Standard monomial theory: invariant theoretic approach
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2008
|
| Schriftenreihe: | Encyclopaedia of mathematical sciences
137 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIV, 265 S. |
| ISBN: | 9783540767565 9783642095436 |
Internformat
MARC
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| 245 | 1 | 0 | |a Standard monomial theory |b invariant theoretic approach |c Venkatramani Lakshmibai ; Komaranapuram N. Raghavan |
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| 490 | 1 | |a Encyclopaedia of mathematical sciences |v 137 | |
| 490 | 1 | |a Encyclopaedia of mathematical sciences / Invariant theory and algebraic transformation groups |v 8 | |
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Datensatz im Suchindex
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| adam_text | VENKATRAMANI LAKSHMIBAI KOMARANAPURAM N. RAGHAVAN STANDARD MONOMIAL
THEORY INVARIANT THEORETIC APPROACH 4Y SPRINGER CONTENTS INTRODUCTION 1
1.1 THE SUBJECT MATTER IN A NUTSHELL 1 1.1.1 WHATISCIT? 1 1.1.2 WHAT IS
SMT? 2 1.1.3 THE SMT APPROACH TO CIT 2 1.2 THE SUBJECT MATTER IN DETAIL
2 1.2.1 PROOF BY THE SMT APPROACH 3 1.2.2 SL N (K),SO N (K) ACTIONS 5
1.3 WHY THIS BOOK? 6 1.4 A BRIEF HISTORY OF SMT 7 1.5 SOME FEATURES OF
THE SMT APPROACH 7 1.6 THE ORGANIZATION OF THE BOOK 9 GENERALITIES ON
ALGEBRAIC VARIETIES 11 2.1 SOME BASIC DEFINITIONS 11 2.2 ALGEBRAIC
VARIETIES 12 2.2.1 AFFINE VARIETIES 12 GENERALITIES ON ALGEBRAIC GROUPS
17 3.1 ABSTRACT ROOT SYSTEMS 17 3.2 ROOT SYSTEMS OF ALGEBRAIC GROUPS 19
3.2.1 LINEAR ALGEBRAIC GROUPS 19 3.2.2 PARABOLIC SUBGROUPS 21 3.3
SCHUBERT VARIETIES 22 3.3.1 WEYL AND DEMAZURE MODULES 24 3.3.2 LINE
BUNDLES ON-G/Q 25 3.3.3 EQUATIONS DEFINING A SCHUBERT VARIETY 27
GRASSMANNIAN . 29 4.1 THE PLUCKER EMBEDDING 29 4.1.1 THE PARTIALLY
ORDERED SET / /,* 30 X CONTENTS 4.2 4.3 4.4 4.5 4.6 4.1.2 PLIICKER
EMBEDDING AND PLIICKER COORDINATES 4.1.3 PLIICKER QUADRATIC RELATIONS
4.1.4 MORE GENERAL QUADRATIC RELATIONS 4.1.5 THE CONE GD, N OVER GD,N
4.1.6 IDENTIFICATION OF G/PD WITH GD, N SCHUBERT VARIETIES OF G /* 4.2.1
BRUHAT DECOMPOSITION 4.2.2 DIMENSION OF XJ 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 DENNING
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 DETERMINANTAL VARIETIES 5.1 RECOLLECTION OF FACTS 5.1.1
EQUATIONS DEFINING SCHUBERT VARIETIES IN THE GRASSMANNIAN 5.1.2
EVALUATION OF PLIICKER 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 D, 5.2.2 IDENTIFICATION OF D, WITH Y^ 5.2.3 THE
BIJECTION 6 5.2.4 THE PARTIAL ORDER ; 5.2.5 COGENERATION OF AN IDEAL
5.2.6 THE MONOMIAL ORDER AND GROBNER BASES SYMPLECTIC GRASSMANNIAN 6.1
SOME BASIC FACTS ON SP(V) 6.1.1 SCHUBERT VARIETIES IN G/BG 6.2 THE
VARIETY G/P N 6.2.1 IDENTIFICATION OF SYM M N WITH OQ 6.2.2 CANONICAL
DUAL PAIR 6.2.3 THE BIJECTION 9 6.2.4 THE DUAL WEYL G-MODULE WITH
HIGHEST WEIGHT CO N 6.2.5 IDENTIFICATION OF D,(SYM M N ) WITH Y PN (CP)
30 31 32 33 33 34 34 35 35 36 36 36 37 38 39 39 39 40 41 44 46 47 47 48
48 49 49 49 49 51 51 52 53 55 56 59 60 61 62 62 63 64 CONTENTS XI 6.2.6
ADMISSIBLE PAIRS AND CANONICAL PAIRS 65 6.2.7 CANONICAL PAIRS 65 6.2.8
THE INCLUSION R : L N ^ N ^Y W P X W P 66 6.2.9 A STANDARD
MONOMIAL BASIS FOR D T (SYM M N ) 67 6.2.10 DE CONCINI-PROCESI S BASIS
FOR D T {SYM M N ) 68 7 ORTHOGONAL GRASSMANNIAN 71 7.1 THE EVEN
ORTHOGONAL GROUP SO{2N) 71 7.1.1 SCHUBERT VARIETIES IN G/B G 74 7.2 THE
VARIETY G/P N 77 7.2.1 IDENTIFICATION OF SK M N WITH O^ 78 7.2.2
CANONICAL DUAL PAIR 78 7.2.3 THE BIJECTION 6 79 7.2.4 THE DUAL WEYL
G-MODULE WITH HIGHEST WEIGHT W N 79 7.2.5 IDENTIFICATION OF D T (SK M N
) WITH Y G (CP) 80 7.2.6 A STANDARD MONOMIAL BASIS FOR D T (SK M N ) 82
8 THE STANDARD MONOMIAL THEORETIC BASIS 85 8.1 SMT FOR THE EVEN
ORTHOGONAL GRASSMANNIAN 86 8.2 SMT FOR THE SYMPLECTIC GRASSMANNIAN 89 9
REVIEW OF GIT 95 9.1 G-SPACES 95 9.1.1 REDUCTIVE GROUPS 95 9.2 AFFINE
QUOTIENTS 98 9.2.1 AFFINE ACTIONS 99 9.3 CATEGORICAL QUOTIENTS 101 9.3.1
EXAMPLES 102 9.4 GOOD QUOTIENTS 103 9.4.1 SOME RESULTS ON GOOD QUOTIENTS
105 9.5 STABLE AND SEMI-STABLE POINTS 108 9.5.1 STABLE, SEMISTABLE, AND
POLYSTABLE POINTS 108 9.5.2 OTHER CHARACTERIZATIONS OF STABILITY,
SEMISTABILITY 110 9.6 PROJECTIVE QUOTIENTS 114 9.7 L-LINEAR ACTIONS 117
9.8 HILBERT-MUMFORD CRITERION 117 10 INVARIANT THEORY 121 10.1
PRELIMINARY LEMMAS 121 10.2 SL D (#)-ACTION 124 10.2.1 THE FUNCTIONS F T
124 10.2.2 THE FIRST AND SECOND FUNDAMENTAL THEOREMS 126 10.3 GL N
(/Q-ACTION: . 128 10.3.1 THE FIRST AND SECOND FUNDAMENTAL THEOREMS 129
10.4 0 N (#)-ACTION 132 XII CONTENTS 10.5 SP2I(K)-ACTION 136 11 SL N
(/O-ACTION 137 11.1 QUADRATIC RELATIONS 138 11.1.1 THE PARTIALLY ORDERED
SET H R ,D 139 11.2 THE ^-ALGEBRA S 140 11.2.1 THE SL N (FF)-ACTION 141
11.3 STANDARD MONOMIALS IN THE ^-ALGEBRA S 142 11.3.1 QUADRATIC
RELATIONS 143 11.3.2 LINEAR INDEPENDENCE OF STANDARD MONOMIALS 145
11.3.3 THE ALGEBRA S(D) 146 11.3.4 A STANDARD MONOMIAL BASIS FOR R(D)
148 11.3.5 STANDARD MONOMIAL BASES FOR M(D), S(D) 149 11.4 NORMALITY AND
COHEN-MACAULAYNESS OF THE A -ALGEBRA S 150 11.4.1 THE ALGEBRA ASSOCIATED
TO A DISTRIBUTIVE LATTICE 150 11.4.2 FLAT DEGENERATIONS OF CERTAIN
A -ALGEBRAS 151 11.4.3 THE DISTRIBUTIVE LATTICE D 152 11.4.4 FLAT
DEGENERATION OF SPEC R (D) TO THE TORIC VARIETY SPEC A(D) 154 11.5 THE
RING OF INVARIANTS K[X] SL (K) 155 12 S0 N (A ACTION 159 12.1
PRELIMINARIES 160 12. 12. 12. 12. . 1 THE LAGRANGIAN GRASSMANNIAN
VARIETY 161 .2 SCHUBERT VARIETIES IN L M 161 .3 THE OPPOSITE BIG CELL IN
L M 162 .4 THE FUNCTIONS F RT ON OQ 163 12.1.5 THE OPPOSITE CELL IN X(W)
164 12.1.6 SYMMETRIC DETERMINANTAL VARIETIES 164 12.1.7 THE SET H M 165
12.2 THE ALGEBRA S 167 12.2.1 STANDARD MONOMIALS AND THEIR LINEAR
INDEPENDENCE 168 12.2.2 LINEAR INDEPENDENCE OF STANDARD MONOMIALS 169
12.3 THE ALGEBRA S(D) 169 12.3.1 QUADRATIC RELATIONS 170 12.3.2 A
STANDARD MONOMIAL BASIS FOR R(D) 171 12.3.3 STANDARD MONOMIAL BASES FOR
S(D) 172 12.4 COHEN-MACAULAYNESS OF S 173 12.4.1 A DOSET ALGEBRA
STRUCTURE FOR R(D) 175 12.5 THE EQUALITY R S N(K) = S 176 12.6
APPLICATION TO MODULI PROBLEM 180 12.7 RESULTS FOR THE ADJOINT ACTION OF
SL2(K) 181 CONTENTS XIII 13 APPLICATIONS OF STANDARD MONOMIAL THEORY 187
13.1 TANGENT SPACE AND SMOOTHNESS 187 13.1.1 THE ZARISKI TANGENT SPACE
187 13.1.2 SMOOTH AND NON-SMOOTH POINTS 188 13.1.3 THE SPACE T(W, R) 188
13.1.4 A CANONICAL AFFINE NEIGHBORHOOD OF A T-FIXED POINT 188 13.1.5 THE
AFFINE VARIETY Y(W, R) 189 13.1.6 EQUATIONS DEFINING Y(W, T) IN O~ 189
13.1.7 JACOBIAN CRITERION FOR SMOOTHNESS 189 13.1.8 7-STABLE CURVES 190
13.2 SINGULARITIES OF SCHUBERT VARIETIES IN THE FLAG VARIETY 190 13.2.1
IDEAL OF Y(W, R) 191 13.2.2 A CRITERION FOR SMOOTHNESS 193 13.2.3
COMPONENTS OF THE SINGULAR LOCUS 193 13.3 SINGULAR LOCI OF SCHUBERT
VARIETIES IN THE GRASSMANNIAN 195 13.3.1 MULTIPLICITY AT A SINGULAR
POINT 196 13.4 RESULTS FOR SCHUBERT VARIETIES IN A MINUSCULE G/P 200
13.4.1 HOMOGENEITY OF J P (W) 201 13.4.2 A BASIS FOR (M TIW ) R /(M R ,
W Y +L 202 13.5 APPLICATIONS TO OTHER VARIETIES 202 13.5.1 LADDER
DETERMINANTAL VARIETIES 202 13.5.2 THE VARIETIES V;, 1 I Z 204
13.5.3 QUIVER VARIETIES 205 13.6 VARIETY OF COMPLEXES 209 13.6.1 A
PARTIAL ORDER ON { (K I, FO, ***, H)} 210 13.7 DEGENERATIONS OF SCHUBERT
VARIETIES TO TORIC VARIETIES 210 13.7.1 GENERALITIES ON DISTRIBUTIVE
LATTICES 210 13.7.2 AN IMPORTANT EXAMPLE 211 13.7.3 GENERALITIES ON
TORIC VARIETIES 212 13.7.4 AN EXAMPLE 213 13.7.5 THE ALGEBRA ASSOCIATED
TO A DISTRIBUTIVE LATTICE 213 13.7.6 VARIETIES DEFINED BY BINOMIALS 214
13.7.7 DEGENERATIONS OF SCHUBERT VARIETIES IN THE GRASSMANNIAN TO TORIC
VARIETIES 215 APPENDIX: PROOF OF THE MAIN THEOREM OF SMT 219 A.I
NOTATION 219 A.2 ADMISSIBLE PAIRS AND THE FIRST BASIS THEOREM 220 A.2.1
MORE NOTATION 220 A.2.2 CHEVALLEY MULTIPLICITY 220 A.2.3 MINUSCULE AND
CLASSICAL TYPE PARABOLICS 220 A.2.4 ADMISSIBLE PAIRS 221 A.3 THE THREE
EXAMPLES 221 A.3.1 EXAMPLE A 222 A.3.2 EXAMPLE B 222 XIV CONTENTS A.3.3
EXAMPLE C 223 A.4 TABLEAUX AND THE STATEMENT OF THE MAIN THEOREM 224 A.5
PREPARATION 225 A.6 THE TABLEAU CHARACTER FORMULA 226 A.7 THE STRUCTURE
OF ADMISSIBLE PAIRS 226 A.8 THE PROCEDURE 227 A.9 THE BASIS 229 A.10 THE
FIRST BASIS THEOREM 230 A.I 1 LINEAR INDEPENDENCE 232 A. 12 ARITHMETIC
COHEN-MACAULAYNESS AND ARITHMETIC NORMALITY 235 A.12.1 ARITHMETIC
COHEN-MACAULAYNESS 236 REFERENCES 243 INDEX 249 INDEX OF NOTATION 260
AUTHOR INDEX 263
|
| adam_txt |
VENKATRAMANI LAKSHMIBAI KOMARANAPURAM N. RAGHAVAN STANDARD MONOMIAL
THEORY INVARIANT THEORETIC APPROACH 4Y SPRINGER CONTENTS INTRODUCTION 1
1.1 THE SUBJECT MATTER IN A NUTSHELL 1 1.1.1 WHATISCIT? 1 1.1.2 WHAT IS
SMT? 2 1.1.3 THE SMT APPROACH TO CIT 2 1.2 THE SUBJECT MATTER IN DETAIL
2 1.2.1 PROOF BY THE SMT APPROACH 3 1.2.2 SL N (K),SO N (K) ACTIONS 5
1.3 WHY THIS BOOK? 6 1.4 A BRIEF HISTORY OF SMT 7 1.5 SOME FEATURES OF
THE SMT APPROACH 7 1.6 THE ORGANIZATION OF THE BOOK 9 GENERALITIES ON
ALGEBRAIC VARIETIES 11 2.1 SOME BASIC DEFINITIONS 11 2.2 ALGEBRAIC
VARIETIES 12 2.2.1 AFFINE VARIETIES 12 GENERALITIES ON ALGEBRAIC GROUPS
17 3.1 ABSTRACT ROOT SYSTEMS 17 3.2 ROOT SYSTEMS OF ALGEBRAIC GROUPS 19
3.2.1 LINEAR ALGEBRAIC GROUPS 19 3.2.2 PARABOLIC SUBGROUPS 21 3.3
SCHUBERT VARIETIES 22 3.3.1 WEYL AND DEMAZURE MODULES 24 3.3.2 LINE
BUNDLES ON-G/Q 25 3.3.3 EQUATIONS DEFINING A SCHUBERT VARIETY 27
GRASSMANNIAN '. 29 4.1 THE PLUCKER EMBEDDING 29 4.1.1 THE PARTIALLY
ORDERED SET / /,* 30 X CONTENTS 4.2 4.3 4.4 4.5 4.6 4.1.2 PLIICKER
EMBEDDING AND PLIICKER COORDINATES 4.1.3 PLIICKER QUADRATIC RELATIONS
4.1.4 MORE GENERAL QUADRATIC RELATIONS 4.1.5 THE CONE GD, N OVER GD,N
4.1.6 IDENTIFICATION OF G/PD WITH GD, N SCHUBERT VARIETIES OF G /* 4.2.1
BRUHAT DECOMPOSITION 4.2.2 DIMENSION OF XJ 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 DENNING
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 DETERMINANTAL VARIETIES 5.1 RECOLLECTION OF FACTS 5.1.1
EQUATIONS DEFINING SCHUBERT VARIETIES IN THE GRASSMANNIAN 5.1.2
EVALUATION OF PLIICKER 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 D, 5.2.2 IDENTIFICATION OF D, WITH Y^ 5.2.3 THE
BIJECTION 6 5.2.4 THE PARTIAL ORDER ; 5.2.5 COGENERATION OF AN IDEAL
5.2.6 THE MONOMIAL ORDER AND GROBNER BASES SYMPLECTIC GRASSMANNIAN 6.1
SOME BASIC FACTS ON SP(V) 6.1.1 SCHUBERT VARIETIES IN G/BG 6.2 THE
VARIETY G/P N 6.2.1 IDENTIFICATION OF SYM M N WITH OQ 6.2.2 CANONICAL
DUAL PAIR 6.2.3 THE BIJECTION 9 6.2.4 THE DUAL WEYL G-MODULE WITH
HIGHEST WEIGHT CO N 6.2.5 IDENTIFICATION OF D,(SYM M N ) WITH Y PN (CP)
30 31 32 33 33 34 34 35 35 36 36 36 37 38 39 39 39 40 41 44 46 47 47 48
48 49 49 49 49 51 51 52 53 55 56 59 60 61 62 62 63 64 CONTENTS XI 6.2.6
ADMISSIBLE PAIRS AND CANONICAL PAIRS 65 6.2.7 CANONICAL PAIRS 65 6.2.8
THE INCLUSION R\ : L N ^ N ^Y W P " X W P " 66 6.2.9 A STANDARD
MONOMIAL BASIS FOR D T (SYM M N ) 67 6.2.10 DE CONCINI-PROCESI'S BASIS
FOR D T {SYM M N ) 68 7 ORTHOGONAL GRASSMANNIAN 71 7.1 THE EVEN
ORTHOGONAL GROUP SO{2N) 71 7.1.1 SCHUBERT VARIETIES IN G/B G 74 7.2 THE
VARIETY G/P N 77 7.2.1 IDENTIFICATION OF SK M N WITH O^ 78 7.2.2
CANONICAL DUAL PAIR 78 7.2.3 THE BIJECTION 6 79 7.2.4 THE DUAL WEYL
G-MODULE WITH HIGHEST WEIGHT W N 79 7.2.5 IDENTIFICATION OF D T (SK M N
) WITH Y G (CP) 80 7.2.6 A STANDARD MONOMIAL BASIS FOR D T (SK M N ) 82
8 THE STANDARD MONOMIAL THEORETIC BASIS 85 8.1 SMT FOR THE EVEN
ORTHOGONAL GRASSMANNIAN 86 8.2 SMT FOR THE SYMPLECTIC GRASSMANNIAN 89 9
REVIEW OF GIT 95 9.1 G-SPACES 95 9.1.1 REDUCTIVE GROUPS 95 9.2 AFFINE
QUOTIENTS 98 9.2.1 AFFINE ACTIONS 99 9.3 CATEGORICAL QUOTIENTS 101 9.3.1
EXAMPLES 102 9.4 GOOD QUOTIENTS 103 9.4.1 SOME RESULTS ON GOOD QUOTIENTS
105 9.5 STABLE AND SEMI-STABLE POINTS 108 9.5.1 STABLE, SEMISTABLE, AND
POLYSTABLE POINTS 108 9.5.2 OTHER CHARACTERIZATIONS OF STABILITY,
SEMISTABILITY 110 9.6 PROJECTIVE QUOTIENTS 114 9.7 L-LINEAR ACTIONS 117
9.8 HILBERT-MUMFORD CRITERION 117 10 INVARIANT THEORY 121 10.1
PRELIMINARY LEMMAS 121 10.2 SL D (#)-ACTION 124 10.2.1 THE FUNCTIONS F T
124 10.2.2 THE FIRST AND SECOND FUNDAMENTAL THEOREMS 126 10.3 GL N
(/Q-ACTION: .' 128 10.3.1 THE FIRST AND SECOND FUNDAMENTAL THEOREMS 129
10.4 0 N (#)-ACTION 132 XII CONTENTS 10.5 SP2I(K)-ACTION 136 11 SL N
(/O-ACTION 137 11.1 QUADRATIC RELATIONS 138 11.1.1 THE PARTIALLY ORDERED
SET H R ,D 139 11.2 THE ^-ALGEBRA S 140 11.2.1 THE SL N (FF)-ACTION 141
11.3 STANDARD MONOMIALS IN THE ^-ALGEBRA S 142 11.3.1 QUADRATIC
RELATIONS 143 11.3.2 LINEAR INDEPENDENCE OF STANDARD MONOMIALS 145
11.3.3 THE ALGEBRA S(D) 146 11.3.4 A STANDARD MONOMIAL BASIS FOR R(D)
148 11.3.5 STANDARD MONOMIAL BASES FOR M(D), S(D) 149 11.4 NORMALITY AND
COHEN-MACAULAYNESS OF THE A"-ALGEBRA S 150 11.4.1 THE ALGEBRA ASSOCIATED
TO A DISTRIBUTIVE LATTICE 150 11.4.2 FLAT DEGENERATIONS OF CERTAIN
A"-ALGEBRAS 151 11.4.3 THE DISTRIBUTIVE LATTICE D 152 11.4.4 FLAT
DEGENERATION OF SPEC R (D) TO THE TORIC VARIETY SPEC A(D) 154 11.5 THE
RING OF INVARIANTS K[X] SL " (K) 155 12 S0 N (A ACTION 159 12.1
PRELIMINARIES 160 12. 12. 12. 12. . 1 THE LAGRANGIAN GRASSMANNIAN
VARIETY 161 .2 SCHUBERT VARIETIES IN L M 161 .3 THE OPPOSITE BIG CELL IN
L M 162 .4 THE FUNCTIONS F RT ON OQ 163 12.1.5 THE OPPOSITE CELL IN X(W)
164 12.1.6 SYMMETRIC DETERMINANTAL VARIETIES 164 12.1.7 THE SET H M 165
12.2 THE ALGEBRA S 167 12.2.1 STANDARD MONOMIALS AND THEIR LINEAR
INDEPENDENCE 168 12.2.2 LINEAR INDEPENDENCE OF STANDARD MONOMIALS 169
12.3 THE ALGEBRA S(D) 169 12.3.1 QUADRATIC RELATIONS 170 12.3.2 A
STANDARD MONOMIAL BASIS FOR R(D) 171 12.3.3 STANDARD MONOMIAL BASES FOR
S(D) 172 12.4 COHEN-MACAULAYNESS OF S 173 12.4.1 A DOSET ALGEBRA
STRUCTURE FOR R(D) 175 12.5 THE EQUALITY R S N(K) = S 176 12.6
APPLICATION TO MODULI PROBLEM 180 12.7 RESULTS FOR THE ADJOINT ACTION OF
SL2(K) 181 CONTENTS XIII 13 APPLICATIONS OF STANDARD MONOMIAL THEORY 187
13.1 TANGENT SPACE AND SMOOTHNESS 187 13.1.1 THE ZARISKI TANGENT SPACE
187 13.1.2 SMOOTH AND NON-SMOOTH POINTS 188 13.1.3 THE SPACE T(W, R) 188
13.1.4 A CANONICAL AFFINE NEIGHBORHOOD OF A T-FIXED POINT 188 13.1.5 THE
AFFINE VARIETY Y(W, R) 189 13.1.6 EQUATIONS DEFINING Y(W, T) IN O~ 189
13.1.7 JACOBIAN CRITERION FOR SMOOTHNESS 189 13.1.8 7-STABLE CURVES 190
13.2 SINGULARITIES OF SCHUBERT VARIETIES IN THE FLAG VARIETY 190 13.2.1
IDEAL OF Y(W, R) 191 13.2.2 A CRITERION FOR SMOOTHNESS 193 13.2.3
COMPONENTS OF THE SINGULAR LOCUS 193 13.3 SINGULAR LOCI OF SCHUBERT
VARIETIES IN THE GRASSMANNIAN 195 13.3.1 MULTIPLICITY AT A SINGULAR
POINT 196 13.4 RESULTS FOR SCHUBERT VARIETIES IN A MINUSCULE G/P 200
13.4.1 HOMOGENEITY OF J P (W) 201 13.4.2 A BASIS FOR (M TIW ) R /(M R ,
W Y +L 202 13.5 APPLICATIONS TO OTHER VARIETIES 202 13.5.1 LADDER
DETERMINANTAL VARIETIES 202 13.5.2 THE VARIETIES V;, 1 I Z 204
13.5.3 QUIVER VARIETIES 205 13.6 VARIETY OF COMPLEXES 209 13.6.1 A
PARTIAL ORDER ON { (K I, FO, ***, H)} 210 13.7 DEGENERATIONS OF SCHUBERT
VARIETIES TO TORIC VARIETIES 210 13.7.1 GENERALITIES ON DISTRIBUTIVE
LATTICES 210 13.7.2 AN IMPORTANT EXAMPLE 211 13.7.3 GENERALITIES ON
TORIC VARIETIES 212 13.7.4 AN EXAMPLE 213 13.7.5 THE ALGEBRA ASSOCIATED
TO A DISTRIBUTIVE LATTICE 213 13.7.6 VARIETIES DEFINED BY BINOMIALS 214
13.7.7 DEGENERATIONS OF SCHUBERT VARIETIES IN THE GRASSMANNIAN TO TORIC
VARIETIES 215 APPENDIX: PROOF OF THE MAIN THEOREM OF SMT 219 A.I
NOTATION 219 A.2 ADMISSIBLE PAIRS AND THE FIRST BASIS THEOREM 220 A.2.1
MORE NOTATION 220 A.2.2 CHEVALLEY MULTIPLICITY 220 A.2.3 MINUSCULE AND
CLASSICAL TYPE PARABOLICS 220 A.2.4 ADMISSIBLE PAIRS 221 A.3 THE THREE
EXAMPLES 221 A.3.1 EXAMPLE A 222 A.3.2 EXAMPLE B 222 XIV CONTENTS A.3.3
EXAMPLE C 223 A.4 TABLEAUX AND THE STATEMENT OF THE MAIN THEOREM 224 A.5
PREPARATION 225 A.6 THE TABLEAU CHARACTER FORMULA 226 A.7 THE STRUCTURE
OF ADMISSIBLE PAIRS 226 A.8 THE PROCEDURE 227 A.9 THE BASIS 229 A.10 THE
FIRST BASIS THEOREM 230 A.I 1 LINEAR INDEPENDENCE 232 A. 12 ARITHMETIC
COHEN-MACAULAYNESS AND ARITHMETIC NORMALITY 235 A.12.1 ARITHMETIC
COHEN-MACAULAYNESS 236 REFERENCES 243 INDEX 249 INDEX OF NOTATION 260
AUTHOR INDEX 263 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Lakshmibai, Venkatramani Raghavan, Komaranapuram N. |
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| author_facet | Lakshmibai, Venkatramani Raghavan, Komaranapuram N. |
| author_role | aut aut |
| author_sort | Lakshmibai, Venkatramani |
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| callnumber-subject | QA - Mathematics |
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| ctrlnum | (OCoLC)182664252 (DE-599)BVBBV023172801 |
| dewey-full | 516.353 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.353 |
| dewey-search | 516.353 |
| dewey-sort | 3516.353 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV023172801 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T19:58:27Z |
| indexdate | 2024-07-09T21:12:15Z |
| institution | BVB |
| isbn | 9783540767565 9783642095436 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016359450 |
| oclc_num | 182664252 |
| open_access_boolean | |
| owner | DE-19 DE-BY-UBM DE-11 DE-188 DE-91G DE-BY-TUM |
| owner_facet | DE-19 DE-BY-UBM DE-11 DE-188 DE-91G DE-BY-TUM |
| physical | XIV, 265 S. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Springer |
| record_format | marc |
| series | Encyclopaedia of mathematical sciences |
| series2 | Encyclopaedia of mathematical sciences Encyclopaedia of mathematical sciences / Invariant theory and algebraic transformation groups |
| spelling | Lakshmibai, Venkatramani Verfasser (DE-588)122537416 aut Standard monomial theory invariant theoretic approach Venkatramani Lakshmibai ; Komaranapuram N. Raghavan Berlin [u.a.] Springer 2008 XIV, 265 S. txt rdacontent n rdamedia nc rdacarrier Encyclopaedia of mathematical sciences 137 Encyclopaedia of mathematical sciences / Invariant theory and algebraic transformation groups 8 Schubert, Variétés de Schubert varieties Standardmonomentheorie (DE-588)7611309-7 gnd rswk-swf Invariantentheorie (DE-588)4162209-1 gnd rswk-swf Schubert-Mannigfaltigkeit (DE-588)4512043-2 gnd rswk-swf Derivation Algebra (DE-588)4134656-7 gnd rswk-swf Standardmonomentheorie (DE-588)7611309-7 s Schubert-Mannigfaltigkeit (DE-588)4512043-2 s Invariantentheorie (DE-588)4162209-1 s DE-604 Derivation Algebra (DE-588)4134656-7 s Raghavan, Komaranapuram N. Verfasser (DE-588)134199553 aut Invariant theory and algebraic transformation groups Encyclopaedia of mathematical sciences 8 (DE-604)BV014336202 8 Encyclopaedia of mathematical sciences 137 (DE-604)BV024126459 137 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016359450&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lakshmibai, Venkatramani Raghavan, Komaranapuram N. Standard monomial theory invariant theoretic approach Encyclopaedia of mathematical sciences Schubert, Variétés de Schubert varieties Standardmonomentheorie (DE-588)7611309-7 gnd Invariantentheorie (DE-588)4162209-1 gnd Schubert-Mannigfaltigkeit (DE-588)4512043-2 gnd Derivation Algebra (DE-588)4134656-7 gnd |
| subject_GND | (DE-588)7611309-7 (DE-588)4162209-1 (DE-588)4512043-2 (DE-588)4134656-7 |
| title | Standard monomial theory invariant theoretic approach |
| title_auth | Standard monomial theory invariant theoretic approach |
| title_exact_search | Standard monomial theory invariant theoretic approach |
| title_exact_search_txtP | Standard monomial theory invariant theoretic approach |
| title_full | Standard monomial theory invariant theoretic approach Venkatramani Lakshmibai ; Komaranapuram N. Raghavan |
| title_fullStr | Standard monomial theory invariant theoretic approach Venkatramani Lakshmibai ; Komaranapuram N. Raghavan |
| title_full_unstemmed | Standard monomial theory invariant theoretic approach Venkatramani Lakshmibai ; Komaranapuram N. Raghavan |
| title_short | Standard monomial theory |
| title_sort | standard monomial theory invariant theoretic approach |
| title_sub | invariant theoretic approach |
| topic | Schubert, Variétés de Schubert varieties Standardmonomentheorie (DE-588)7611309-7 gnd Invariantentheorie (DE-588)4162209-1 gnd Schubert-Mannigfaltigkeit (DE-588)4512043-2 gnd Derivation Algebra (DE-588)4134656-7 gnd |
| topic_facet | Schubert, Variétés de Schubert varieties Standardmonomentheorie Invariantentheorie Schubert-Mannigfaltigkeit Derivation Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016359450&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV014336202 (DE-604)BV024126459 |
| work_keys_str_mv | AT lakshmibaivenkatramani standardmonomialtheoryinvarianttheoreticapproach AT raghavankomaranapuramn standardmonomialtheoryinvarianttheoreticapproach |