Coxeter matroids:
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Boston [u.a.]
Birkhäuser
2003
|
| Schriftenreihe: | Progress in mathematics
216 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references (p. [253]-257) and index |
| Beschreibung: | xxii, 264 p. ill. |
| ISBN: | 0817637648 3764337648 |
Internformat
MARC
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| 100 | 1 | |a Borovik, Alexandre |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Coxeter matroids |c Alexandre V. Borovik ; I. M. Gelfand ; Neil White |
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 2003 | |
| 300 | |a xxii, 264 p. |b ill. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Progress in mathematics |v 216 | |
| 500 | |a Includes bibliographical references (p. [253]-257) and index | ||
| 650 | 4 | |a Coxeter-Gruppe | |
| 650 | 4 | |a Matroid | |
| 650 | 4 | |a Matroids | |
| 650 | 4 | |a Matroid - Coxeter-Gruppe | |
| 700 | 1 | |a Gelʹfand, Izrailʹ M. |d 1913-2009 |e Verfasser |0 (DE-588)118831364 |4 aut | |
| 700 | 1 | |a White, Neil |e Verfasser |4 aut | |
| 830 | 0 | |a Progress in mathematics |v 216 |w (DE-604)BV000004120 |9 216 | |
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Datensatz im Suchindex
| _version_ | 1804132718014889984 |
|---|---|
| adam_text | Contents
Introduction vii
Preface xi
1 Matroids and Flag Matroids 1
1.1 Matroids 1
1.1.1 Definition in terms of bases 2
1.1.2 Examples 2
1.1.3 Circuits 4
1.2 Representable matroids 5
1.3 Maximality Property 7
1.4 Increasing Exchange Property 9
1.5 Sufficient systems of exchanges 10
1.5.1 Strong Exchange Property 11
1.6 Matroids as maps 12
1.7 Flag matroids 12
1.7.1 Flags 12
1.7.2 Flag matroids 13
1.7.3 Matroid quotients 13
1.7.4 Equivalence of Maximality Property and concordance of
constituents 14
1.7.5 Representable flag matroids 15
1.7.6 Higgslift 17
1.8 Flag matroids as maps 18
1.9 Exchange properties for flag matroids 19
1.9.1 Increasing Exchange Property for flag matroids 19
1.9.2 Failure of the Strong Exchange Property for flag matroids .. 19
1.10 Root system 20
1.10.1 Roots 20
1.10.2 Transpositions and reflections 21
1.10.3 Geometric representation of flags 22
xvi Contents
1.10.4 Orderings associated with the root system 23
1.11 Polytopes associated with flag matroids 24
1.11.1 Polytopes associated with flag matroids 24
1.11.2 Main Theorem 25
1.12 Properties of matroid polytopes 27
1.12.1 Adjacency in matroids 27
1.12.2 Groups generated by transpositions 27
1.12.3 Components of matroids and the transposition graph 28
1.12.4 2 dimensional faces of matroid polytopes 29
1.12.5 Dimension of the matroid polytope 30
1.13 Minkowski sums 30
1.14 Exercises for Chapter 1 33
2 Matroids and Semimodular Lattices 37
2.1 Lattices as generalizations of projective geometry 38
2.2 Semimodular lattices 38
2.3 Jordan Holder permutation 39
2.4 Geometric lattices 42
2.4.1 Bases of lattices 42
2.4.2 Closure operators 43
2.4.3 Geometric lattice determined by a matroid 43
2.5 Representations of matroids 44
2.6 Representation of flag matroids 47
2.6.1 Retractions 48
2.6.2 Matroid maps from chains 49
2.7 Every flag matroid is representable 50
2.8 Exercises for Chapter 2 52
3 Symplectic Matroids 55
3.1 Definition of symplectic matroids 55
3.1.1 Hyperoctahedral group and admissible permutations 55
3.1.2 Admissible orderings 56
3.1.3 Symplectic matroids 57
3.2 Root systems of type Cn 58
3.2.1 Roots 58
3.2.2 Simple systems of roots 58
3.2.3 Correspondences 59
3.3 Polytopes associated with symplectic matroids 60
3.3.1 Geometric representation of admissible sets 60
3.3.2 Gelfand Serganova Theorem for symplectic matroids 61
3.4 Representable symplectic matroids 63
3.4.1 Isotropic subspaces 63
3.4.2 Symplectic matroids from isotropic subspaces 64
3.4.3 Examples 65
3.4.4 Operations on representations 66
Contents xvii
3.5 Homogeneous symplectic matroids 67
3.6 Symplectic flag matroids 69
3.6.1 Examples 70
3.6.2 Representable symplectic flag matroids 71
3.7 Greedy Algorithm 73
3.8 Independent sets 74
3.9 Symplectic matroid constructions 75
3.10 Orthogonal matroids 75
3.10.1 Dn admissible orderings 75
3.10.2 Orthogonal matroids 76
3.10.3 Representable orthogonal matroids 77
3.10.4 Orthogonal flag matroids 77
3.11 Open problems 77
3.12 Exercises for Chapter 3 78
4 Lagrangian Matroids 81
4.1 Lagrangian matroids 81
4.1.1 Transversals 81
4.1.2 Symmetric Exchange Axiom 82
4.1.3 Represented Lagrangian matroids 83
4.1.4 Homogeneous Lagrangian matroids 84
4.2 Circuits and strong exchange 84
4.2.1 Dual matroid 84
4.2.2 Circuits 85
4.2.3 Circuits and cocircuits 86
4.2.4 Strong Exchange Property 87
4.2.5 Circuit characterizations of Lagrangian matroids 88
4.3 Maps on orientable surfaces 91
4.3.1 Maps on compact surfaces 91
4.3.2 Matroids, representations and maps 92
4.4 Exercises for Chapter 4 98
5 Reflection Groups and Coxeter Groups 101
5.1 Hyperplane arrangements 101
5.1.1 Chambers of a hyperplane arrangement 101
5.1.2 Galleries 103
5.2 Polyhedra and polytopes 105
5.3 Mirrors and reflections 106
5.3.1 Systems of mirrors and of reflections 107
5.3.2 Finite reflection groups 108
5.4 Root systems 109
5.4.1 Mirrors and their normal vectors 109
5.4.2 Root systems 110
5.4.3 Positive and simple systems Ill
5.4.4 Classification of root systems 112
xviii Contents
5.5 Isotropy groups 113
5.6 Parabolic subgroups 113
5.7 Coxeter complex 114
5.7.1 Chambers 114
5.7.2 Generation by simple reflections 116
5.7.3 Action of W on W 117
5.8 Labeling of the Coxeter complex 117
5.9 Galleries 118
5.9.1 Bending 120
5.10 Generators and relations 122
5.10.1 Coxeter group 122
5.11 Convexity 123
5.12 Residues 125
5.12.1 The mirror system of a residue 126
5.12.2 Residues are convex 127
5.12.3 Gate property of residues 128
5.12.4 Opposite chamber in a residue 129
5.13 Foldings 129
5.14 Bruhat order 130
5.14.1 Characterization of the Bruhat order 131
5.14.2 Bruhat ordering on W/Wj 133
5.15 Splitting the Bruhat order 135
5.15.1 Some properties of the length function /(to) 135
5.15.2 The property Z 136
5.16 Generalized permutahedra 138
5.17 Symmetric group as a Coxeter group 141
5.17.1 Coxeter complex of the symmetric group 141
5.17.2 Permutahedron 142
5.17.3 Length in Symn 142
5.17.4 Bruhat order in Symn 143
5.18 Exercises for Chapter 5 144
6 Coxeter Matroids 151
6.1 Coxeter matroids 151
6.1.1 The Maximality Property 152
6.1.2 Matroid maps 152
6.1.3 Flag matroids are Coxeter matroids 153
6.1.4 The Strong Exchange Property 154
6.1.5 The Increasing Exchange Property 154
6.2 Root systems 155
6.2.1 Orbits of W on V 155
6.2.2 Orderings of W ¦ a j 156
6.3 The Gelfand Serganova Theorem 157
6.3.1 A Useful reformulation of the Gelfand Serganova Theorem. 159
6.3.2 A corollary 159
Contents xix
6.4 Coxeter matroids and polytopes 159
6.5 Examples 160
6.6 W matroids 161
6.7 Characterization of matroid maps 168
6.8 Adjacency in matroid polytopes 169
6.9 Combinatorial adjacency 170
6.10 The matroid polytope 172
6.11 Exchange groups of Coxeter matroids 174
6.11.1 Dimension of the matroid polytope 175
6.12 Flag matroids and concordance 175
6.12.1 Shifts 176
6.12.2 Concordance 177
6.12.3 Constituents of a flag matroid 178
6.13 Combinatorial flag variety 179
6.13.1 Definition of the combinatorial flag variety 179
6.13.2 Weak map ordering 181
6.13.3 Expansion 181
6.14 Shellable simplicial complexes 183
6.15 Shellability of the combinatorial flag variety 186
6.16 Open problems 187
6.17 Exercises for Chapter 6 189
7 Buildings 199
7.1 Gaussian decomposition 199
7.2 BN pedis 202
7.2.1 Definition of a BN paix 202
7.2.2 Standard generators are involutions 203
7.2.3 Length function 203
7.2.4 Bruhat decomposition 204
7.2.5 Refinement of Axiom BN1 205
7.3 Deletion Property 206
7.4 Deletion property and Coxeter groups 208
7.5 Reflection representation of W 211
7.5.1 Construction 211
7.5.2 The Coxeter graph 213
7.5.3 Irreducibility of the reflection representation 213
7.5.4 Finite Coxeter groups are Euclidean reflection groups 214
7.5.5 Positive and negative roots 215
7.5.6 The reflection representation is faithful 215
7.6 Classification of finite Coxeter groups 216
7.6.1 Labeled graphs and associated bilinear forms 216
7.6.2 Classification of positive definite graphs 216
7.7 Chamber systems 220
7.7.1 Chamber systems 220
7.7.2 Coxeter complex 220
xx Contents
7.7.3 Residues and parabolic subgroups 220
7.7.4 The geometric realization 221
7.7.5 Hag complex of a vector space 222
7.8 W metric 223
7.8.1 W metrics and associated chamber systems 223
7.8.2 Order complex of a semimodular lattice admits a W metric . 224
7.9 Buildings 226
7.9.1 Definition of buildings 226
7.9.2 Generalized wi gons 226
7.9.3 Buildings of projective spaces 228
7.9.4 Building associated with a BN paii 230
7.9.5 Strongly transitive automorphism groups 231
7.10 Representing Coxeter matroids in buildings 233
7.10.1 Retractions 233
7.10.2 Apartments are convex 234
7.10.3 Geodesic galleries and reduced words 235
7.10.4 Retractions give matroid maps 236
7.11 Vector space representations and building representations 237
7.11.1 An, Bn, Cn and Dn representations 237
7.11.2 Buildings from flags of subspaces 238
7.11.3 Vector space representations of W matroids are building
representations 239
7.12 Residues in buildings 240
7.12.1 Residues are convex 240
7.12.2 Residues are buildings 240
7.12.3 Intersection of residues 241
7.12.4 Intersection of a residue and an apartment 241
7.13 Buildings of type An_i = Symn 241
7.14 Combinatorial flag varieties, revisited 243
7.14.1 Gaussian schemes 243
7.14.2 Retractions 245
7.14.3 Representation morphism 245
7.14.4 Partial metric on Q.*w 246
7.14.5 The case W = An 248
7.15 Open Problems 248
7.16 Exercises for Chapter 7 250
References 253
Index 259
|
| any_adam_object | 1 |
| author | Borovik, Alexandre Gelʹfand, Izrailʹ M. 1913-2009 White, Neil |
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| id | DE-604.BV019307087 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:57:16Z |
| institution | BVB |
| isbn | 0817637648 3764337648 |
| language | English |
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| series | Progress in mathematics |
| series2 | Progress in mathematics |
| spelling | Borovik, Alexandre Verfasser aut Coxeter matroids Alexandre V. Borovik ; I. M. Gelfand ; Neil White Boston [u.a.] Birkhäuser 2003 xxii, 264 p. ill. txt rdacontent n rdamedia nc rdacarrier Progress in mathematics 216 Includes bibliographical references (p. [253]-257) and index Coxeter-Gruppe Matroid Matroids Matroid - Coxeter-Gruppe Gelʹfand, Izrailʹ M. 1913-2009 Verfasser (DE-588)118831364 aut White, Neil Verfasser aut Progress in mathematics 216 (DE-604)BV000004120 216 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012775053&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Borovik, Alexandre Gelʹfand, Izrailʹ M. 1913-2009 White, Neil Coxeter matroids Progress in mathematics Coxeter-Gruppe Matroid Matroids Matroid - Coxeter-Gruppe |
| title | Coxeter matroids |
| title_auth | Coxeter matroids |
| title_exact_search | Coxeter matroids |
| title_full | Coxeter matroids Alexandre V. Borovik ; I. M. Gelfand ; Neil White |
| title_fullStr | Coxeter matroids Alexandre V. Borovik ; I. M. Gelfand ; Neil White |
| title_full_unstemmed | Coxeter matroids Alexandre V. Borovik ; I. M. Gelfand ; Neil White |
| title_short | Coxeter matroids |
| title_sort | coxeter matroids |
| topic | Coxeter-Gruppe Matroid Matroids Matroid - Coxeter-Gruppe |
| topic_facet | Coxeter-Gruppe Matroid Matroids Matroid - Coxeter-Gruppe |
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