Flag varieties: an interplay of geometry, combinatorics, and representation theory
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New Delhi
Hindustan Book Agency
2009
|
| Series: | Texts and readings in mathematics
53 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. [259] - 262 |
| Physical Description: | XIV, 272 S. graph. Darst. 24 cm |
| ISBN: | 9788185931920 |
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MARC
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Record in the Search Index
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|---|---|
| adam_text | Titel: Flag varieties
Autor: Lakshmibai, Venkatramani
Jahr: 2009
Contents
Preface v
Introduction xi
1 Preliminaries 1
1.1 Commutative Algebra........................................1
1.2 Affine Varieties................................................6
1.3 Projective Varieties............................................11
1.4 Schemes - Affine and Projective..............................13
1.5 The Scheme Spec(A)..........................................15
1.6 The Scheme Proj(5)..........................................17
1.7 Sheaves of Ox-Modules......................................18
1.8 Attributes of Varieties....................22
2 Structure Theory of Semisimple Rings 25
2.1 Semisimple Modules.....................25
2.2 Semisimple Rings.......................31
2.3 Brauer Groups and Central Simple Algebras........39
2.4 The Group Algebra, K G ..................41
2.5 The Center of K[G] .....................43
Exercises ..............................45
3 Representation Theory of Finite Groups 47
3.1 Representations of G.....................47
3.2 Characters of Representations................49
3.3 Ordinary Representations..................52
3.4 Tensor Product of Representations.............53
3.5 Contragradient Representations...............54
3.6 Restrictions and Inductions.................55
vii
viU CONTENTS
3.7 Character Group of G....................56
Exercises ..............................°
4 Representation Theory of the Symmetric Group 59
4.1 The Symmetric Group Sn....................................59
4.2 Frobenius-Young Modules..................55
4.3 Specht Modules................................................71
4.4 General Case.........................74
4.5 Proof of Theorem 4.3.7........................................76
4.6 Representation Theory of An................79
Exercises ..............................82
5 Symmetric Polynomials 83
5.1 Notation and Motivation...................83
5.2 Several Bases for S (n)....................84
5.3 Kostka Numbers ; Determinantal Formulas........85
5.4 Results on .........................87
Exercises ..............................93
6 Schur-Weyl Duality and the Relationship Between
Representations of Sd and GLn (C) 95
6.1 Generalities..........................95
6.2 Schur-Weyl Duality......................97
6.3 Characters of the Schur Modules..............99
6.4 Schur Module Representations of SLn (C).........101
6.5 Representations of GLn (C).................102
Exercises ..............................104
7 Structure Theory of Complex Semisimple Lie
Algebras 107
7.1 Introduction to Semisimple Lie Algebras..........107
7.2 The Exponential Map in Characteristic Zero........109
7.3 Structure of Semisimple Lie Algebras............110
7.4 Jordan Decomposition in Semisimple Lie Algebras .... 114
7.5 The Lie Algebra sln (C)......................
7.6 Cartan Subalgebras......................216
7.7 Root Systems......................................218
7.8 Structure Theory of $[n (C).................120
Exercises ....................................122
CONTENTS ix
8 Representation Theory of Complex Semisimple Lie
Algebras 123
8.1 Representations of g.....................123
8.2 Weight Spaces ........................127
8.3 Finite Dimensional Modules.................131
8.4 Fundamental Weights ....................134
8.5 Dimension and Character Formulas.............135
8.6 Irreducible sln (C)-Modules.................136
Exercises ..............................136
9 Generalities on Algebraic Groups 137
9.1 Algebraic Groups and Their Lie Algebras.........137
9.2 The Tangent Space......................138
9.3 Jordan Decomposition in G.................139
9.4 Variety Structure on G/H..................144
9.5 The Flag Variety.......................149
9.6 Structure of Connected Solvable Groups..........151
9.7 Borel Fixed Point Theorem.................153
9.8 Variety of Borel Subgroups.................158
Exercises ..............................159
10 Structure Theory of Reductive Groups 161
10.1 Cartan Subgroups......................161
10.2 The Weyl Group.......................162
10.3 Regular and Singular Tori..................164
10.4 Semisimple Rank 1......................166
10.5 One Parameter Subgroups..................168
10.6 Reductive Groups.......................172
10.7 Almost Simple Groups....................175
10.8 Schubert Varieties k Bruhat Decomposition........177
10.9 Standard Parabolic Subgroups ................180
Exercises ..............................181
11 Representation Theory of Semisimple Algebraic
Groups 183
11.1 Weight Vectors........................183
11.2 Geometric Realization of V(A) ...............189
X
CONTENTS
12 Geometry of the Grassmannian, Flag and their
Schubert Varieties via Standard Monomial Theory 197
12.1 Grassmannian Variety....................197
12.2 Gd n Identified with a Homogeneous Space.........201
12.3 Schubert Varieties ......................202
12.4 Standard Monomials.....................204
12.5 Equations Defining Schubert Varieties...........207
12.6 Unions of Schubert Varieties.................208
12.7 Vanishing Theorems.....................211
12.8 Results for T.........................214
Exercises ..............................222
13 Singular Locus of a Schubert Variety in the Flag
Variety SLn/B 225
13.1 Generalities..........................225
13.2 Singular Loci of Schubert Varieties.............226
Exercises ..............................231
14 Applications 233
14.1 Determinantal Varieties...................233
14.2 Classical Invariant Theory..................237
14.3 Toric Degeneration of a Schubert Variety .........240
Exercises ..............................243
A Chevalley Groups 245
A.l A Basis for £.........................245
A.2 Kostant s Z-form.......................247
A.3 Admissible Lattices......................249
A.4 The Chevalley Groups...................250
A.5 Simplicity of Groups of Adjoint Type............253
A.6 Chevalley Groups and Algebraic Groups..........254
A.7 Generators, Relations of Universal Group.........256
Bibliography 259
List of Symbols
Index
263
267
|
| any_adam_object | 1 |
| author | Lakshmibai, Venkatramani Brown, Justin |
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| classification_rvk | SK 240 SK 340 |
| classification_tum | MAT 173f MAT 147f MAT 202f |
| ctrlnum | (OCoLC)418736988 (DE-599)GBV592206653 |
| 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 |
| format | Book |
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| isbn | 9788185931920 |
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| spelling | Lakshmibai, Venkatramani Verfasser (DE-588)122537416 aut Flag varieties an interplay of geometry, combinatorics, and representation theory V. Lakshmibai ; Justin Brown New Delhi Hindustan Book Agency 2009 XIV, 272 S. graph. Darst. 24 cm txt rdacontent n rdamedia nc rdacarrier Texts and readings in mathematics 53 Literaturverz. S. [259] - 262 Fahnenmannigfaltigkeit (DE-588)4431550-8 gnd rswk-swf Darstellungstheorie (DE-588)4148816-7 gnd rswk-swf Algebraische Gruppe (DE-588)4001164-1 gnd rswk-swf Fahnenmannigfaltigkeit (DE-588)4431550-8 s Algebraische Gruppe (DE-588)4001164-1 s Darstellungstheorie (DE-588)4148816-7 s DE-604 Brown, Justin Verfasser (DE-588)14281783X aut Texts and readings in mathematics 53 (DE-604)BV011380811 53 DE-601 pdf/application http://zbmath.org/?q=an:1166.14001 Zentralblatt MATH Inhaltstext HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027160810&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lakshmibai, Venkatramani Brown, Justin Flag varieties an interplay of geometry, combinatorics, and representation theory Texts and readings in mathematics Fahnenmannigfaltigkeit (DE-588)4431550-8 gnd Darstellungstheorie (DE-588)4148816-7 gnd Algebraische Gruppe (DE-588)4001164-1 gnd |
| subject_GND | (DE-588)4431550-8 (DE-588)4148816-7 (DE-588)4001164-1 |
| title | Flag varieties an interplay of geometry, combinatorics, and representation theory |
| title_auth | Flag varieties an interplay of geometry, combinatorics, and representation theory |
| title_exact_search | Flag varieties an interplay of geometry, combinatorics, and representation theory |
| title_full | Flag varieties an interplay of geometry, combinatorics, and representation theory V. Lakshmibai ; Justin Brown |
| title_fullStr | Flag varieties an interplay of geometry, combinatorics, and representation theory V. Lakshmibai ; Justin Brown |
| title_full_unstemmed | Flag varieties an interplay of geometry, combinatorics, and representation theory V. Lakshmibai ; Justin Brown |
| title_short | Flag varieties |
| title_sort | flag varieties an interplay of geometry combinatorics and representation theory |
| title_sub | an interplay of geometry, combinatorics, and representation theory |
| topic | Fahnenmannigfaltigkeit (DE-588)4431550-8 gnd Darstellungstheorie (DE-588)4148816-7 gnd Algebraische Gruppe (DE-588)4001164-1 gnd |
| topic_facet | Fahnenmannigfaltigkeit Darstellungstheorie Algebraische Gruppe |
| url | http://zbmath.org/?q=an:1166.14001 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027160810&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV011380811 |
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