Lakshmibai / Raghavan Standard Monomial Theory

Invariant Theoretic Approach
2008
ISBN: 978-3-540-76757-2
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 137, 266 Seiten

Reihe: Encyclopaedia of Mathematical Sciences

Schubert varieties provide an inductive tool for studying flag varieties. This book is mainly a detailed account of a particularly interesting instance of their occurrence: namely, in relation to classical invariant theory. More precisely, it is about the connection between the first and second fundamental theorems of classical invariant theory on the one hand and standard monomial theory for Schubert varieties in certain special flag varieties on the other.

Autoren/Hrsg.

Lakshmibai, V.

Raghavan, K. N.

Inhaltsverzeichnis

1;Preface;6
2;Contents;8
3;1 Introduction;14
3.1;1.1 The subject matter in a nutshell;14
3.2;1.2 The subject matter in detail;15
3.3;1.3 Why this book?;19
3.4;1.4 A brief history of SMT;20
3.5;1.5 Some features of the SMT approach;20
3.6;1.6 The organization of the book;22
4;2 Generalities on algebraic varieties;23
4.1;2.1 Some basic definitions;23
4.2;2.2 Algebraic varieties;24
5;3 Generalities on algebraic groups;29
5.1;3.1 Abstract root systems;29
5.2;3.2 Root systems of algebraic groups;31
5.3;3.3 Schubert varieties;34
6;4 Grassmannian Variety;41
6.1;4.1 The Pl cker embedding;41
6.2;4.2 Schubert varieties of Gd,;46
6.3;4.3 Standard monomial theory for Schubert varieties in Gd,;48
6.4;4.4 Standard monomial theory for a union of Schubert varieties;51
6.5;4.5 Vanishing theorems;53
6.6;4.6 Arithmetic Cohen-Macaulayness, normality and factoriality;56
7;5 Determinantal varieties;59
7.1;5.1 Recollection of facts;59
7.2;5.2 Determinantal varieties;61
8;6 Symplectic Grassmannian;67
8.1;6.1 Some basic facts on Sp(V );68
8.2;6.2 The variety G/ Pn;72
9;7 Orthogonal Grassmannian;82
9.1;7.1 The even orthogonal group SO(2n);82
9.2;7.2 The variety G/ Pn;88
10;8 The standard monomial theoretic basis;95
10.1;8.1 SMT for the even orthogonal Grassmannian;96
10.2;8.2 SMT for the symplectic Grassmannian;99
11;9 Review of GIT;104
11.1;9.1 G- spaces;104
11.2;9.2 Affine quotients;107
11.3;9.3 Categorical quotients;110
11.4;9.4 Good quotients;112
11.5;9.5 Stable and semi-stable points;117
11.6;9.6 Projective quotients;123
11.7;9.7 L- linear actions;126
11.8;9.8 Hilbert-Mumford criterion;126
12;10 Classical Invariant Theory;130
12.1;10.1 Preliminary lemmas;130
12.2;10.2 SLd( K)- action;133
12.3;10.3 GLn( K)- action:;137
12.4;10.4 On( K)- action;141
12.5;10.5 Sp2;145
12.6;(K)- action;145
13;11 SLn( K)- action;146
13.1;11.1 Quadratic relations;147
13.2;11.2 The K- algebra S;149
13.3;11.3 Standard monomials in the K- algebra S;151
13.4;11.4 Normality and Cohen-Macaulayness of the K- algebra S;159
13.5;11.5 The ring of invariants K[X];164
14;12 SOn( K)- action;168
14.1;12.1 Preliminaries;169
14.2;12.2 The algebra S;176
14.3;12.3 The algebra S(D);178
14.4;12.4 Cohen-Macaulayness of S;182
14.5;12.5 The equality RSOn(;185
14.6;= S;185
14.7;12.6 Application to moduli problem;189
14.8;12.7 Results for the adjoint action of SL2( K);190
15;13 Applications of standard monomial theory;195
15.1;13.1 Tangent space and smoothness;195
15.2;13.2 Singularities of Schubert varieties in the flag variety;198
15.3;13.3 Singular loci of Schubert varieties in the Grassmannian;203
15.4;13.4 Results for Schubert varieties in a minuscule G/P;208
15.5;13.5 Applications to other varieties;210
15.6;13.6 Variety of complexes;217
15.7;13.7 Degenerations of Schubert varieties to toric varieties;218
16;Appendix: Proof of the main theorem of SMT;226
16.1;A.1 Notation;226
16.2;A.2 Admissible pairs and the first basis theorem;227
16.3;A.3 The three examples;228
16.4;A.4 Tableaux and the statement of the main theorem;231
16.5;A.5 Preparation;232
16.6;A.6 The tableau character formula;233
16.7;A.7 The structure of admissible pairs;233
16.8;A.8 The procedure;234
16.9;A.9 The basis;236
16.10;A.10 The first basis theorem;237
16.11;A.11 Linear independence;239
16.12;A.12 Arithmetic Cohen-Macaulayness & Arithmetic normality for Schubert varieties;242
17;References;249
17.1;Index;255
17.2;Index of notation;266
18;Author index;269