ Id,n be as in the proof of Proposition 13.7.7.1. Then λ = τ ∨ φ, μ = τ ∧ φ. Corollary 13.7.7.3 For τ ∈ Id,n , let Hτ = {i ∈ Id,n | i ≤ τ }. Then Hτ is a distributive lattice.

216

13 Applications of standard monomial theory

Proposition 13.7.7.4 Let τ, φ ∈ Id,n be non-comparable. Let  pτ p φ = cα,β pα pβ

(*)

be the expression for pτ pφ as a sum of standard monomials. Then pτ ∨φ pτ ∧φ occurs with coefficient 1 in (*). Proof. Let notations be as above. Then, in view of Chapter 5, Proposition 5.1.0.2 for any (α, β) on the right-hand side of (∗), we have α ≥ τ ∨ φ, β ≤ τ ∧ φ. Further, if α = τ ∨ φ, then β = τ ∧ φ (by weight considerations; note that for θ ∈ Id,n , say θ = (i1 , · · · , id ), weight of pθ = −(i1 + · · · + id )). Hence restricting (*) to X(τ ∨ φ), we obtain pτ pφ = cpτ ∨φ pτ ∧φ ,

on X(τ ∨ φ).

Since standard monomials basis is characteristic-free, we conclude that the coefficient c of pτ ∨φ pτ ∧φ on the right-hand side of (*) is ±1. In order to prove that c = 1, we compare the coefficients of the monomial m = xi1 1 . . . xid d xj1 1 . . . xjd d on both sides (note that a Plücker coordinate pa1 ,...,ad (being the d × d-minor of the generic n × n matrix (xij ) with row indices a1 , . . . , ad and column indices 1, . . . , d) is a polynomial in the xij ’s). Let pα pβ appearing on the right-hand side of (*) be such that α = τ ∨ φ. This implies α > τ ∨ φ, and β < τ ∧ φ. Let α = (α1 , . . . , αd ) and β = (β1 , . . . , βd ). We have (α1 , . . . , αd ) > (k1 , . . . , kd ) ((k1 , . . . , kd ) being as in the proof of Proposition 13.7.7.1). Let t be the smallest integer ≤ d such that αt > kt . This implies, in view of Corollary 13.7.7.2 (notations being as in that Corollary) αp = kp , βp = lp , p < t, kt ∈ {α1 , . . . , αd }, kt , lt ∈ {β1 , . . . , βd }. Hence in the expression for pα pβ as a polynomial in the xij ’s, xkt t xlt t will not be a factor in any of the monomials. Hence the monomial m does not occur in pα pβ . Now the term xi1 1 . . . xid d (resp. xj1 1 . . . xjd d ) being the product of the diagonal entries in pτ (resp. pφ ), it occurs with coefficient 1 in pτ (resp. pφ ). Hence m occurs with coefficient 1 in pτ pφ . This, together with the fact that m does not occur in any pα pβ , α = τ ∨ φ, implies that m should occur with coefficient 1 in pτ ∨φ pτ ∧φ (note that in pτ ∨φ pτ ∧φ the monomial m is realized as the product m1 m2 , where m1 (resp. m2 ) is the product of the diagonal entries in pτ ∨φ (resp. pτ ∧φ ). From this, it follows that c = 1. Let L = Hτ , Vτ = Spec RL . Then by [39] (see also Remark 13.7.5.4) we get Theorem 13.7.7.5 Vτ is an affine toric variety. Combining the above Theorem with Theorem 11.4.2.2 of Chapter csln, we obtain a ) to Vτ : degeneration of X(τ

13.7 Degenerations of Schubert varieties to toric varieties

217

Theorem 13.7.7.6 (cf. [33]) There exists a flat family over Spec K[t] with generic ) and special fiber (t = 0) equal to Vτ . fiber (t invertible) equal to X(τ By similar arguments, we also obtain flat toric degenerations of Schubert varieties in a minuscule G/P as well as a flat toric degeneration of the flag variety (see [33] for details).

Appendix: Proof of the main theorem of SMT

Here we prove the main theorem of standard monomial theory—see Theorem A.4.0.1 below. We do this not in complete generality as in Littelmann [77], for that would take us too far afield, but in a limited context that suffices for the invariant theoretic applications in the book. More specifically, we would like to just prove enough to derive as special cases Theorems 8.1.0.2 and 8.2.0.7 of Chapter 8. The proof we give here follows [103]. Unfortunately, it is still not ab initio in two respects. Firstly, we use Theorem A.6.0.1 without giving its proof. Since the proof of this theorem uses the Demazure character formula, in effect we are implicitly assuming this formula (the validity of this formula is related to the normality of Schubert varieties [10, Comments to Chapter 3]). However, the reliance on Theorem A.6.0.1 is not serious and can be avoided—an argument like the one in [77, page 559, Proof of Theorem 2] can be adapted to the present context and the proof would then use only the Weyl character formula rather than the Demazure character formula. Secondly, the preparatory §A.5 contains no proofs. The statements here with proofs and more can all be found in [69, §3].

A.1 Notation The following notation remains fixed throughout this appendix: K G T W sβ β∨ B ≥

an algebraically closed field of arbitrary characteristic; a semisimple simply connected algebraic group over K; a maximal torus of G; the Weyl group of G (with respect to T ); the reflection corresponding to a root β; the coroot corresponding to a root β; a Borel subgroup of G containing T ; Bruhat order on W .

220

Appendix: Proof of the main theorem of SMT

A.2 Admissible pairs and the first basis theorem A.2.1 More notation In this section, in addition to the notation in §A.1 above, we further fix: P  WP WP ≥ L pτ X(w)

a maximal parabolic subgroup of G containing B; the fundamental weight corresponding to P ; the Weyl group of P (with respect to T ); the set of minimal length coset representatives of W/WP ; Bruhat order on W/WP induced from that on W ; the ample generator of Pic G/P ; extremal weight vector in H 0 (G/P , L) of weight −τ (ω). the Schubert variety in G/P corresponding to w ∈ W/WP

In the sequel, we shall denote WP also by Wω or also Wi , where i is given by ω = ωi . When we write sα τ >j τ , it means: τ is an element of W/WP , α a simple root, and , α ∨  = j (we encounter only the cases j = 1, 2). A.2.2 Chevalley multiplicity Let τ , ϕ be elements of W P such that X(ϕ) is a Schubert divisor in X(τ ), i.e., X(ϕ) ⊆ X(τ ) and X(ϕ) is of codimension 1 in X(τ ). Then τ = sβ ϕ for a unique positive root β. The (positive) integer mτ (ϕ) := (ϕ( ), β ∨ ) is called the Chevalley multiplicity of X(ϕ) in X(τ ). A.2.3 Minuscule and classical type parabolics The fundamental weight  (respectively P ) is said to be minuscule if |(, α ∨ )| ≤ 1 for every root α. This is equivalent to saying that all the weights of the corresponding fundamental representation are extremal, that is, they are Weyl group translates of the highest weight. Thus the weights form one orbit under the action of the Weyl group. This condition has an equivalent geometric formulation: if X(ϕ) is a Schubert divisor in X(τ ) (in G/P ), then the Chevalley multiplicity mτ (ϕ) is ≤ 1. Every fundamental weight of the special linear group is minuscule—see §A.3.1 below. We say that  (respectively P ) is of classical type if |(, α ∨ )| ≤ 2 for every root α. This condition has an equivalent geometric formulation: if X(ϕ) is a Schubert divisor in X(τ ) (in G/P ), then the Chevalley multiplicity mτ (ϕ) is ≤ 2. The terminology is justified by the following facts: every fundamental weight of a classical group is of classical type; conversely, if every fundamental weight is of classical type then the group is classical. Although we do not make use of this definition below, we recall here that  (respectively P ) is called quasi-minuscule if either it is minuscule or there is only one non-extremal weight in the corresponding fundamental representation and that is the zero weight. The full list of minuscule, quasi-minuscule, and classical type fundamental weights of all simply connected simple groups is tabulated in Figure A.1 below. The ordering of the weights is as in [8].

A.3 The three examples Type of Group

Classical type fundamental weights Others

Total number

··

··

n

n

1

2 , . . . , n−1

n

Cn (n ≥ 3)

1

2

3 , . . . , n

n

Dn (n ≥ 4)

1 , n−1 , n

2 (Adj)

3 , . . . , n−2

n

E6

1 , 6

2 (Adj)

3 , 5

5

E7

7

1 (Adj)

2 , 6

4

E8

··

8 (Adj)

1

2

F4

··

4

1 (Adj)

2

G2

··

1

··

1

Quasi-minuscule Minuscule

Non-minuscule

An

1 , . . . , n

Bn

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Fig. A.1. Table of fundamental weights of classical type

A.2.4 Admissible pairs In this subsection, we assume that P is a maximal parabolic subgroup of classical type. An ordered pair (τ, θ ) of elements of W P is admissible if either τ = θ (in which case it is called a trivial admissible pair) or τ  θ and there exists a sequence τ = τ0 > τ1 > . . . > τr = θ of elements in W P such that, for 1 ≤ j ≤ r, τj −1 = sαj τj >2 τj for some simple root αj (there could be repetitions among the αj ). In the case of a minuscule parabolic, the admissible pairs are clearly all trivial. The importance of admissible pairs stems from the fact that they form a natural indexing set for a basis of the space H 0 (G/P , L) of sections of L: the first basis theorem, namely Proposition A.10.0.3 below, asserts that there is a basis {v(σ )} of the dual of H 0 (G/P , L), where the σ varies over admissible pairs; we denote by {p(σ )} the basis of H 0 (G/P , L) dual to {v(σ )}. In a given context, if there is more than one parabolic subgroup in the picture, we use the term admissible pair of shape P (or of shape  ) to make it clear which parabolic is being referred to. The formulation of the definition of admissible pairs was a key step in the development of standard monomial theory. In Littelmann’s language of paths [74, 75], an admissible pair is just an LS path of shape a fundamental weight of classical type.

A.3 The three examples Described in this section are three examples of data fixed as in §A.1, §A.2 above. These are the only cases that are relevant for the invariant theoretic applications

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treated in this book. In each of the three examples, ( , ) denotes an inner product on X(T ) ⊗ R (where X(T ) is the character group of the torus T ) that is invariant under the action of the Weyl group; and εj denotes the character of T that maps each element of T to is j th diagonal entry. A.3.1 Example A For the first example, we take V to be an n-dimensional vector space over the algebraically closed base field K, and fix a basis e1 , . . . , en of V . Let: G T B P

to be the special linear group SL(V ) (consisting of linear automorphisms of V of determinant 1) be the set of diagonal matrices (with respect to e1 , . . . , en ) in G; be the set of upper triangular matrices (with respect to e1 , . . . , en ) in G; to be any maximal parabolic subgroup of G containing B.

Then W gets identified with the symmetric group Sn —its action on the diagonal matrices is by permutation of the entries; and there exists an integer d, 1 ≤ d ≤ n, so that is ε1 + · · · + εd ; is the Grassmannian of d-dimensional subspaces in an n-dimensional space; gets identified with Sd × Sn−d ; is the set I (d, n) of subsets (of distinct entries) of cardinality d of {1, . . . , n}; an element of I (d, n) is written τ = {1 ≤ τ1 < · · · < τd ≤ n}; τ ( ) is ετ1 + · · · + ετd for τ ∈ W/WP ; ≥ is defined by: τ ≥ θ if τj ≥ θj for 1 ≤ j ≤ d.

 G/P WP W/WP

The roots are ±(εi − εj ) for 1 ≤ i < j ≤ n. For any root β, we clearly have , β ∨  = 2(, β)/(β, β) = (, β) is 1, 0, or −1 (no matter what d is). Thus P is minuscule and the only admissible pairs are the trivial ones.

A.3.2 Example B Let V be an 2n-dimensional vector space with a symmetric non-degenerate bilinear form ( , ) ; we assume (only for this example) that the characteristic of the underlying field K is not 2. Choose a basis e1 , . . . , e2n of V such that (ei , ej ) is 1 if i = j ∗ and 0 otherwise, where j ∗ := 2n + 1 − j for 1 ≤ j ≤ 2n. Let: G T B εj  P

be the special orthogonal group SO(V ) (consisting of the elements of SL(V ) that preserve the form). be the set of diagonal matrices (with respect to e1 , . . . , e2n ) in G; be the set of upper triangular matrices (with respect to e1 , . . . , e2n ) in G; be the character of T sending an element to the j th element of the diagonal; be the fundamental weight 12 (ε1 + · · · + εn ); be the maximal parabolic subgroup of G corresponding to  .

A.3 The three examples

223

Let C denote the subgroup of (Z/2Z)n consisting of those n-tuples whose sum is 0 in Z/2Z (equivalently those containing an even number of 1’s). The symmetric group Sn acts on it by group homomorphisms. Then: W gets identified with the semi-direct product C  Sn ; WP gets identified with the Sn ; W/WP gets identified with C; a typical element of C is written τ = (τ1 , . . . , τn ), where each τj is 0 or 1; G/P is the even orthogonal Grassmannian (the connected component containing the span of e1 , . . . , en in the space of n-dimensional isotropic subspaces of V ); τ ( ) is 12 ((−1)τ1 ε1 + · · · + (−1)τn εn ) for τ in W/WP = C; ≥ defined by: τ ≥ θ if τj ≥ θj for 1 ≤ j ≤ n (set 1  0 by definition). The roots are ±(εi ± εj ) for 1 ≤ i < j ≤ n. It is clear that, for any root β, we have , β ∨  = 2(, β)/(β, β) = (, β) is 1, 0, or −1 Thus P is minuscule and the only admissible pairs are the trivial ones. A.3.3 Example C Let V be a finite dimensional vector space. Let ( , ) be a non-degenerate alternating bilinear form on V . Then the dimension of V is necessarily even, say 2n. Choose a basis e1 , . . . , e2n of V such that  1 if i = j ∗ and i < j (ei , ej ) = −1 if i = j ∗ and i > j where j ∗ := 2n − j + 1, for 1 ≤ j ≤ 2n. Let: G T B εj  P

be the symplectic group Sp(V ) (consisting of the elements of SL(V ) (or even GL(V )) that preserve the form). be the set of diagonal matrices (with respect to e1 , . . . , e2n ) in G; be the set of upper triangular matrices (with respect to e1 , . . . , e2n ) in G; be the character of T sending an element to its j th diagonal entry; be the fundamental weight ε1 + · · · + εn ; be the maximal parabolic subgroup of G corresponding to  .

Let C denote the group of (Z/2Z)n . The symmetric group Sn acts on it by group homomorphisms. Then: W gets identified with the semi-direct product C  Sn ; gets identified with Sn ; WP W/WP gets identified with C; a typical element of C is written τ = (τ1 , . . . , τn ), where each τj is 0 or 1; G/P is the symplectic Grassmannian ( the space of n-dimensional isotropic subspaces); τ ( ) is (−1)τ1 ε1 + · · · + (−1)τn εn for τ in W/WP = C; ≥ defined by: τ ≥ θ if τj ≥ θj for 1 ≤ j ≤ n (set 1  0 by definition).

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The roots are ±(εi ± εj ) for 1 ≤ i < j ≤ n and ±2εi , 1 ≤ i ≤ n. It is clear that, for any root β, we have , β ∨  = 2(, β)/(β, β) is 2, 1, 0, −1, or −2. Thus P is of classical type. Let us now determine the admissible pairs. The simple roots are εi − εi+1 for 1 ≤ i ≤ n − 1 and 2εn . For τ in C, τ ( ), (2εn )∨  = (τ ( ), εn ) is either 1 or −1. The simple reflection corresponding to α := εi − εi+1 acts on C by interchanging the entries in positions i and i + 1. If τ in C is such that its entry in position i is 0 and the entry in position i + 1 is 1, then τ ( ), α ∨  = 2. It follows easily from these observations that an ordered pair (τ, θ ) of elements C is admissible if and only if τ ≥ θ , both have the same number of 0’s, and this number is > 0.

A.4 Tableaux and the statement of the main theorem We follow the notation as in §A.1,§A.2. Let λ be a dominant integral weight. We assume that λ is a sum d1 +· · ·+dm of fundamental weights of classical type. The sequence d1 , . . . , dm remains fixed. For the invariant theoretic applications treated in this book, it suffices to consider the case when d1 = . . . = dm . Many assertions made below without proof are however easily seen to be true in this special case. We are thus justified somewhat in skipping these proofs. A tableau of shape λ is an m-tuple σ = (σ1 , . . . , σm ) where σj = (τj , θj ) is an admissible pair in W/Wdj . A lift σ˜ = (σ˜ 1 , . . . , σ˜ m ), σ˜ j = (τ˜j , θ˜j ), of σ to W is standard if τ˜1 ≥ θ˜1 ≥ . . . τ˜m ≥ θ˜m in the Bruhat order. A tableau is standard if it has a standard lift. For m = 1, a tableau is just an admissible pair (and hence standard). An element w of W dominates a standard tableau σ if there exists a standard lift σ˜ of σ with w ≥ σ˜ 1 . For a standard tableau σ , there exists by [69, Corollary 4.5 ] such a standard lift that, for any standard lift σˆ , we have τ˜j ≤ τˆj and θ˜j ≤ θˆj for every j . This lift σ˜ is clearly unique. It is called the minimal lift of σ . The element τ˜1 of the minimal lift is the unique minimal element among those that dominate σ . It is called the minimal dominating element of σ . For an element w of W , let X(w) denote the Schubert variety in G/B corresponding to w, i.e., the closure of the B-orbit (with the canonical reduced structure) of the T -fixed point of G/B corresponding to w. For a dominant weight μ, the associated line bundles on G/B and X(w) are denoted Lμ . There is a natural restriction map H 0 (G/B, Lμ ) → H 0 (X(w), Lμ ) between the spaces of sections of Lμ . For a fundamental weight  of classical type and P the corresponding maximal parabolic, let {p(σ )}, as σ varies over admissible pairs of shape  , denote the basis of H 0 (G/P , L ) = H 0 (G/B, L ) given by the first basis theorem—see remark in §A.2.4 above. Theorem A.4.0.1 Let λ = d1 + · · · + dm be any expression for λ as a sum (possibly repeated) of fundamental weights of classical type. For a standard tableau σ = (σ1 , . . . , σm ) of shape λ, let p(σ ) denote the image of p(σ1 ) ⊗ · · · ⊗ p(σm ) under the natural map

A.5 Preparation

225

H 0 (G/B, Ld1 ) ⊗ · · · ⊗ H 0 (G/B, Ldm ) −→ H 0 (G/B, Lλ ). Then the p(σ ) as σ varies over standard tableaux of shape λ is a basis for the space H 0 (G/B, Lλ ) of sections. More generally, for w an element of the Weyl group, as σ varies over the standard tableaux of shape λ dominated by w, the restrictions of p(σ ) to X(w) form a basis for the space of sections H 0 (X(w), Lλ ) on X(w). Corollary A.4.0.2 The p(σ ) as σ varies over standard tableaux dominated by w form a basis for the homogeneous co-ordinate ring of X(w) for the projective embedding X(w) → P(H 0 (G/B, Lλ )∗ ). Proof. It follows from the construction of p(τ ), for τ an admissible pair, that p(τ ) lives in the linear span in H 0 (G/B, Lλ ) (= Vλ∗ ) of the image in P(H 0 (G/B, Lλ )∗ ) of X(w). The rest follows.

A.5 Preparation We need to recall from [69, §3] the following basic—and perhaps not so basic— facts. Let GZ be a Z-form of G, i.e., a simply connected semisimple Chevalley group scheme over Z such that GZ ⊗ K = G. We may choose T and B in G so that there exist TZ a maximal torus subgroup scheme of GZ and BZ a Borel subgroup scheme of GZ containing TZ such that TZ ⊗ K = T and BZ ⊗ K = B. We talk of roots, weights, Weyl group, etc. with respect to TZ and BZ and that would amount to the same as with respect to T and B. Thus the combinatorial definitions in §A.2, §A.4 above make sense for the data over Z and amount to the same as for the original data over the field K. Let λ be a dominant integral weight. Let PZ be a parabolic subgroup scheme of GZ containing BZ such that the character λ lifts to a character of PZ . Let Vλ,Q be the finite dimensional irreducible representation of GQ := GZ ⊗ Q with highest weight λ. Fix a highest weight vector e = vλ in Vλ,Q —this is determined uniquely up to a non-zero scalar factor in Q. Let τ be an element of W/WP (we can omit the subscript Z as seen in the previous paragraph). Then τ can be represented by a Z-valued point of the normalizer N (TZ ) of TZ in GZ . Set eτ := τ e—this is welldetermined up to a sign factor. Set Vλ,Z (τ ) := UZ+ eτ , where UZ is the Kostant Zform of the universal enveloping algebra U of the lie algebra of GQ , and UZ+ the positive part of UZ . Set Vλ,Z := Vλ,Z (w0 ), where w0 is the longest element of the Weyl group. Then Vλ,Z is a free Z-module such that VZ ⊗ Q = Vλ,Q . It is stable under UZ . Further VZ = UZ− e—here UZ− is of course the negative part of UZ . In particular, every eτ is a primitive element of VZ . We call Vλ := Vλ,Z ⊗ K the Weyl module. and Vλ,τ := Vλ,Z (τ ) ⊗ K the Demazure module (also denoted Vλ (τ ) sometimes). We have a canonical isomorphism of the image of X(τ ) under the embedding H 0 (G/B, Lλ )∗ ∼ = Vλ ; and the linear span G/B → P(Vλ ) is Vλ,τ . Thus, if R(τ ) = n R(τ )n is the homogeneous co-ordinate

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ring of the image of X(τ ) in P(Vλ ), then the first graded piece R(τ )1 is identified with Vλ (τ )∗ .

A.6 The tableau character formula The weight of an admissible pair (τ, θ ) is defined to be (τ ( ) + θ ( ))/2, where  is the shape of the admissible pair. The weight of a tableau is the sum of the weights of its component admissible pairs. Theorem A.6.0.1 (The tableau character formula) Let w be an element of the Weyl group and λ a dominant integral weight. The character of the Demazure module Vλ,w equals the formal sum of the weights of those standard tableaux of shape λ that are dominated by w. Proof. Proof follows from [76, page 324, Corollary 1] and the Demazure character formula [10, §3.3].

A.7 The structure of admissible pairs Lemma A.7.0.1 Let  be a fundamental weight of classical type, α and β be simple roots, and τ be an element of W/W . If τ ( ), α ∨  = 2 and τ ( ), β ∨  > 0, then α and β are orthogonal. Similarly if τ ( ), α ∨  = −2 and τ ( ), β ∨  < 0, then α and β are orthogonal. Proof. We prove only the first statement, the proof of the second being similar. We have sα τ ( ) = τ ( )−2α. If α and β are not orthogonal, we get sα τ ( ), β ∨  ≥ 3, which contradicts our hypothesis that  is of classical type. The following proposition is readily verified in the three examples of our interest discussed in §A.3. So there is no need for us (in the context of the present book) to prove it. The reader who is interested in a proof is referred to [61, ], [69, ], or [103, §2]. The ε, ϕ, e, and f values defined in the proposition have a precise meaning in Littelmann’s language of paths [74, 75]: namely eα and fα are the “root operators” and εα (π) := max{k ≥ 0 | eαk (π) = 0}, and ϕα (π) := max{k ≥ 0 | fαk (π) = 0}. Even without knowing precisely the definition of the root operators, the reader can nevertheless see that there is a pattern to the values they take. Proposition A.7.0.2 Let σ = (τ, θ ) be an admissible pair, α a simple root, and s the simple reflection corresponding to α. Then exactly one of the following twelve mutually exclusive possibilities holds, and accordingly we say that the α-type of σ is (A), or (B), . . . , or (M): (A) sτ = τ and sθ = θ ; in this case, we set εα (τ, θ ) = 0 and ϕα (τ, θ ) = 0. (B) sτ = τ and sθ >2 θ ; in this case, we set εα (τ, θ ) = 0, ϕα (τ, θ ) = 1, and fα (τ, θ ) = (τ, sθ ).

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227

(C) sτ = τ and sθ <2 θ ; in this case, we set εα (τ, θ ) = 1 and ϕα (τ, θ ) = 0, and eα (τ, θ ) = (τ, sθ ). (D) sτ >1 τ and sθ <1 θ ; in this case, we set εα (τ, θ ) = 0 and ϕα (τ, θ ) = 0. (E) sτ >1 τ and sθ >1 θ ; in this case, we set εα (τ, θ ) = 0, ϕα (τ, θ ) = 1, and fα (τ, θ ) = (sτ, sθ ). (F) sτ >2 τ and sθ >2 θ ; in this case, we set εα (τ, θ ) = 0 and ϕα (τ, θ ) = 0. (G) sτ >2 τ and sθ = θ ; in this case, we set εα (τ, θ ) = 0, ϕα (τ, θ ) = 1, and fα (τ, θ ) = (sτ, θ ). (H) sτ >2 τ and sθ >2 θ ; in this case, we set εα (τ, θ ) = 0, ϕα (τ, θ ) = 2, fα (τ, θ ) = (sτ, θ ), and fα2 (τ, θ ) = (sτ, sθ ). (J) sτ <1 τ and sθ <1 θ ; in this case, we set εα (τ, θ ) = 1, ϕα (τ, θ ) = 0, and eα (τ, θ ) = (sτ, sθ ). (K) sτ <2 τ and sθ >2 θ ; in this case, we set εα (τ, θ ) = 1, ϕα (τ, θ ) = 1, eα (τ, θ ) = (sτ, θ ), and fα (τ, θ ) = (τ, sθ ). (L) sτ <2 τ and sθ <2 θ ; in this case, we set εα (τ, θ ) = 1, ϕα (τ, θ ) = 0, and eα (τ, θ ) = (sτ, θ ). (M) sτ <2 τ and sθ <2 θ ; in this case, we set εα (τ, θ ) = 2, ϕα (τ, θ ) = 0, eα (τ, θ ) = (τ, sθ ), and eα2 (τ, θ ) = (sτ, sθ ). The following does not occur: (I) sτ <1 τ and sθ >1 θ . It is convenient to let the root operators eα and fα act as follows on the Weyl group W itself: for w in W and α a simple root, fα w is the bigger of w and sα w in the Bruhat order, and eα w is the smaller of w and sα w. We let the action of eα and fα on W/Wα be induced from their action on W . (Caution: The above action of the root operators on elements of W/W does not always coincide with their action on the elements thought of as trivial admissible pairs.) The following proposition, whose proof is immediate, will be used repeatedly. Proposition A.7.0.3 (i) If w ≥ w  then fα w ≥ fα w  and eα w ≥ eα w  . (ii) If w  w  and sα w ≥ w, then fα w  fα w  .

A.8 The procedure Let λ be a dominant integral weight. We assume that λ is a sum d1 + · · · + dm of fundamental weights of classical type. The sequence d1 , . . . , dm remains fixed. A pointed tableau (σ, si1 · · · sim ) consists of a standard tableau σ of shape λ and a reduced expression si1 · · · sim for an element w which dominates σ . Described in this section is a procedure for associating to a pointed tableau (σ, si1 · · · sim ) a monomial (p ) (p ) differential operator Yi1 1 · · · Yin n , where p1 , . . . , pn are non-negative integers— here the Yi are the generators of UZ− , with Yi being the Chevalley generator corresponding to the simple reflection si of the Lie algebra g of the group GQ = GZ ⊗ Q. The p1 , . . . , pn will turn out positive if w is the minimal dominating element of σ . (p)

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The Procedure. Let (σ, si1 · · · sim ) be a pointed tableau. The procedure works by induction on n. If n = 0, the associated monomial is the identity. Now suppose that n ≥ 1. Throughout this section j denotes such an integer variable that 1 ≤ j ≤ m. We write s, ε, e, and f instead of si1 , εi1 , ei1 , and fi1 respectively. Let (τj , θj ) be the admissible pair σj . Let J be the least integer such that either sθJ > θJ or sτJ +1 > τJ +1 —if sτj ≤ τj and sθj ≤ θj for all j , then J is defined to be m; if sτ1 > τ1 , then J is defined to be 0. Let p1 := 1≤j ≤J ε(σj ), and ζ be the tableau defined by ζj := σj for j > J and ζj := eε(σj ) σj for j ≤ J . It follows from Proposition A.8.0.1 below that (ζ, si2 · · · sim ) is a pointed tableau. The integers p2 , . . . , pn are obtained by applying the induction hypothesis to this pointed tableau. It follows immediately from the definition of J that (i) sτj ≤ τj for j ≤ J , sθj ≤ θj for j < J . Let (τj , θj ) be the admissible pair ζj . Using (1) and going through all the possibilities listed in Proposition A.7.0.2 for admissible pairs, we find (2) For j ≤ J , (τj , θj ) = (eτj , eθj ), and so sτj ≥ τj and sθj ≥ θ + j  . (3) For j ≤ J , τj = f τj . For j < J , θj = f θj . Either θJ = θJ or θJ = f θJ according as sθj > θJ or sθJ ≤ θJ . Proposition A.8.0.1 Set w := si1 · · · sin . Then (I) ζ is standard. (II) sw dominates ζ . (III) If w is the minimal dominating element of σ , then p1 is positive and sw = si1 · · · sin is the minimal dominating element of ζ . Proof. (I) Let τ˜1 ≥ . . . ≥ θ˜m be a standard lift of σ . Set τ˜j := eτ˜j and θ˜j := eθ˜j for j ≤ J ; τ˜  := τ˜j and θ˜  := θ˜j for j > J . It follows from item (2) above that j

j

(τ˜j , θ˜j ) is a lift of ζj for all j . To prove that this lift is standard, note that, by the definition of J , either eτ˜J +1 = τ˜J +1 , in which case, by Proposition A.7.0.3 (1), θ˜j := eθ˜J ≥ eτ˜J +1 = τ˜J +1 , or eθ˜J = θ˜J , in which case, θ˜j := eθ˜J = θ˜J = τ˜J +1 . That τ˜1 ≥ θ˜1 ≥ . . . ≥ τ˜m ≥ θ˜m now follows again from Proposition A.7.0.3 (1). (II) It follows from Proposition A.7.0.3 (1) that if, in the proof of statement (I), the standard lift τ˜1 ≥ . . . θ˜m of σ is so chosen that w dominates it, then sw = ew dominates the standard lift τ˜1 ≥ . . . ≥ θ˜m of ζ . (III) Suppose that p1 = 0, that is, ε(σj ) = 0 for j ≤ J . Then ζ = σ . From (II) it follows that ew = si2 · · · sin , which is strictly less than w, dominates σ . Suppose that w  dominates ζ . Let τ˜1 ≥ . . . ≥ θ˜m be such a standard lift of ζ that w  ≥ τ˜1 . Set τ˜j := f τ˜j and θ˜j := f θ˜j for j < J , τ˜j := τ˜j , θ˜j := θ˜j for j > J , τ˜J := f τ˜J , and finally, according as sθJ > θJ or sθJ ≤ θJ , either θ˜J := θ˜J or θ˜J := f θ˜J . It follows from Proposition A.7.0.3 (1) that τ˜1 ≥ . . . ≥ θ˜m and also that f w  ≥ τ˜1 . It follows from item (3) above τ˜1 ≥ . . . θ˜m is a lift of σ . Thus f w 

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229

dominates σ and so f w  ≥ w = si1 · · · sin . By Proposition A.7.0.3 (1), we have ef w  ≥ ew = si2 · · · sin . But clearly w  ≥ ef w  . Thus w  ≥ si2 · · · sin .

A.9 The basis We now state our main theorem. Let λ be a dominant integral weight. We assume that λ is a sum d1 + · · · + dm of fundamental weights of classical type. The sequence d1 ,. . . ,dm remains fixed. Fix a highest weight vector vλ of Vλ . Let w be an element of the Weyl group. For σ a standard tableau dominated by w, consider the pointed tableau (σ, si1 · · · sin ), where si1 · · · sin is an arbitrarily chosen reduced expression for w—the expression could vary with σ . Set u(σ ) := (p ) (p ) (p ) (p ) Yi1 1 · · · Yin n vλ , where Yi1 1 · · · Yin n is the differential operator associated by the procedure of §A.8 to the pointed tableau (σ, si1 · · · sin ). The vector u(σ ) depends in general on w and si1 · · · sin , but we do not indicate this dependence explicitly in the notation for the vector. If λ is a fundamental weight, u(σ ) is independent of w and si1 · · · sin as we show in §A.10, and we denote it by v(σ ) rather than u(σ ). Theorem A.9.0.1 Let λ be a sum of fundamental weights of classical type, Vλ the Weyl module VZ ⊗ K introduced in §A.5, and w an element of the Weyl group. The vectors u(σ ), as σ varies over standard tableaux of shape λ that are dominated by w, form a basis for the Demazure module Vλ,w . (p )

(p )

Proof. We have u(σ ) := Yi1 1 · · · Yin n vλ for a reduced expression sI1 · · · sin of w. So it is clear that u(σ ) belongs to Vλ,w . The Tableau character formula (Theorem A.6.0.1) tells us that the number of u(σ ) equals the dimension of Vλ,w . It is therefore enough to show that u(σ ) span Vλ,w or that u(σ ) are linearly independent. It is proved in Proposition A.10.0.3 that, in case λ is a fundamental weight, the vectors u(σ ) span Vλ,w . Let us now indicate the proof of linear independence of u(σ ) in case λ is not a fundamental weight. We have λ = d1 + · · · + dm with m ≥ 2. Let V be the tensor product representation V1 ⊗ · · · ⊗ Vm where Vj := Vdj . Fix v1 , . . . , vm highest weight vectors respectively in V1 , . . . , Vm . We can take Vλ to be the G-submodule of V generated by the highest weight vector vλ = v1 ⊗ · · · ⊗ vm . Let X denote the set of tableaux of shape λ, standard or otherwise. Associate to x = (x1 , . . . , xm ) in X the element v(x) := v(x1 ) ⊗ · · · ⊗ v(xm ) where v(xj ) is the basis element in Vj corresponding to the admissible pair xj —see Propositions A.10.0.1, A.10.0.3. The set {v(x) | x ∈ X} is then a basis of V . (k ) (k ) The result of the action of a differential operator Yj1 1 · · · Yjr r on vλ = v1 ⊗· · ·⊗ vm can be expressed uniquely as an integral linear combination of basis elements v(x). Let  c(σ, x)v(x), c(σ, x) non-negative integers) u(σ ) = x∈X

be the expression for u(σ ) as such a linear combination.

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We show in Proposition A.11.0.1 that c(σ, σ ) = 1 and that if c(σ, x) = 0, then σ ≥ x in the lexicographic partial order. The linear independence of u(σ ) follows. Let us now see how Theorem A.9.0.1 and its proof lead to a quick proof of the main Theorem A.4.0.1 of standard monomial theory. For a fundamental weight  of classical type, let {p(σ )} be the basis of H 0 (G/B, L ) dual to the basis {v(σ )} of V —here σ varies over admissible pairs of shape  . Proof. It follows from Proposition A.11.0.1 below that p(σ ), u(σ ) = 1 and that p(σ ), u(σ  ) = 0 only if σ  ≥ σ . This finishes the proof and also shows that the transition matrix between the dual of the basis {u(σ )} and the standard monomial theoretic basis {p(σ )} is unipotent triangular.

A.10 The first basis theorem We assume throughout this section that λ =  is a fundamental weight of classical type with P as the associated maximal parabolic sub group. A standard tableau σ of shape  is therefore an admissible pair in W/W . Proposition A.10.0.1 The vector u(σ ) defined in §A.9 depends only on σ . It is independent of the choice of the dominating element w and also of the choice of the reduced expression for w. (To emphasise this fact we denote the vector by v(σ ) rather than u(σ ).) Proof. Consider two pointed tableaux (σ, si1 · · · sin ) and (σ, sj1 · · · sjN ). Let u (respectively v) be the vector u(σ ) calculated by applying the procedure of §A.8 to (σ, si1 · · · sin ) (respectively to (σ, sj1 · · · sjN )). We need to show that u = v. We use a double induction, the first on length (τ ) + length (θ ), where σ = (τ, θ ), and the second on (n, N ). We say (n , N  ) ≤ (n, N ) if n ≤ n and N  ≤ N . The induction hypothesis will be that the statement holds for all such (n , N  ) that (n , N  )  (n, N ). Suppose that length(τ ) + length(θ ) = 0. Then (τ, θ ) = (1, 1). The procedure applied to any pointed tableau (σ, si1 · · · sin ) will yield the monomial 1, so that u = v = v . Now suppose that ei1 σ = 0. Then u = u(σ, si2 · · · sin ), and so u = v by the hypothesis of induction on (n, N). We therefore assume from now on that ei1 σ = 0 and ej1 σ = 0. Note that the function length(τ ) + length(θ ) strictly decreases when we pass from σ to either ei1 (σ ) or ej1 (σ ). So, by the induction hypothesis, u(ei1 (σ )) and u(ej1 σ ) are unambiguously determined, and we can choose any pointed tableaux to calculate these. We tacitly use this comment in what follows. (p )

εi (σ )

Suppose that i1 = j1 . Then u := Yi1 1 u(ei11 σ ) =: v, and we are done. From now on we use α and β instead respectively of αi1 and αj1 , and assume that α = β. Suppose that α and β are orthogonal. Set p := εα (σ ) and q = εβ (σ ) (these are q p p q non-zero by our assumption). We have p = εα (eβ (σ )), q = εβ (eα σ ), and eα eβ σ = q p

(p)

p

(p) (q)

q p

(q) (p)

p q

eβ eα σ , so that u := Yα u(eα σ ) = Yα Yβ u(eβ eα σ ) = Yβ Yα u(eα eβ σ ) = Yβ u(eβ σ ) =: v. We therefore assume from now on that α, β ∨  = 0. (q)

q

A.10 The first basis theorem

231

From the assumption that eα (σ ) = 0, eβ (σ ) = 0 we get sα τ ≤ τ , sβ τ ≤ τ . Thus the possibilities for the pair (τ ( ), α ∨ , τ ( ), β ∨ ) are (0, 0), (0, −1), (0, −2), (−1, −1), (−1, −2), and (−2, −2) (the possibilities (−1, 0), (−2, 0), and (−2, −1) can be eliminated by interchanging α and β). Further, if the possibility (0, 0) (respectively (0, −1)) occurs, then the pair (θ ( ), α ∨ , θ ( ), β ∨ ) can only be (−2, −2) (respectively (−2, −1)). Using the hypothesis that  is of classical type, it can be seen (see Lemma A.7.0.1) that α and β are orthogonal except perhaps in the cases (−1, −1) and (0, −2), so all but these cases are eliminated from consideration. In the case (−1, −1) the corresponding pair for θ can, by Proposition A.7.0.2 (2), only be (−1, −1). In the case (0, −2), the corresponding pair for θ can only be (−2, 0) or (−2, 2) (the pair (−2, −2) is eliminated because of our assumption that α and β are not orthogonal). We treat these cases one by one. The arguments in all cases are similar. Suppose that the pairs for τ and θ are both (−1, −1). It can readily be verified that α, β ∨  = β, α ∨  = −1, τ >1 sα τ >2 sβ sα τ >1 sα sβ sα τ and θ >1 sα θ >2 (2) sβ sα θ >1 sα sβ sα θ . We have u := Yα u(sα τ, sα θ ) = Yα Yβ Yα u(sα sβ sα τ, sα sβ sα θ ), (2)

and similarly v := Yβ Yα Yβ u(sβ sα sβ τ, sβ sα sβ θ ). But sα sβ sα = sβ sα sβ , and (2) (2) Yα Yβ Yα = Yβ Yα Yβ . (This relation is obtained from the Serre relations as fol(2)

(2)

(2)

(2)

lows: 0 = Yα (Yα Yβ − Yβ Yα Yβ + Yβ Yα ) = (Yβ Yα − Yα Yβ Yα + Yα Yβ )Yβ .) So we are done. Suppose that (τ ( ), α ∨ , τ ( ), β ∨ ) = (0, −2) and that (θ ( ), α ∨ , θ ( ), β ∨ ) = (−2, 0). It can readily be verified that α, β ∨  = β, α ∨  = −1, τ >2 sβ τ >2 sα sβ τ , and θ >2 sα θ >2 sβ sα θ . We have u := Yα u(τ, sα θ ) = (2) (2) Yα Yβ Yα u(sα sβ τ, sβ sα θ ), and v := Yβ u(sβ τ, θ ) = Yβ Yα Yβ u(sα sβ τ, sβ sα θ ). (2)

(2)

Using the relation Yα Yβ Yα = Yβ Yα Yβ (see the previous paragraph) we are done. Suppose that (τ ( ), α ∨ , τ ( ), β ∨ ) = (0, −2) and that (θ ( ), α ∨ , θ ( ), β ∨ ) = (−2, 2). We have β, α ∨  = −1, but α, β ∨  can now be either −1 or −2, and accordingly there are two cases. First suppose that α, β ∨  = −1. It can readily be verified that τ >2 sβ τ >2 sα sβ τ , and sβ θ >2 θ >2 sα θ . We have u := Yα u(τ, sα θ ) = Yα Yβ Yα u(sα sβ τ, sα θ ), and v := Yβ u(sβ τ, θ ) = (2) (2) (2) Yβ Yα u(sα sβ τ, sα θ ). But Yβ Yα − Yα Yβ Yα + Yα Yβ = 0 is a Serre relation, and Yβ u(sα sβ τ, sα θ ) = 0, for if it were non-zero its α-weight would be 3, and we are done. Now suppose that we are in the situation of the previous paragraph, except that α, β ∨  is now −2. It can readily be verified that τ >2 sβ τ >2 sα sβ τ >2 sβ sα sβ τ and sβ θ >2 θ >2 sα θ >2 sβ sα θ . We have u := Yα u(τ, sα θ ) = (2) (2) (2) Yα Yβ Yα Yβ u(sβ sα sβ τ, sβ sα θ ) v := Yβ u(sβ τ, θ ) = Yβ Yα Yβ u(sβ sα sβ τ, sβ sα θ ). (2)

Adding the relations Yα ((adYβ )3 Yα = 0) and ((adYα )2 Yβ = 0)Yβ , and ignor(2) (2)

ing terms with Yβ3 or Yα3 since they act as zero on V , we see that Yβ Yα Yβ (2) Yα Yβ Yα Yβ .

So u = v and we are done.

=

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Corollary A.10.0.2 If x, y are admissible pairs and α is a simple root satisfying x = fαt y = 0, then Yα(t) v(y) is an integral multiple of v(x). If, furthermore, eα y = 0, (t) then Yα v(y) = v(x). Proof. Let us assume that t ≥ 1 for if t = 0 the statements are trivially true. By choosing such an element w of the Weyl group that dominates x and for which there is a reduced expression of the type sα · · · , we see from Proposition A.10.0.1 (p ) p that v(x) = u(x, sα · · · ) = Yα 1 v(eα 1 x), where p1 = εα (x). Similarly we have p1 −t εα (y) ε (y) v(y) = Yα v(eα y). But clearly eαα y = eαεα (x) x. And p1 = t if eα y = 0. So both statements are proved. Proposition A.10.0.3 (First Basis Theorem) Let w be an element of the Weyl group. The vectors v(σ ), as σ varies over admissible pairs in W/W that are dominated by w, span the Demazure module V,w (and hence also form a basis—see Theorem A.6.0.1). Proof. We proceed by induction on the length n of w. For n = 1, there is only one v(σ ), namely, the highest weight vector v , and this spans the one dimensional module V,1 . Let n ≥ 1. Let sα be such a simple reflection that sα w < w. The induction hypothesis is that v(ζ ), ζ dominated by sα w, span V,sα w . Since V,w = SL2,α V,sα w (SL2,α being the copy of SL(2) in G associated to α), it is enough to show that the (t) vectors Yα v(ζ ) are spanned by v(σ ), σ dominated by w. Fix a ζ dominated by sα w. First suppose that fαt ζ = 0. Then by Corol(t) lary A.10.0.2, Yα v(ζ ) is an integral multiple of v(fαt ζ ). From Proposition A.7.0.2, it is easily seen that fαt ζ is dominated by fα (sα w) = w, and so we are done. (t) We therefore assume that fαt ζ = 0. We may also assume that Yα v(ζ ) = 0, since (t) otherwise there is nothing to prove. Since the α-weight of Yα v(ζ ) is bounded below by −2, the only way this can happen is if t = 1 and ζ has α-weight 0. For such a ζ , we (2) Claim: Yα Xα v(ζ ) = Yα v(ζ ). We first finish the proof assuming the claim. The vector Xα v(ζ ) belongs to V,sα w . So it is a linear combination of v(y), y dominated by sα w and of α-weight 2. But, for such a y, fα2 y = 0 is dominated by w, and (2) Yα v(y) = v(fα2 y) by Corollary A.10.0.2. (2) It only remains to show the claim. We have Yα Xα v(ζ ) = 1/2Yα Xα Yα v(ζ ) (since Xα Yα = Yα Xα on a vector of α-weight 0). Now Yα Xα Yα v(ζ ) = −Hα Yα v(ζ )+ Xα Yα2 v(ζ ). We have Yα2 v(ζ ) = 0 and Hα Yα v(ζ ) = −2Yα v(ζ ) by α-weight considerations, and so the claim is proved.

A.11 Linear independence We prove in this section the linear independence of the u(σ ) in Theorem A.9.0.1 in the case λ is a non-fundamental weight. The linear independence follows immediately from Proposition A.11.0.1 below.

A.11 Linear independence

233

We impose on the set of admissible pairs the lexicographic partial order and on the set X of all tableaux (standard or otherwise) of shape λ the induced lexicographic order. More precisely, for admissible pairs (τ, θ ), (τ  , θ  ) in the same W/W , we write (τ, θ )  (τ  , θ  ) if τ  τ  or if τ = τ  and θ  θ  . For x, x  in X, we write x ≥ x  if xj  xj for the least such j that xj = xj . Proposition A.11.0.1 With notation as in the proof of Theorem A.9.0.1, c(σ, σ ) = x and c(σ, x) = 0 only if σ ≥ x. In other words,  u(σ ) = v(σ ) + c(σ, x)v(x) {x|σ x}

Proof. We proceed by induction on the length n of the reduced expression in the pointed tableau (σ, si1 · · · sin ) used to calculate u(σ ). For n = 0, we have uσ = vσ = vλ and so the assertion holds. Let now n ≥ 1. Let J, p1 , and ζ be as in procedure of §A.8. We write just Y , e, and f respectively for Yi1 , ei1 and fi1 . The induction hypothesis is that  c(ζ, y)v(y) u(ζ ) = v(ζ ) + {y|ζ y}

and so we have



u(σ ) := Y (p1 ) u(ζ ) = Y (p1 ) v(ζ ) +

c(ζ, y)Y (p1 ) v(y)

{y|ζ y}

For y in X, let Z(y) t = (t1 , . . . , tm ) of non denote the set of such m-tuples negative integers that tj = p1 and, for all j , Y (tj ) v(yj ) = 0. Given such a t, let Y t v(y) denote the element Y (t1 ) v(y1 ) ⊗ · · · ⊗ Y (tm ) v(ym ). We have  Y t v(y) Y (p1 ) v(y) = t∈Z(y)

It follows from Corollary A.10.0.2 that if f tj yj = 0, then Y (tj ) v(yj ) is an integral multiple of v(f tj yj ). But it can happen that f tj yj = 0 and yet Y (tj ) v(yj ) = 0. In any case Y (tj ) vyj is uniquely an integral linear combination of basis vectors v(xj ) of Vj associated to admissible pairs xj in W/Wdj . We can therefore write  Y t v(y) = b(y, t, x)v(x) x∈B(y,t)

where the b(y, t, x) are integers and B(y, t) is the appropriate subset of X. With this notation, it is enough to prove the following Claim: there exists such an element t 0 of Z(ζ ) that (1) b(ζ, t 0 , σ ) = 1, (2) for any such t in Z(ζ ) that t = t 0 , we have σ  x for x in B(ζ, t), and (3) for any such y, t, and x that ζ  y, t ∈ Z(y), and x ∈ B(y, t), we have σ  x. We first prove (1). Set t 0 := (ε(σ1 ), . . . , ε(σJ ), 0, . . . , 0). We have t0,j = p1 from the definition of p1 . We have f t0,j ζj = σj from the definition of ζ . Since

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eζj = 0 for j ≤ J , it follows from Corollary A.10.0.2 that Y t 0 v(ζ ) = v(σ ). This proves (1). We next prove (2) and (3) both together. Suppose that ζ ≥ y and t ∈ Z(y), and that t = t 0 if ζ = y. Let s be the least such integer that either ζs = ys or t0,s = ts . By assumption, such an s exists. We clearly have σj = xj for j < s. So it is enough to show that σs  xs . To prove that σs  xs , we first make some simplifications. Suppose that ts = 0. t Then either t0,s > 0 or ζs  ys . In either case we have σs = fα0,s ζs  ys = xs (recall p p that, for p > 0, if fα (τ, θ ) = 0, then fα (τ, θ )  (τ, θ ), by Proposition A.7.0.2 (2)), t0,j = tj = p1 , and and we are done. If s > J , then t0,s = ts = 0 (because t0,j = 0 for j > J ), so we may assume that s ≤ J . If s = J , we may further assume that either ζs = ys and t0,s > ts , or ζs  ys and t0,s ≥ ts . To sum up, we have to show that σs  xs under each of the following sets of conditions: • • • •

s s s s

< J , ζs < J , ζs = J , ζs = J , ζs

= ys , t0,s =

ts  ys = ys , t0,s  ts  ys , t0,s ≥ ts

We want to now transfer the burden of proof to the next proposition which is purely about admissible pairs. To this end, we note that, by statement (2) in §A.8, for s ≤ J , we have sτs ≥ τs and sθs ≥ θs , where ζs = (τs , θs ) (in other words, ζs has α-type (A), (B), (E), (G), or (H) of Proposition A.7.0.2). We note also that t0,s = ϕα (ζs ) for s < J . It is easily checked that the four sets of conditions above correspond respectively to the four sets of conditions in the proposition below. The proof will therefore be complete once the proposition below is proved. Proposition A.11.0.2 Let α be a simple root, ζ and y be admissible pairs (in the same W/W ). Assume that ζ has α-type (A), (B), (E), (G), or (H) of Proposition A.7.0.2. Let t0 ≥ 0 and t > 0 be integers satisfying ϕα (ζ ) ≥ t0 and t Yα(t) v(y) = 0. Set σ = fα0 (ζ ). If any of the following four sets of conditions holds, then σ  x, where x is any such admissible pair that v(x) occurs with non-zero (t) (t) coefficient c(x) in the expression Yα v(y) = c(x)v(x) for Yα v(y) as an integral linear combination of the basis vectors v(x). (i) (ii) (iii) (iv)

ζ ζ ζ ζ

= y, t0  y, t0 = y, t0  y, t0

= ϕα (ζ ), t = t0 = ϕα (ζ ) t ≥t

Proof. The case when (3) holds is the easiest, so we consider that first. By Corollary A.10.0.2, x = fαt y. Now the conclusion that σ  x follows from the following statement, which can readily be verified from the list of possibilities for admissible pairs in Proposition A.7.0.2 (2): for an admissible pair z, if ϕα (z) ≥ r  r, then r fα 0 z  fαr z.

A.12 Arithmetic Cohen-Macaulayness and arithmetic normality

235

We now make some reductions. We use the following notation in the sequel: for an admissible pair z, we write τ (z) and θ (z) for its components in W/W —in other words, z = (τ (z), θ (z)). Suppose (1), (2), or (4) holds (in fact, what follows applies even if (3) holds, but that is now irrelevant). Then τ (σ ) = fα τ (ζ ) (because t ≥ 1 and the α-type of ζ is not (D) or (F)). Since Yα v(y) belongs to the Demazure module corresponding to fα τ (y), we have fα τ (y) ≥ τ (x). Now, if τ (ζ )  τ (y), we have, by Proposition A.7.0.3 (2), τ (σ ) = fα τ (ζ )  fα τ (y) ≥ τ (x), and we are done. So we may assume that the following holds: ( ) τ (ζ ) = τ (y) and τ (σ ) = fα τ (ζ ) = fα τ (y) = τ (x). In particular, θ (ζ ) ≥ θ (y) and if ζ  y then θ (ζ )  θ (y). In case (4), if t0 = ϕα (ζ ), then (2) holds, so we replace (4) by the following more stringent condition: (4) ζ  y, ζ is of type α-type (H), and t0 = t = 1. Now we prove the assertion under the additional assumption that fαt y = 0. By Corollary A.10.0.2, we have x = fαt y. If (1) holds, then (3) holds also, and so we are done. If (2) holds, we have, by Proposition A.7.0.3 (2), θ (σ ) = fα θ (ζ )  fα θ (y) ≥ θ (x), and we are done. If (4) holds, it is easily checked that θ (σ ) = θ (ζ )  θ (y) = θ (x), and we are done. We are therefore justified in assuming from now on that fαt y = 0. This means in particular that the α-weight of y is non-positive, and since the α-weight of Yα v(y) is bounded below by −2, we conclude that t = 1 and y has α-weight 0 (y can have α-type only (A), (D), or (F) of Proposition A.7.0.2), and so x has α-weight −2, that is, α-type of x can only be (M). By ( ), we have τ (y) = τ (ζ ) and fα τ (y) = τ (x), so that τ (y) = eα τ (x) and τ (x) = τ (y) − 2α. On the other hand, by equating the weights of x and Yα v(y), we get τ (x) + θ (x) = τ (y) + θ (y) − 2α. From these two equations, we conclude that θ (x) = θ (y) , that is, θ (x) = θ (y) in W/W . Suppose that (2) or (4) holds. By ( ) we have θ (ζ )  θ (y). Combining this with the conclusion of the preceding paragraph, we get θ (σ ) ≥ θ (ζ )  θ (y) = θ (x), which along with ( ) shows that σ  x. Suppose that (1) holds. If t0  t, then (3) holds, so we are done. So assume that t  t0 . It was shown above that we may assume that t = 1 and that x has α-type (M). So t0 = 0, and ζ has α-type (A), which means in particular that fα τ (ζ ) = τ (ζ ). Combining this with the assumption ( ), we see that τ (ζ ) = τ (x) but this is absurd, for the same element of W/W cannot be the first component of admissible pairs of α-type (A) and (M).

A.12 Arithmetic Cohen-Macaulayness & Arithmetic normality for Schubert varieties In this section, we comment on the Cohen-Macaulayness and normality of Schubert varieties in G/P , with P being a maximal parabolic subgroup of classical type

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(covering our application to Invariant Theory—the three examples A, B, and C of Chapter 1). For such a G/P , the Cohen-Macaulayness and normality for Schubert varieties may be concluded just using the standard monomial basis as given by Theorem A.4.0.1. As mentioned in the Introduction, in recent times, among the several techniques of proving the Cohen-Macaulayness of algebraic varieties, particularly those that are related to Schubert varieties, two techniques have proved to be quite effective, namely, Frobenius splitting technique and deformation technique. Frobenius splitting technique is used in [104], for example, for proving the (arithmetic) CohenMacaulayness of Schubert varieties. The deformation technique consists in constructing a flat family over A1 , with the given variety as the generic fiber (corresponding to t ∈ K invertible). If the special fiber (corresponding to t = 0) is Cohen-Macaulay, then one may conclude the Cohen-Macaulayness of the given variety. Going by the works of [12, 17, 21, 33, 46], it is reasonable to believe that if there is a “standard monomial basis” for the coordinate ring of the given variety, then the deformation technique will work well in general (using the “straightening relations”). A proof for arithmetic Cohen-Macaulayness for Schubert varieties in the Grassmannian was presented in Chapter 4, using local cohomology. We present yet another proof using the deformation technique. A.12.1 Arithmetic Cohen-Macaulayness Degenerations of Schubert varieties in the Grassmannian Let G = SL(n). Fix r, 1 ≤ r ≤ n − 1. Consider the polynomial algebra P = K[xβ , β ∈ Tτ ], where τ ∈ Ir,n and Tτ = {φ ∈ Ir,n | φ ≤ τ } Definition A.12.1.1 For τ ∈ Ir,n , let Iτ be the ideal of P generated by the set of monomials {xγ xδ | γ and δ are non-comparable elements of Tτ }. We define the discrete algebra associated to τ (or to the Schubert variety X(τ ) (in the Grassmannian Gr,n )) to be the K-algebra P /Iτ , and denote it by Dτ . ) (the The main result of this subsection is that Spec Dτ is a degeneration of X(τ cone over the Schubert variety X(τ )), i.e. there exists a one parameter flat family ) (i.e., a family parametrized by Spec K[[t]]) with generic fiber (t invertible) X(τ and special fiber (t = 0) Spec Dτ . We first recall the following relation (cf. (*)) from Chapter 4. If γ and δ are non-comparable elements of Tt , then we have a quadratic relation in Rτ (the homogeneous coordinate ring of X(τ ) for the Plücker embedding) of the form  aλμ pλ pμ , aλμ ∈ K ∗ , (*) pγ pδ = λ,μ

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237

where the monomials in the right hand side are standard, i.e. λ ≥ μ; further, λ > both γ and δ, and μ < both γ and δ. We shall denote by Q the set of all such quadratic relations for various pairs of non-comparable elements γ , δ in Tτ . Lemma A.12.1.2 There exists a function θ : Tτ → Z+ such that given any quadratic relation pγ pδ = λ,μ aλμ pλ pμ in Q, we have θ (λ)+θ (μ) > θ (γ )+θ (δ) for all λ, μ appearing in the sum. Proof. Since Tτ is a finite partially ordered set, we can define an order function ord : Tτ → Z+ such that β > β  implies ord β > ord β  . For β ∈ Tτ set θ (β) = N ord β , where N is a positive integer. We claim that for a sufficiently large N , θ is the required function. Indeed, using the relation (*) above, we have ord γ < ord λ and ord δ < ord λ; therefore, choosing a sufficiently large N, we can achieve the inequality N ord λ + N ord μ > N ord γ + N ord δ for all quadratic relations in Q. We recall the following well known Lemma from commutative algebra. Lemma A.12.1.3 Let M be a finitely generated module over the power series ring A = K[[t]] and let L be the quotient field of A. Then M is a free module if and only if dimK M ⊗A K ≤ dimL M ⊗A L. Proof. If M is free, then obviously the inequality holds. Conversely, suppose that the inequality holds. If dimk M ⊗A K = m, then by Nakayama’s lemma M can be generated by m elements and therefore we have an exact sequence 0 → N → Am → M → 0 of A-modules. Tensoring the sequence by L and using the inequality, we conclude that N ⊗A L = 0. Since N is torsion free, N = 0, and hence M is isomorphic to Am . Theorem A.12.1.4 Let τ ∈ Ir,n . There exists a flat family over Spec K[[t]] whose ) and the special fiber (t = 0) is Spec Dτ , where generic fiber (t invertible) is X(τ Dτ is the discrete algebra associated to τ . Proof. Let Rτ be the homogeneous coordinate ring of X(τ ) (for the Plu¨cker embedding) and let Q be as above (namely, the set of quadratic relations of the form (∗)). For each quadratic relation pγ pδ = λ,μ aλμ pλ pμ in Q, consider the polynomial  aλμ xλ xμ Fγ ,δ = xγ xδ − λ,μ

in P = K[xφ , φ ∈ Tτ ], and let J be the ideal of P generated by all such polynomials. If x φ denotes the canonical image of xφ in P /J , then it is clear that P /J is generated

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by standard monomials in x φ , φ ∈ Tτ . Moreover, we have a canonical surjective map fτ : P /J → Rτ which sends x φ to pφ . Since Rτ has a basis consisting of standard monomials in pφ ’s, it follows that fτ is injective. Hence P /J is isomorphic to Rτ . Now using Lemma A.12.1.2, choose a function θ : Tτ → Z+ such that for each quadratic relation pγ pδ = λ,μ aλμ pλ pμ in Q, θ (λ) + θ (μ) − θ (γ ) − θ (δ) > 0. Further, let A = K[[t]]. Corresponding to each quadratic relation in Q as above, consider the polynomial  Fγ ,δ,t = xγ xδ − aλμ t θ(λ)+θ(μ)−θ(γ )−θ(δ) xλ xμ λ,μ

in PA := A[xφ , φ ∈ Tτ ]. Denoting by I the ideal generated by the polynomials Fγ ,δ,t in PA , we set Rτ = PA /I. Let L be the quotient field of A and let PL = L[xφ , φ ∈ Tτ ]. If J˜ is the ideal of PL generated by the polynomials Fγ ,δ and I˜ is the ideal generated by the polynomials Fγ ,δ,t , then it is easy to see that PL /I˜ ∼ = PL /J˜, the isomorphism being induced by the L-automorphism of PL which sends xβ to t −θ(β) xβ . Thus we have Rα ⊗A L = PA /I ⊗A L ∼ = PL /I˜ ∼ = PL /J˜ ∼ = P /J ⊗K L ∼ = Rτ ⊗K L and Rτ ⊗A K ∼ = Rτ /tRτ ∼ = Dτ . This shows that the generic fiber (t invertible) of Rτ is Rτ and the special fiber (t = 0) is Dτ . Now we show that Rτ is flat over A. From the definition of Dτ it is clear that Dτ is generated by standard monomials in the canonical images x φ of xφ , φ ∈ Tτ . Therefore, using the above isomorphisms, it follows that for every m ≥ 0, we have, dimK (Rτ )m ⊗A K = dimK (Dτ )m ≤ dimK (Rτ )m = dimL (Rτ )m ⊗K L = dimL (Rτ )m ⊗A L, where (Rτ )m , (Dτ )m , (Rτ )m denote the m-th graded components of the respective rings. Thus, in view of Lemma A.12.1.3, each graded component of Rτ is a free A-module, and hence Rτ is a flat A-algebra. Corollary A.12.1.5 (i) Dτ has a basis consisting of standard monomials in x φ , φ ∈ Tτ . (ii) Dτ is reduced. (iii) If P = Proj P , where P = K[xφ , φ ∈ Tτ ], then the degree of X(τ ) in the Plücker embedding is the same as the degree of Proj Dτ in P. Proof. (i) Since for each m ≥ 0, (Rτ )m is a free A-module, we have dimK (Rτ )m ⊗A K = dimL (Rτ )m ⊗A L, and therefore the inequalities (∗) in the proof of the above theorem imply that dimK (Dτ )m = dimK (Rτ )m , for each m ≥ 0.

(**)

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239

Since distinct standard monomials on X(τ ) of degree m form a basis of (Rτ )m , we obtain dimK (Dτ )m = #{standard monomials on X(τ ) of degree m}. This together with the fact that (Dτ )m is generated by standard monomials (of degree m) implies that (Dτ )m has a basis consisting of standard monomials of degree m. (ii) Let f = ai fi , where ai ∈ K, ai = 0 and fi are standard monomials in Dτ . From the definition of Dτ it is clear that any product of standard monomials in Dτ is either zero or a standard monomial. Therefore, using (i) it follows that f m = 0 for every m ≥ 0. This proves that Dτ is reduced. (iii) The equality (∗∗) in the proof of (i) implies that the Hilbert polynomial of X(τ ) is the same as that of Dτ , which proves the assertion. Let T be a subset of Ir,n . By a chain of length m in T , we shall mean a sequence Λ = (λ(0), λ(1), . . . , λ(m)) of elements of T such that λ(i) > λ(i + 1), 0 ≤ i ≤ m − 1. We say that Λ is maximal if it is not a subchain of any chain in T different from Λ. For a chain Λ in Tτ , τ ∈ Ir,n let L(Λ) denote the linear space in P = Proj P defined by the equations xφ = 0, φ ∈ Tτ \ Λ, where Tτ \ Λ denotes the set of elements in Tτ not appearing in Λ. Proposition A.12.1.6 If M(Tτ ) denotes the set of maximal chains in Tτ , then Proj Dτ =



L(Λ) scheme theoretically,

Λ∈M(Tτ )

i.e. the linear spaces L(Λ), Λ ∈ M(Tτ ), are the irreducible components of Proj Dτ . Proof. The ideal of the scheme theoretic union on the right hand side is I=



IΛ ,

Λ∈M(Tτ )

where IΛ is the ideal of L(Λ) (note that IΛ is generated by the set {xφ | φ ∈ Tτ \Λ}). We shall show that I is the ideal defining Proj Dτ , i.e. I is generated by the set of monomials S = {xγ xδ | γ , δ are non-comparable elements of Tτ }. In fact, if γ , δ are non-comparable in Tτ , then both of them together cannot belong to a maximal chain in Tτ . This shows that xγ xδ ∈ IΛ for all Λ ∈ M(Tτ ), and hence S ⊂ I . Now, if F ∈ I , it is easy to see that every monomial of F is in I . Thus, to complete the proof it suffices to show that every monomial F ∈ I is divisible by an element of S. In fact, since F ∈ IΛ for all Λ ∈ M(Tτ ), it is clear that F is not a standard monomial, and hence it is divisible by an element of S. Corollary A.12.1.7 The degree of X(τ ) in the Plücker embedding is equal to the number of maximal chains in Tτ . The proof is immediate from Theorem A.12.1.4 and the above Proposition.

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Stanley-Reisner ring ([106]) ) using Theorem In this subsection, we prove the Cohen-Macaulayness of X(τ A.12.1.4. Given a finite partially ordered set H , let A = K[xα , α ∈ H ], IH the ideal generated by {xγ xδ | γ , δ are non-comparable elements of H }. Then the ring RH := A/IH is called the Stanley-Reisner ring associated to H . Theorem A.12.1.8 Spec Dτ is Cohen-Macaulay. Proof. We have, for H = Tτ , RH = Dτ , and the Cohen-Macaulayness of StanleyReisner ring RTτ follows from [6]. Combining Theorem A.12.1.4 with the above Theorem, we obtain )(= Spec Rτ ) is Cohen-Macaulay. Theorem A.12.1.9 X(τ Using deformation technique and standard monomial basis, DeConcini-Lakshmibai prove the arithmetic Cohen-Macaulayness of Schubert varieties in G/P , P being a maximal parabolic subgroup of classical type. Arithmetic normality As above, let P be a maximal parabolic subgroup of classical type of G; let X be a Schubert variety in G/P . Consider the projective embedding X → P(V ) where V = H 0 (G/P , L)∗ . As a consequence of Theorem A.4.0.1, we have that the restriction map H 0 (P(V ), Lm ) → H 0 (X, Lm ), m ∈ N is surjective. Hence if we know the (geometric) normality of X, then we would obtain the arithmetic normality of X (note that if a projective variety X → Pn is normal, then it is arithmetically normal if and only if the restriction map H 0 (Pn , Lm ) → H 0 (X, Lm ) is surjective for all m, L being the tautological line bundle on Pn ). We  is Cohen-Macaulay. Hence normality of X would follow once we have by [21], X know that X is regular in codimension one, i.e., the singular locus of X has codimension at least two (in view of Serre Criterionfor normality - a ring A is CohenMacaulay if and only if A has S2 and R1 ). But one knows this to be true, thanks to Chevalley (cf. [15]). We have included below a proof (cf. [69]) of this result of Chevalley. We first fix some notation: Let G, B, T be as in § A.1. Let X(τ ) be a Schubert variety in G/B. Let X(w) be a Schubert divisor in X(τ ) moved by a simple root α, i.e., τ = sα w, and l(τ ) = l(w) + 1. Let SL(2, α) denote the copy of SL(2) in G associated to α. Set Bα = B ∩ SL(2, α) Let (∗)

Zw = SL(2, α) ×Bα X(w)

A.12 Arithmetic Cohen-Macaulayness and arithmetic normality

241

i.e., Zw is the quotient variety modulo the equivalence relation in SL(2, α) × X(w): (g, x) ∼ (gb, b−1 x), g ∈ SL(2, α), b ∈ Bα , x ∈ X(w) Note that if we define an action of Bα on SL(2, α) × X(w) by (g, x) · b = (gb, b−1 x), g ∈ SL(2, α), b ∈ Bα , x ∈ X(w) then this action is a free action and Zw is the orbit space (SL(2, α) × X(w))/Bα . We have a canonical morphism (∗∗)

p : Zw → P1 = SL(2, α)/Bα

Note that Zw is the fiber space with fiber X(w), associated to the principal fibration SL(2, α) → SL(2, α)/Bα with structure group Bα . Set (∗ ∗ ∗)

ψ : Zw → X(τ )

Theorem A.12.1.10 Schubert varieties are regular in codimension one. Proof. It suffices to show that given w, w  ∈ W such that X(w  ) is a Schubert divisor in X(w), the point ew is a smooth point of X(w). We prove this by decreasing induction on dim X(w). Since X(w  ) is a Schubert divisor in X(w), we have that w = sα w  for some positive root α. If α is simple, then taking a lift nα in N (T ) for sα ∈ W (= N(T )/T , N(T ) being the normalizer of T ), we have that multiplication on the left by nα defines an automorphism of X(w) which maps ew to ew (note that since sα w < w, X(w) is stable for left multiplication by elements of the minimal parabolic P with SP = {α}). Hence the result follows in this case since X(w) is smooth at ew . Let then α be not simple. Fix a simple root β such that w  < sβ w  . Then the facts that l(sβ w  ) = l(w  ) + 1 = l(w) implies that w < sβ w (for w > sβ w would imply w ≥ sβ w  which is not possible, since w = sβ w  ). Let τ = sβ w. Then by induction hypothesis, X(τ ) is smooth at esβ w ; further, since SL(2, β)ew  X(w), we have that ψ −1 (esβ w ) is a point, namely, the point (sβ , ew ) (here, ψ is as in (***)). Hence by Zariski’s main Theorem, ψ induces an isomorphism in a neighborhood of (sβ , ew ); in particular, (sβ , ew ) is smooth on Zw . This implies that the fiber, say Y of p : Zw → P1 through (sβ , ew ) is smooth at (sβ , ew ). Now multiplication by sβ induces an isomorphism of X(w) onto Y under which ew is mapped onto (sβ , ew ). Hence we obtain that X(w) is smooth at ew . This completes the proof of the Theorem.

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Index Page numbers appear in three fonts, two special fonts in addition to the ordinary one. For example, consider the index entry “extremal weight vectors”. The appearance of the page number 24 means that the definition of extremal weight vectors can be found on that page. As another example, consider the index entry “first basis theorem”. The appearance of the page number 232 means that the statement of the theorem can be found on that page. of even orthogonal Grassmannian 79, 87 space of sections on 79 of Grassmannian 34 space of sections of 78 of symplectic Grassmannian 90 space of sections on 63 arithmetic Cohen-Macaulayness of Grassmannian Schubert varieties 45, 240 of Schubert varieties in classical G/P 240 arithmetic normality 15 of Grassmannian Schubert varieties 240 of Schubert varieties in classical G/P 240

abstract weight of a root system 18 adjoint action 20 admissible elements 66, 67, 82 standard monomials in 67 admissible pairs 173, 221 importance of 221 in symplectic Grassmannian 65, 92, 224 bijection with admissible elements 67, 92 bijection with dual admissible elements 69 bijection with minors p(A, B) 163 Plücker co-ordinate associated to 93 minimal dominating elements of 224 structure of 226–227 trivial 173, 221 weight of 226 affine algebraic set 12 topological dimension of 14 affine algebraic variety 13 coordinate ring of 13 local ring at a point 12, 13 affine morphism 99, 103 affine quotient 98–101 is a categorical quotient 102 is a good quotient 104 universal mapping property of 100 affine space 12 algebra of dual numbers 128, 131, 135 algebraic groups 19, 17–28 ample generator of Picard group in Grassmannian space of sections of 62 of G/P for P maximal parabolic 27

basic invariants 2 for GL action 2, 129 for O action 3, 133 for SL action 5, 141, 141 for SL action on vectors 124 for SL2 adjoint action 182 for SO action 5, 167 straightening relations 170 for Sp action 3, 136 big cell 23-24, SEE ALSO opposite cell birational morphism, sufficiency for 123 book (this one!) organization of 9 why at all? 6 Boolean algebra 211 Borel subgroup 19, 96 branch locus 178

249

250

INDEX

Bruhat (B-orbit) decomposition 24, 34, 47, 87, 90 Bruhat-Chevalley order 24 in terms of Demazure modules 25 in terms of projections 24 in terms of reduced expressions 24 on Weyl group of type Cn 59 on Weyl group of type Dn 76 canonical dual pairs 62, 78 relation with canonical pairs 66 canonical pairs 65 relation with canonical dual pairs 66 Cartan integers 18 categorical quotient 102, 101–103 for G action, where G is GL, SL, O, SL, SL (vector invariants), Sp SEE G invariants sufficient condition for 123 uniqueness of 102 universal mapping property axiom 102 chain lattice 151 character group 19 Chevalley multiplicity mτ (ϕ) 220 CIT SEE classical invariant theory classical invariant theory SEE ALSO first fundamental theorem and second fundamental theorem connection with SMT 1 De Concini-Procesi approach to 3 what is it? 1–2 classical type G/P , Schubert varieties in are arithmetically Cohen-Macaulay 240 are arithmetically normal 240 are normal 240 closed orbits and proper morphisms 107 closed orbits exist 104 cogenerated ideal 52 Cohen-Macaulayness SEE ALSO arithmetic Cohen-Macaulayness of determinantal varieties 51 of Grassmannian Schubert varieties 51 of ladder determinantal varieties 204 of quiver varieties 205 of Schubert varieties via SMT 8

of skew symmetric determinantal variety 82 of SL invariants 158 of SO invariants 180 of sundry other algebras 152, 155, 174, 175, 176, 240 of symmetric determinantal variety 65 Serre-Grothendieck criterion for 45 techniques of proving 8–9 cohomology vanishing theorems 43, 41–44 complete variety 107 basic properties of 107 completely reducible group 96 e.g., torus 19 iff linearly reductive 96 completely reducible module 96 D, the “ordered” set (SO action) 171 as a doset 175 algebra R(D) associated to SEE R(D), the algebra, (SO action) D, the poset (SL action) 147 algebra R(D) associated to SEE R(D), the algebra (SL action) as a distributive lattice 152–153 rank of 153 De Concini-Procesi basis for skew symmetric determinantal variety 83 for symmetric determinantal variety 69 deformation technique 8, 236, 240 Demazure character formula 219, 226 Demazure module as dual of space of sections 27 Demazure module Vλ,τ 25, 225 (dual SMT) basis for 229, 232 character of SEE Demazure character formula Deodhar’s inequality 190 determinantal varieties Dt 49, 47–54 are Cohen-Macaulay 51 are normal 51, 132 as categorical quotients for GL action 129 as quiver varieties 208 defining ideals of 52

INDEX Gröbner basis for 53 dimensions of 51 identified as opposite cells 50, 138 singular loci of 54 singular locus of 195 standard monomial theory for 51, 138 straightening relations on 139 comment about 140 homogeneity of 139 discrete poset algebra K{P } 174 distributive lattice 150, 211 algebra R(L) associated to 151 condition on straightening relations 151 is a flat deformation of A(L) 152 is normal and Cohen-Macaulay 152 algebra A(L) associated to 150 is a flat degeneration of R(L) 152 spectrum is a (normal) toric variety 152 chain product lattice 211 ideal I (L) associated to 150, 213 is a binomial prime ideal 152 is toric 213 is sublattice of Boolean algebra 212 dominant characters 20 dominant weight 19, 27 dominant Weyl chamber 19 doset 173 doset algebra 173 axioms for straightening relation 174 discrete (denoted K{D}) 174 criterion for Cohen-Macaulayness of 174 standard monomials in 174 Dτ , the discrete algebra 236 irreducible components of Proj of 239 is a degeneration of cone over X(τ ) 237 is Cohen-Macaulay 240 is reduced 238 standard monomial basis for 238 dual admissible elements 69 standard monomials in 69 dual canonical pair 82

251

dual Weyl module 26, 63, 79 Dynkin diagram 18 elementary double move 65 embedding dimension 11 ε-degree SEE I (n), v-degree of an element in even orthogonal Grassmannian 77, 86, 223 W/WP of, and other details 223 ample generator of Picard group 79, 87 space of sections on 79 is the homogeneous space SO(2n)/Pn 86 opposite cell identified with Sk Mn 80 Schubert varieties in 86 basis for tangent space at eτ 193 basis of standard monomials for 88 Bruhat (B-orbit) decomposition for 87 indexing set J (n) for 87 standard monomials in 88 w-dominated 88 torus fixed points of 87 extremal weight vectors 24, 27 extremal weights 27, 220 factoriality 14, 15 at a point 14 of Grassmannian Schubert varieties 46 finite generation of rings of invariants 2, 98 first basis theorem 221, 232, 230–232 first fundamental theorem 3 for GL invariants 132 for O invariants 135 for SL invariants 158 for SL vector invariants 128 for SO invariants 180 for Sp invariants 136 SMT approach to 2 what is it? 1 flag variety G/B 23, SEE ALSO flag variety (generalised) G/Q canonical neighborhood at eτ of 188

252

INDEX

equations defining Y (w, τ ) in 189, 191 flat toric degeneration of 217 line bundles on 25–27 opposite cell and twisted opposite cell 23 Picard group of 25–26 Schubert varieties X(w) in 224 are regular in codimension 1 241 basis for tangent space at eτ 192 criteria for smoothness at eτ 192 dimension of tangent space at eτ 191 non-homogeneity of pθ on opp. cell 198 singular loci of (in SLn /B) 193–194 smoothness criterion for (in SLn /B) 193 standard monomial theory for 225 tangent space T (w, τ ) at eτ 188 twisted opposite cells Y (w, τ ) of 189 flag variety (generalised) G/Q 22, SEE ALSO flag variety G/B, Grassmannian, even orthogonal Grassmannian, and symplectic Grassmannian flags of subsets 206 free action 104 Frobenius splitting 8, 9, 236 fundamental theorems of invariant theory SEE first fundamental theorem and second fundamental theorem fundamental weights 20 form basis for weight lattice 18 minuscule 220 examples of 222, 223 of classical type 220 table of 221 quasi-minuscule 220 G-linear actions 108 G-module 95 G/P -II Lemma, application of 155 geometric invariant theory 95–120 geometric quotient 104 example: G/H 104 for set X s of stable points 115

geometrically reductive group 96, SEE ALSO reductive group GL invariants basic invariants ϕij 2, 129 definition of the action 2, 128 equal functions on det. variety Dt 129 theorems about 132 good quotient 103, 103–108 for set X ss of semistable points 114 is a categorical quotient 104  107 is a geometric quotient on Xmax Gröbner basis 53 for determinantal varieties 53 for opposite Schubert cell 53 Grassmannian Gd,n 29, 29–46, 47–49 as an orbit space 30 as the homogeneous space SLn /Pd 33 cone over 33 as a categorical quotient 103 as categorical quotient (K d )⊕n //SL(d) 126 opposite cell of 62, 78 Picard group, ample generator of 34 space of sections on 62, 78 Plücker embedding 30–31 homogeneous co-ordinate ring of 31, 47 Schubert varieties in SEE Grassmannian Schubert varieties torus fixed points of 34 Grassmannian Schubert varieties 34 are arithmetically Cohen-Macaulay 45, 240 are arithmetically normal 46 are Cohen-Macaulay 51 are indexed by Id,n 34, 47 are normal 46, 51 are regular in codimension 1 46 Bruhat (B-orbit) decomposition of 34, 47 cohomology vanishing theorems 43, 41–44 criterion for smoothness 195 defining equations for 38–39, 48 degeneration of cone over to Dτ 237 degree in Plücker embedding 238, 239

INDEX dimensions of 35, 47, 195 factoriality & projective factoriality 46 intersections of 35, 41 are reduced 40 irreducible components of sing. locus of 195 multiplicity mτ (w) at a point eτ 200 pθ on twisted opposite cell U − eτ 197–199 degree dτ (θ ) of 199 examples of 197, 198 homogeneity of 199 Plücker co-ordinate evaluated on 47 Plücker embedding SEE Grassmannian Schubert varieties, standard monomials on singularities of 195–200 standard monomials on 36 form a basis 37, 47 generation by 37 linear independence of 36 straightening relations on 37, 47, 236–238 the Pieri formula 40 unions of 41, 42 are reduced 40 standard monomial basis for 39–40 standard monomials on 39 H , the index set 141 is a ranked poset 142 partial order upon 142 height, of a prime 11 Hilbert’s basis theorem 2 Hilbert’s fourteenth problem 97 Hilbert’s Nullstellensatz 12 Hilbert-Mumford criterion 118, 117–120 for K ∗ 119 homogeneous co-ordinate ring 14 homogeneous coordinates 14 hyperplane 12, 14 hyperplane bundle 15, 16 hypersurface 12, 14 ideal of vanishing 13 Id,n , the poset 30

253

identified as minimal representatives 34 indexes Schubert varieties 34 is a distributive lattice 215 so are ideals in it 215 join and meet of elements in 215 I (n), the index set 90 v-degree of an element in 91 partial order on 90 indecomposable G-module 27 initial term 53 integrality axiom for a root system 17 integrally closed 12 intersections of Schubert varieties 35, 40, 41 invariants SEE appropriate group, e.g., SL invariants irreducibility of ladder det. varieties 204 irreducible G-modules 96 geometric realization of 26–27 isotropic subspace 86, 89 Iwahori decomposition 120 Jacobian criterion for smoothness 189 J (n), the index set 87 partial order on 87 join (x ∨ y of lattice elements) 210 Krull dimension

11

L-linear actions 117 ladder determinantal varieties 203, 202–204 are irreducible, normal, C-M 204 relation with opposite cells 204 singular loci of 204 Lagrangian Grassmannian SEE symplectic Grassmannian lattice 150, SEE ALSO chain lattice, distributive lattice chain product lattice 211 distributive SEE distributive lattice isomorphic to another 211 sublattice 211 length of a Weyl group element 18 Levi subgroup 22 line bundles SEE ALSO Picard group ample 26, 27

254

INDEX

on flag variety G/B 25–27 space of sections 26, 26–27, SEE ALSO standard monomial theory, main theorem of very ample 26, 27 linear action 96 G-linear action 108 linearly reductive group 96, 96 iff completely reducible 96 local property 12 locally closed subvariety 16 LS path 221 maximal parabolic subgroup SEE fundamental weights maximal representatives 22 index Schubert varieties 23 M(D), the algebra grading for 147 standard basis for 149 M(D), the algebra (SL action) 147 meet (x ∧ y of lattice elements) 210 minimal representatives 22 index Schubert varieties 23 minuscule G/P 200 list of 200 Schubert varieties in basis for tangent space at eτ 192 flat toric degenerations of 217 homogeneity of ideal of opp. cell 201 multiplicity mτ (w) at a point eτ 202 tangent space at eτ to 200 moduli space M 181, 180–181 multiple point SEE singular point Mumford’s conjecture 98 Nagata’s counterexample 97 negative roots 18 Noether Normalisation Lemma 11 non-singular point 188 non-smooth point SEE singular point normality 12, 13, 15, SEE ALSO arithmetic normality, 44, SEE ALSO arithmetic normality of determinantal varieties 51, 132 of Grassmannian Schubert varieties 46, 51

of ladder determinantal varieties 204 of quiver varieties 205 of Schubert varieties in classical G/P 240 of skew symm. det. variety 82 of sundry other algebras 152, 155, 179 of symm. det. variety 65, 135 Serre’s criterion for 46, 160, 179, 240 O invariants basic invariants ϕij 3, 133 definition of the action 3, 133 equals ring of symm. det. variety 133, 166 theorems about 135 one (in a lattice) 211 opposite Borel subgroup 23 opposite cell 23-24, SEE ALSO opposite Schubert cell in even orthogonal Grassmannian as σ fixed points of opposite cell of Grassmannian 80 identified with Sk Mn 80 in Grassmannian 48–49, 62, 78 in symplectic Grassmannian 162–163 as fixed points of involution σ of opposite cell of Grassmannian 63 identified with Sym Mn 64, 163 opposite Schubert cell (C AUTION : not a “cell”), 24, SEE ALSO opposite cell determinantal variety as example of 50 in Grassmannian Schubert variety 49 defining ideal of 52 Gröbner basis for 53 skew symm. det. variety as example of 81 symm. det. variety as example of 64 organization of the book 9 orthogonal flag variety Schubert cell in 74 Schubert varieties in, relation with Schubert varieties in flag variety 74 orthogonal group O(V ) 3

INDEX parabolic subgroups 21 partial flag variety G/Q 206 Schubert varieties in 206 pattern avoidance 193 Pfaffian 79, 88, 136 Pfaffian qi 79, 88 Picard group SEE ALSO line bundles and ample generator of Picard group of flag variety G/B 25–26 Pieri formula 40 Plücker co-ordinate associated to an admissible pair 93 Plücker co-ordinate pi 30 Plücker embedding SEE Grassmannian Gd,n and Grassmannian Schubert varieties Plücker relations 31–33 generate ideal of presentation for SL vector invariants 128 properties of 216 polystable point 109, 110 positive roots 18 relative to a Borel subgroup 20 P -Q stability of BwB 194 principal fiber bundle 104 procedure, used in proof of SMT 228 projective algebraic set 14 projective algebraic variety 14 cone over 15, 15 projective factoriality 15 of Grassmannian Schubert varieties 46 projective normality SEE arithmetic normality projective quotient 114–116 projective space 14 proper morphism 107 and closed orbits 107 quadrics in Pn−1 116 quasi-affine variety 16 quasi-finite morphism, sufficiency for 122 quasi-projective variety 16 quiver representations, variety of 205 quiver varieties 205, 205–209 are opposite cells as schemes 209 Cohen-Macaulayness of 205

255

examples of 207–210 determinantal varieties 208 small generic case of 207–208 universal degeneracy schemes 208 variety of complexes 208–210 normality of 205 vanishing ideals of 209 are prime 209 radical R(G) of an algebraic group 19, 96 radical ideal 13 rank of a root system 18 of an algebraic group 19 rational function field 12, 13 rational representation 95 R(D), the algebra (SL action) 147 properties of 154–155 standard basis for 148–149 R(D), the algebra (SO action) 171 is a doset algebra 175 is Cohen-Macaulay 175 standard monomial basis for 171, 172 reduced expression 18 reduced word 18 reductive group 96 acting on varieties SEE affine quotient AND projective quotient is geometrically reductive 98 is linearly reductive in characteristic 0 96 reflection (relative to a vector) 17 regular local ring 11 regular ring 11 regular weight 19, 27 regularity in codim 1 of Schubert var. 46, 241 reshuffle 174 root lattice 18, 20 root operators eα , fα 226 acting on the Weyl group 227 root space 21 root subgroup 21 root system SEE ALSO Weyl group, root lattice, weight lattice, and such other terms abstract 17-19

256

INDEX

basis for 18 of a parabolic subgroup 21 of a semisimple group 20 of type A2 18 of type An−1 (SLn ) 20 of type An−1 (group SLn ) 222 of type Cn (group Sp(2n)) 56–57, 224 of type Dn (group SO(2n)) 72–73, 223 rank of 18 R(w), the algebra 161 presentation for 162 standard monomial basis for 162 standard monomial in 162 straightening relations in 162 S, the algebra (SL action) 141 Cohen-Macaulayness of 155, 150–155 dimension of 155 equality with SL invariants 155 isomorphism with S(D) 149 normality of 155, 150–155 presentation for 150 standard basis for 149 standard monomials in 142 linear independence of 145 straightening relations on 143 S, the algebra (SO action) 168 equals SO invariants 179, 176–179 is birational to SO invariants 177 is Cohen-Macaulay 176, 173–176 is isomorphic to S(D) 173 is normal 179 is regular in codimension 1 178 presentation for 173 standard basis for 172, 173 standard monomials in 168 linear independence of 169 straightening relations on 170 Schubert cell 23, 34 in orthogonal flag variety 74 in symplectic flag variety 59 Schubert varieties 23 dimensions of 23 equations defining 27–28 in Grassmannians SEE Grassmannian Schubert varieties

in other flag varieties SEE appropriate flag variety, Schubert varieties in indexed by W min or W max 23 S(D), the algebra (SL action) 147 isomorphism with S 149 standard basis for 149 S(D), the algebra (SO action) 171 is isomorphic to S 173 standard monomial basis for 172 second fundamental theorem 3 for GL invariants 132 for O invariants 135 for SL invariants 158 for SL vector invariants 128 for SO invariants 180 for Sp invariants 136 SMT approach to 2 what is it? 2 semi-continuity theorem 105 semisimple group 19 semistable point 109, 110 for SL action on vectors 124 w.r.t. a line bundle 117 separated morphism 107 simple reflections 18 simple roots 18 relative to a Borel subgroup 20 singular locus 188 of determinantal varieties 195 of determinantal variety 54 of Grassmannian Schubert varieties 195–200 of ladder determinantal varieties 204 of Schubert varieties in SLn /B 193–194 of variety of complexes 210 singular point 188 skew symmetric determinantal variety Dt (Sk Mm ) as opposite Schubert cell 81 co-ordinate ring of De Concini-Procesi basis for 83 equals Sp invariants 136 standard monomial basis for 83 dimension of 82 is Cohen-Macaulay 82 is normal 82

INDEX Schubert varieties in, relation with Grassmannian Schubert varieties 81 summary of theorems 71 SL invariants basic invariants ϕij , u(I ), ξ(J ) 5 basic invariants p(A, B), u(I ), ξ(J ) 141 definition of the action 5 equality with S 155 theorems about 158 idea of proof of 5 summary of 137–138 SL2 adjoint invariants 181–185 basic invariants U (i, j ), U (i, j, k) 182 standard monomials 183-184 form a basis 185, 185 SLn /B SEE flag variety G/B SL vector invariants basic invariant fτ 124 definition of action 124 equals ring of cone over Grass. 103, 126 stable, semi-stable points for action 116, 124 theorems about 128 smooth point 13, 188 Jacobian criterion for 189 smooth variety 13, 15, 188 SMT SEE standard monomial theory SO invariants basic invariants ϕij , u(I ) 5 basic invariants p(A, B), u(I ) 167 definition of the action 5 equals S 179 is birational to the algebra S 177 relation with O invariants 177 theorems about 179–180 idea of proof of 5–6 summary of 159–160 Sp invariants basic invariants ϕij 3, 136 definition of the action 3, 136 equals ring of skew symm. det. variety 136 theorems about 136 special linear group SL(V ) 222

257

choice of max. torus & Borel subgp 222 invariants SEE SL invariants special orthogonal group SO(V ) 222 Chevalley basis for Lie algebra of 73 choice of max. torus & Borel subgp 71, 86, 222 details of root system, etc. 223 invariants SEE SO invariants stable point 109, 110 characterizations of 113, 116 for SL action on vectors 124 w.r.t. a line bundle 117 standard monomial theory applications of 187–217 summary of 187 Cohen-Macaulayness via SMT 8 connection with CIT 1 approach to first and second fundamental theorems 2, 3–4, 5–6 for GL invariants 132 for O invariants 135 for SL invariants 158 for SL vector invariants 128 for SO invariants 180 for Sp invariants 136 for a classical type G/P 5 for a minuscule G/P 4–5, 88–89 for determinantal varieties 51, 138, 139 for even orthogonal Grassmannian 88, 86–88 for Grassmannian 36–39 for Grassmannian (union of Schubert varieties) 39–40 for skew symm. det. varieties 83 for SL2 adjoint invariants 185 for sundry other algebras 148–149 for symmetric determinantal varieties 67 for symplectic Grassmannian 93, 89–93 main theorem of 225, SEE ALSO first basis theorem straightening relations in, a discussion 7–8 what is it? brief history of 1, 2, 4–5, 7 standard monomials

258

INDEX

in symplectic Grassmannians SEE symplectic Grassmannian Stanley-Reisner algebra SEE discrete poset algebra Stanley-Reisner ring 240 straightening relations 3, 237–238 in SMT, general discussion 7–8 on a determinantal variety 139 comment about 140 homogeneity of 139 on basic SL invariants 143 on basic SO invariants 170 on Grassmannian Schubert varieties 37, 47, 236–237 on sundry other algebras 151 strongly stable point SEE stable point sublattice 211 symmetric determinantal variety Dt (Sym Mm ) 164 as opposite Schubert cell 64, 164 co-ordinate ring of as a homogeneous localization 164 De Concini-Procesi basis for 69 equals O invariants 133, 166 presentation for 167 standard monomial basis for 67, 166 standard monomials in 166 straightening relations in 166 dimension of 65 is Cohen-Macaulay 65 is normal 65, 135 Schubert varieties in, relation with Grassmannian Schubert varieties 64 singular locus of 165 summary of theorems 55 symplectic flag variety Schubert cell in 59 Schubert varieties in, relation with Schubert varieties in flag variety 60 symplectic Grassmannian 61, 89, 161, 223 W/WP of, and other details 223 admissible pairs in 224, SEE ALSO admissible pairs ample generator of Picard group 90 space of sections on 63

Bruhat (B-orbit) decomposition 90 is the homogeneous space Sp(2n)/Pn 89 opposite cell identified with Sym Mn 64 Schubert varieties in 89, 161 basis for tangent space at eτ 193 basis of standard monomials for 93 indexing set I (n) for 90 standard monomials in 93 w-dominated 93 standard tableaux 92 w-dominated 92 torus fixed points of 90 symplectic group Sp(V ) 3, 55, 223 Chevalley basis for Lie algebra of 57 choice of max. torus & Borel subgp 56, 89, 90 choice of max. torus & Borel subgp of 223 details of root system, etc. 223 invariants SEE Sp invariants T -stable curves 190 tableau character formula 226 tableaux 224 pointed 227 standard 224 dominated by w 224 in symplectic Grassmannian SEE symplectic Grassmannian weight of 226 tangent space SEE Zariski tangent space tautological line bundle SEE hyperplane bundle toric ideal 212 example: prime binomial ideal 214, 215 spanned by binomials 212 toric variety (affine) 212 remark about definition 212 toric variety is a quotient 104 torus 19, 96 Uβ , root subgroup 21 unions of Schubert varieties SEE Grassmannian Schubert varieties, unions of

INDEX universal degeneracy loci 205 universal degeneracy schemes 208 universally closed morphism 107 v-degree SEE I (n), v-degree of an element in vanishing theorems SEE cohomology vanishing theorems variety of complexes 208, 209 as an opposite cell 210 irreducible components of 210 singular loci of 210 weight lattice 18, 20 weight space 19 Weyl chamber 19 Weyl group SEE ALSO root system, Weyl chamber, and such other terms of a parabolic subgroup 22 of a root system 18-19 of a semisimple group 20

259

of type Cn (group Sp(2n)) 56, 223 W/WP of 59 Bruhat-Chevalley order 59 formula for length of an element 58 of type Dn (group SO(2n)) 72, 223 W/WP of 75 Bruhat-Chevalley order 76 formula for length of an element 73 Weyl involution 25, 27 Weyl module Vλ 25, 27, 225, SEE ALSO Demazure module Witt’s theorem 86, 89 Young diagram 35, 195 Young subgroup 206 Zariski tangent space Tx (X) 187 Zariski topology 12, 14 Zariski’s main theorem 4, 121 application of 179 zero (in a lattice) 211 ZMT SEE Zariski’s main theorem

Index of notation ≺, monomial order 53 , partial order on minors 51 , comparison order on Hm 165 1-PS, one parameter subgroup 118

fτ , basic invariant (for SL action) 124 fτ,ϕ , restriction of Plücker co-ordinate pτ,ϕ to opposite cell 163 G, same as SO(V ) 71 G, same as Sp(V ) 55 Gd,n , Grassmannian 29 g, Lie algebra of G 20 gα , root space 21 Gn (V ), Grassmannian 86 Gx , isotropy subgroup at x 249

A, admissible elements in Ir,n × Ir,n 66 A, set of admissible elements 82 A, set of admissible pairs 161 a(D), homogeneous ideal (SL action) 147 a(D), homogeneous ideal (SO action) 171 A(L), algebra 150 An,2n , admissible elements in In,2n 67 An,2n , the subset of In,2n 82 An,2n , set of dual admissible elements 69 B, Borel subgroup such that B ⊇ T Bu , unipotent part of B 21

H , same as SL(V ) 55, 71 Hm , set 165 H 0 (G/B, L(λ)), space of sections 26 Hr,d , poset 139 H , set with “comparison” order 168 i, Weyl involution 25 IA , toric ideal 212 Id,n , the poset 30 I (n), the index set 90 It , ideal defining Dt 49 I (w, τ ), ideal defining Y (w, τ ) in Oτ− 189 I (Y ), ideal of vanishing of Y 13

20

C(n1 , . . . , nd ), chain product lattice 211 C(w), Schubert cell 23 D, “ordered” set (SO action) 171 D, poset (SL action) 147 Ds,t (L), ladder determinantal variety 202, 203 Dt , determinantal variety 49 Dτ , algebra 236 dτ (θ ) degree of pθ |U − eτ 199 Dt (Sk Mn ), skew symm. det. variety 80 Dt (Sym Mn ), symm. det. variety 164 Dt (Sym Mn ), symm. det. variety 64 E, matrix 55 eα , root operator 226 E, matrix 71 ε, the element (1, . . . , n) fα , root operator

I, even orthogonal Grassmannian J √(n), index set 87 J , radical of J 13

86

K{D}, discrete doset algebra 174 K{P }, discrete poset algebra 174 K(V ), field of rational functions 13 K[X]SLn (K) , ring of invariants 155 Λ, weight lattice 18 Λ+ , set of dominant weights 19 Lie G, Lie algebra of G 20 Lie Sp(2n), Lie algebra 57 M, moduli space 180 M2 , moduli space 180 M(D), algebra (SL action)

87, 91

226

260

147

INDEX OF NOTATION Mn (V ), the symplectic Grassmannian 89 Mr,s,t , set of standard monomials 183 mτ (ϕ), Chevalley multiplicity 220 mτ (w), multiplicity of X(w) at eτ 197 μ(λ, x), integer 118 Nr,s,t , set of standard monomials N (T ), normalizer of T 20

184

− OG , opp. cell in even ortho. Grassmannian 78 − OG , opp. cell in symp. Grassmannian 61 − OH , opposite cell in Grassmannian 60, 77  , map 184 − O − or OG , opposite cell 163 − O , opposite cell 23 Oτ− , neighbourhood around eτ 188 Oτ− , twisted opposite cell 24

p(A, B), basic invariant (SL action) 141 p(A, B), minor 163 p(A, B), standard monomial 183 p(A, B), basic invariant (SO action) 167 p(α), standard monomial 183 P (D), polynomial algebra (SL action) 147 P (D), polynomial algebra (SO action) 171 φ, m-tuple 164 φ, largest d-tuple 50 ϕij , basic invariant (GL action) 129 ϕij , basic invariant (O action) 133 ϕij , basic invariant (Sp action) 136 Pn , max. par. subgroup of G = SO(V ) 77 Pn , max. par. subgroup of G = Sp(V ) 60 P , poset 175 p(σ ), SMT basis element 224 pi , Plücker co-ordinate 30 pw , extremal weight vector 27 qi , Pfaffian 79, 88 qi (Y ), Pfaffian 79

261

Qn , max. par. subgroup of H = SL(V ) 60, 77 R(D), algebra (SL action) 147 R(G), radical 96 R(G), radical of G 19 R(L), algebra associated to L 151 Rp , localisation 11 R(τ ), homogeneous co-ordinate ring of X(τ ) 36 S, algebra (SL action) 141 S, algebra (SO action) 168 sα , reflection associated to α 17 S(D), algebra (SL action) 146 S(D), algebra (SO action) 171 σ , involution 71 σ , involution 55 Sing X, singular locus of X 188 Sk Mn , n × n skew-symm. matrices 71 SO(V ), special orthogonal group 71 Sp(V ), symplectic group 55 Sym Mn , space of n × n symmetric matrices 55 T , maximal torus 19 θ , bijection 51 θ , bijection 165 θ , bijection 62, 79 Tτ , subset of Id,n 236 T (w, τ ), tangent space to X(w) at eτ 188 u(I ), basic invariant (SL action) 141 U (g), universal enveloping algebra 24 UZ (g), Kostant Z-form 24 UZ+ (g), positive part of UZ (g) 24 u(I ), standard monomial 183 U (i, j ), basic invariant (SL2 adj. action) 182 U (i, j, k), basic invariant (SL2 adj. action) 182 u(I ), basic invariant (SO action) 167 U − , unipotent subgroup 188 U − eτ , neighborhood around eτ 188 u(σ ), basis element 229 Vχ , weight space Vλ , Weyl module

19 25, 225

262

INDEX OF NOTATION

V (λ) SAME AS Vλ Vλ,τ , Demazure module 25, 225 Vλ (τ ) SAME AS Vλ,τ V+ , projective variety 14 v(σ ), basis element 230 W , Weyl group 18 min WQ SAME AS WQ WQ , Weyl group of Q 22 max , maximal representatives WQ

 Xmax 106 X ps , set of polystable points 110 X s , set of stable points 110 X s (L), set of stable points w.r.t. L 117 X ss , set of semistable points 110 X ss (L), set of semistable points w.r.t. L 117 X(T ), character group of T 19 X(w), Schubert variety 23, 224

22

 , fundamental weight 18 Λ(x, y), admissible pair 92

Yφ , opposite cell 50 Y (w), opposite Schubert cell 24 Y (w), opposite cell 164 Y (w, τ ), twisted opp. cell at eτ 189

X G , fixed point set 95 X//G, categorical quotient 102 ξ(I ), basic invariant (SL action) 141 x ∨ y, join of x and y 210 x ∧ y, meet of x and y 210

Z, subgroup of H = SL(V ) 60, 77 Z ps , set of polystable points 109 Z s , set of stable points 109 Z ss , set of semistable points 109

min , minimal representatives WQ

22

Author index

This index was machine generated using the authorindex package. The numbers against the name of an author indicate the pages on which there appears a citation to a paper by the author. Admittedly this index is an overkill since the numbers of the pages on which a bibliographic entry is cited appear alongside the entry in the bibliography. Abeasis, S. 205 Abhyankar, S. S. 204 Aigner, M. 211 Billey, S. 187, 193, 194, 200 Björner, A. 240 Borel, A. VII, 17, 19, 21, 22, 33, 95, 98, 104, 107, 118 Bourbaki, N. 57, 73, 162, 188, 220 Brion, M. 27, 193, 219, 226 Buch, A. S. 205 Caldero, P. 8, 137, 187, 210, 236 Carrell, J. B. 190, 193 Chevalley, C. 24–26, 240 Chirivì, R. 8, 137, 187, 210, 236 Cox, D. A. 104 De Concini, C. 6, 8, 85, 93, 137, 159, 160, 162, 173–176, 205, 236, 240 Del Fra, A., see Abeasis, S. 205 Dlab, V. 205 Dolgachev, I. VII, 6 Doubilet, P. 8, 52, 140 Eagon, J. A., see Hochster, M. 6, 205 Eisenbud, D. VII, 11, 152, 196, 197, 214

Eisenbud, D., see De Concini, C. 8, 137, 176 Fogarty, J. 96 Fogarty, J., see Mumford, D. VII, 95, 98, 104, 180 Fulton, W. 104, 205, 208, 212 Fulton, W., see Buch, A. S. 205 Gabriel, P., see Dlab, V. 205 Gonciulea, N. 8, 137, 151, 187, 202, 204, 210, 213, 217, 236 Goodman, R. VII, 6 Haboush, W. J. 2, 98 Hartshorne, R. VII, 11, 15, 26, 41, 46, 120, 197 Herzog, J. 53 Hibi, T. 152, 214, 216 Hochster, M. 6, 205 Hodge, W. V. D. 1, 2, 7, 29, 36 Humphreys, J. E. 17–19, 21, 22, 95 Huneke, C. 8, 137, 236 Jantzen, J. C. 17, 26, 27 Kassel, C. 194 Kempf, G. 212

264

Author index

Kempf, G. R. 28 Kirwan, F., see Mumford, D. VII, 95, 98, 104, 180 Kleiman, S. L. 6 Knudsen, F. F., see Kempf, G. 212 Kostant, B. 24 Kraft, H.-P. VII, 6 Kumar, S. 193 Kumar, S., see Brion, M. 27, 219, 226 Kuttler, J., see Carrell, J. B. 193 Lakshmibai, V. VII, 1–7, 47, 55, 56, 65, 71, 80, 85, 93, 137, 150, 159, 162, 165, 167, 169, 170, 175, 187, 191, 193–195, 200–202, 205, 209, 210, 219, 224–226, 240 Lakshmibai, V., see Billey, S. 187, 193, 200 Lakshmibai, V., see De Concini, C. 6, 8, 137, 159, 160, 162, 173–175, 236, 240 Lakshmibai, V., see Gonciulea, N. 8, 137, 151, 187, 202, 204, 210, 213, 217, 236 Lakshmibai, V., see Huneke, C. 8, 137, 236 Landolfi, J., see Kleiman, S. L. 6 Lascoux, A., see Kassel, C. 194 Littelmann, P. 7, 219, 221, 226 Littelmann, P., see Lakshmibai, V. VII, 6, 7 Magyar, P., see Lakshmibai, V. 202, 205, 209, 210 Manivel, L. 194 Matsumura, H. 2, 6, 11, 45 Mehta, V. B. 9, 137, 160, 181 Mulay, S. B. 204 Mumford, D. VII, 11, 12, 95, 98, 101, 104, 121, 176–180, 196 Mumford, D., see Kempf, G. 212 Musili, C. 4, 6–9, 29, 36, 44, 46, 205, 208, 210 Musili, C., see Lakshmibai, V. 1, 5, 7, 85, 93, 162, 175, 226 Nagata, M. 2, 96–98 Narasimhan, M. S. 180 Newstead, P. E. VII, 95, 99, 108, 113, 117 Olver, P. J. VII, 6

Procesi, C. 8, 185 Procesi, C., see De Concini, C. 8, 137, 176 Procesi, C., see Kraft, H.-P. VII, 6 Proctor, R. A. 59, 60, 76 Raghavan, K. N. 7, 219, 226 Raghavan, K. N., see Lakshmibai, V. 5, 159, 167 Rajeswari, K. N., see Lakshmibai, V. 7, 80, 193 Ramadas, T. R., see Mehta, V. B. 9, 137, 160, 181 Ramanan, S., see Narasimhan, M. S. 180 Ramanathan, A. 8, 27, 236 Ramanathan, A., see Kempf, G. R. 28 Ramanathan, A., see Mehta, V. B. 137 Reisner, G. A. 240 Reutenauer, C., see Kassel, C. 194 Rosenlicht, M. 104 Rota, G.-C., see Doubilet, P. 8, 52, 140 Saint-Donat, B., see Kempf, G. 212 Sandhya, B., see Lakshmibai, V. 193, 194 Sankaran, P., see Lakshmibai, V. 5, 159, 167 Sankaran, P., see Raghavan, K. N. 7, 219, 226 Schlessinger, M. 180 Serre, J.-P. 43, 44 Seshadri, C. S. 4, 7, 85, 88, 95, 120, 201 Seshadri, C. S., see Lakshmibai, V. VII, 1–7, 47, 55, 56, 71, 85, 93, 162, 165, 175, 187, 191, 219, 224–226, 240 Seshadri, C. S., see Musili, C. 205, 208, 210 Shafarevich, I. R., see Popov, V. L. VII, 6 Shukla, P., see Lakshmibai, V. 5, 137, 150, 159, 167, 169, 170 Spanier, E. H. 41 Srinivas, V., see Mehta, V. B. 9 Stein, J., see Doubilet, P. 8, 52, 140 Steinberg, R. 161 Strickland, E., see De Concini, C. 205 Sturmfels, B. 152, 212 Sturmfels, B., see Eisenbud, D. 152, 214 Trivedi, V., see Mehta, V. B. 9, 137 Trung, N. V., see Herzog, J. 53 Vinberg, E. B., see Popov, V. L. VII, 6

Pedoe, D., see Hodge, W. V. D. 1, 2, 7, 29, 36 Popov, V. L. VII, 6, 98

Wachs, M., see Björner, A. 240

Author index Wallach, N. R., see Goodman, R. VII, 6 Warrington, G. S., see Billey, S. 194 Weyl, H. 1, 2, 121, 124, 126, 128, 129, 132, 136

Weyman, J., see Lakshmibai, V. 195, 200–202

265