Standard Monomial Theory (eBook)

Preface 6
Contents 8
1 Introduction 14
1.1 The subject matter in a nutshell 14
1.2 The subject matter in detail 15
1.3 Why this book? 19
1.4 A brief history of SMT 20
1.5 Some features of the SMT approach 20
1.6 The organization of the book 22
2 Generalities on algebraic varieties 23
2.1 Some basic definitions 23
2.2 Algebraic varieties 24
3 Generalities on algebraic groups 29
3.1 Abstract root systems 29
3.2 Root systems of algebraic groups 31
3.3 Schubert varieties 34
4 Grassmannian Variety 41
4.1 The Pl cker embedding 41
4.2 Schubert varieties of Gd, 46
4.3 Standard monomial theory for Schubert varieties in Gd, 48
4.4 Standard monomial theory for a union of Schubert varieties 51
4.5 Vanishing theorems 53
4.6 Arithmetic Cohen-Macaulayness, normality and factoriality 56
5 Determinantal varieties 59
5.1 Recollection of facts 59
5.2 Determinantal varieties 61
6 Symplectic Grassmannian 67
6.1 Some basic facts on Sp(V ) 68
6.2 The variety G/ Pn 72
7 Orthogonal Grassmannian 82
7.1 The even orthogonal group SO(2n) 82
7.2 The variety G/ Pn 88
8 The standard monomial theoretic basis 95
8.1 SMT for the even orthogonal Grassmannian 96
8.2 SMT for the symplectic Grassmannian 99
9 Review of GIT 104
9.1 G- spaces 104
9.2 Affine quotients 107
9.3 Categorical quotients 110
9.4 Good quotients 112
9.5 Stable and semi-stable points 117
9.6 Projective quotients 123
9.7 L- linear actions 126
9.8 Hilbert-Mumford criterion 126
10 Classical Invariant Theory 130
10.1 Preliminary lemmas 130
10.2 SLd( K)- action 133
10.3 GLn( K)- action: 137
10.4 On( K)- action 141
10.5 Sp2 145
(K)- action 145
11 SLn( K)- action 146
11.1 Quadratic relations 147
11.2 The K- algebra S 149
11.3 Standard monomials in the K- algebra S 151
11.4 Normality and Cohen-Macaulayness of the K- algebra S 159
11.5 The ring of invariants K[X] 164
12 SOn( K)- action 168
12.1 Preliminaries 169
12.2 The algebra S 176
12.3 The algebra S(D) 178
12.4 Cohen-Macaulayness of S 182
12.5 The equality RSOn( 185
= S 185
12.6 Application to moduli problem 189
12.7 Results for the adjoint action of SL2( K) 190
13 Applications of standard monomial theory 195
13.1 Tangent space and smoothness 195
13.2 Singularities of Schubert varieties in the flag variety 198
13.3 Singular loci of Schubert varieties in the Grassmannian 203
13.4 Results for Schubert varieties in a minuscule G/P 208
13.5 Applications to other varieties 210
13.6 Variety of complexes 217
13.7 Degenerations of Schubert varieties to toric varieties 218
Appendix: Proof of the main theorem of SMT 226
A.1 Notation 226
A.2 Admissible pairs and the first basis theorem 227
A.3 The three examples 228
A.4 Tableaux and the statement of the main theorem 231
A.5 Preparation 232
A.6 The tableau character formula 233
A.7 The structure of admissible pairs 233
A.8 The procedure 234
A.9 The basis 236
A.10 The first basis theorem 237
A.11 Linear independence 239
A.12 Arithmetic Cohen-Macaulayness & Arithmetic normality for Schubert varieties
References 249
Index 255
Index of notation 266
Author index 269