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Standard Monomial Theory Invariant Theoretic Approach

:	V. Lakshmibai, K. N. Raghavan
:	Standard Monomial Theory Invariant Theoretic Approach
:	Springer-Verlag
:	9783540767572
:	1
:	CHF 104.40
:

:	Arithmetik, Algebra
:	English

:	266
:	Wasserzeichen
:	PC/MAC/eReader/Tablet
:	PDF

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.

	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	239
	242	239
	References	249
	Index	255
	Index of notation	266
	Author index	269