Coxeter Matroids

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ISBN: 9780817637644 | 0817637648
Cover: Hardcover
Copyright: 7/1/2003

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Matroids appear in diverse areas of mathematics, from combinatorics to algebraic topology and geometry. This largely self-contained work provides an intuitive and interdisciplinary treatment of Coxeter matroids, a new and beautiful generalization of matroids which is based on a finite Coxeter group.Key topics and features:* Systematic, clearly written exposition with ample references to current research* Matroids are examined in terms of symmetric and finite reflection groups* Finite reflection groups and Coxeter groups are developed from scratch* The Gelfand-Serganova Theorem is presented, allowing for a geometric interpretation of matroids and Coxeter matroids as convex polytopes with certain symmetry properties* Matroid representations and combinatorial flag varieties are studied in the final chapter* Many exercises throughout* Excellent bibliography and indexAccessible to graduate students and research mathematicians alike, Coxeter Matroids can be used as an introductory survey, a graduate course text, or a reference volume.

Introduction	p. vii
Preface	p. xi
Matroids and Flag Matroids	p. 1
Matroids	p. 1
Definition in terms of bases	p. 2
Examples	p. 2
Circuits	p. 4
Representable matroids	p. 5
Maximality Property	p. 7
Increasing Exchange Property	p. 9
Sufficient systems of exchanges	p. 10
Strong Exchange Property	p. 11
Matroids as maps	p. 12
Flag matroids	p. 12
Flags	p. 12
Flag matroids	p. 13
Matroid quotients	p. 13
Equivalence of Maximality Property and concordance of constituents	p. 14
Representable flag matroids	p. 15
Higgs lift	p. 17
Flag matroids as maps	p. 18
Exchange properties for flag matroids	p. 19
Increasing Exchange Property for flag matroids	p. 19
Failure of the Strong Exchange Property for flag matroids	p. 19
Root system	p. 20
Roots	p. 20
Transpositions and reflections	p. 21
Geometric representation of flags	p. 22
Orderings associated with the root system	p. 23
Polytopes associated with flag matroids	p. 24
Polytopes associated with flag matroids	p. 24
Main Theorem	p. 25
Properties of matroid polytopes	p. 27
Adjacency in matroids	p. 27
Groups generated by transpositions	p. 27
Components of matroids and the transposition graph	p. 28
2-dimensional faces of matroid polytopes	p. 29
Dimension of the matroid polytope	p. 30
Minkowski sums	p. 30
Exercises for Chapter 1	p. 33
Matroids and Semimodular Lattices	p. 37
Lattices as generalizations of projective geometry	p. 38
Semimodular lattices	p. 38
Jordan-Holder permutation	p. 39
Geometric lattices	p. 42
Bases of lattices	p. 42
Closure operators	p. 43
Geometric lattice determined by a matroid	p. 43
Representations of matroids	p. 44
Representation of flag matroids	p. 47
Retractions	p. 48
Matroid maps from chains	p. 49
Every flag matroid is representable	p. 50
Exercises for Chapter 2	p. 52
Symplectic Matroids	p. 55
Definition of symplectic matroids	p. 55
Hyperoctahedral group and admissible permutations	p. 55
Admissible orderings	p. 56
Symplectic matroids	p. 57
Root systems of type C[subscript n]	p. 58
Roots	p. 58
Simple systems of roots	p. 58
Correspondences	p. 59
Polytopes associated with symplectic matroids	p. 60
Geometric representation of admissible sets	p. 60
Gelfand-Serganova Theorem for symplectic matroids	p. 61
Representable symplectic matroids	p. 63
Isotropic subspaces	p. 63
Symplectic matroids from isotropic subspaces	p. 64
Examples	p. 65
Operations on representations	p. 66
Homogeneous symplectic matroids	p. 67
Symplectic flag matroids	p. 69
Examples	p. 70
Representable symplectic flag matroids	p. 71
Greedy Algorithm	p. 73
Independent sets	p. 74
Symplectic matroid constructions	p. 75
Orthogonal matroids	p. 75
D[subscript n]-admissible orderings	p. 75
Orthogonal matroids	p. 76
Representable orthogonal matroids	p. 77
Orthogonal flag matroids	p. 77
Open problems	p. 77
Exercises for Chapter 3	p. 78
Lagrangian Matroids	p. 81
Lagrangian matroids	p. 81
Transversals	p. 81
Symmetric Exchange Axiom	p. 82
Represented Lagrangian matroids	p. 83
Homogeneous Lagrangian matroids	p. 84
Circuits and strong exchange	p. 84
Dual matroid	p. 84
Circuits	p. 85
Circuits and cocircuits	p. 86
Strong Exchange Property	p. 87
Circuit characterizations of Lagrangian matroids	p. 88
Maps on orientable surfaces	p. 91
Maps on compact surfaces	p. 91
Matroids, representations and maps	p. 92
Exercises for Chapter 4	p. 98
Reflection Groups and Coxeter Groups	p. 101
Hyperplane arrangements	p. 101
Chambers of a hyperplane arrangement	p. 101
Galleries	p. 103
Polyhedra and polytopes	p. 105
Mirrors and reflections	p. 106
Systems of mirrors and of reflections	p. 107
Finite reflection groups	p. 108
Root systems	p. 109
Mirrors and their normal vectors	p. 109
Root systems	p. 110
Positive and simple systems	p. 111
Classification of root systems	p. 112
Isotropy groups	p. 113
Parabolic subgroups	p. 113
Coxeter complex	p. 114
Chambers	p. 114
Generation by simple reflections	p. 116
Action of W on W	p. 117
Labeling of the Coxeter complex	p. 117
Galleries	p. 118
Bending	p. 120
Generators and relations	p. 122
Coxeter group	p. 122
Convexity	p. 123
Residues	p. 125
The mirror system of a residue	p. 126
Residues are convex	p. 127
Gate property of residues	p. 128
Opposite chamber in a residue	p. 129
Foldings	p. 129
Bruhat order	p. 130
Characterization of the Bruhat order	p. 131
Bruhat ordering on W/W[subscript J]	p. 133
Splitting the Bruhat order	p. 135
Some properties of the length function l(w)	p. 135
The property Z	p. 136
Generalized permutahedra	p. 138
Symmetric group as a Coxeter group	p. 141
Coxeter complex of the symmetric group	p. 141
Permutahedron	p. 142
Length in Sym[subscript n]	p. 142
Bruhat order in Sym[subscript n]	p. 143
Exercises for Chapter 5	p. 144
Coxeter Matroids	p. 151
Coxeter matroids	p. 151
The Maximality Property	p. 152
Matroid maps	p. 152
Flag matroids are Coxeter matroids	p. 153
The Strong Exchange Property	p. 154
The Increasing Exchange Property	p. 154
Root systems	p. 155
Orbits of W on V	p. 155
Orderings of W - [omega subscript J]	p. 156
The Gelfand-Serganova Theorem	p. 157
A Useful reformulation of the Gelfand-Serganova Theorem	p. 159
A corollary	p. 159
Coxeter matroids and polytopes	p. 159
Examples	p. 160
W-matroids	p. 161
Characterization of matroid maps	p. 168
Adjacency in matroid polytopes	p. 169
Combinatorial adjacency	p. 170
The matroid polytope	p. 172
Exchange groups of Coxeter matroids	p. 174
Dimension of the matroid polytope	p. 175
Flag matroids and concordance	p. 175
Shifts	p. 176
Concordance	p. 177
Constituents of a flag matroid	p. 178
Combinatorial flag variety	p. 179
Definition of the combinatorial flag variety	p. 179
Weak map ordering	p. 181
Expansion	p. 181
Shellable simplicial complexes	p. 183
Shellability of the combinatorial flag variety	p. 186
Open problems	p. 187
Exercises for Chapter 6	p. 189
Buildings	p. 199
Gaussian decomposition	p. 199
BN-pairs	p. 202
Definition of a BN-pair	p. 202
Standard generators are involutions	p. 203
Length function	p. 203
Bruhat decomposition	p. 204
Refinement of Axiom BN1	p. 205
Deletion Property	p. 206
Deletion property and Coxeter groups	p. 208
Reflection representation of W	p. 211
Construction	p. 211
The Coxeter graph	p. 213
Irreducibility of the reflection representation	p. 213
Finite Coxeter groups are Euclidean reflection groups	p. 214
Positive and negative roots	p. 215
The reflection representation is faithful	p. 215
Classification of finite Coxeter groups	p. 216
Labeled graphs and associated bilinear forms	p. 216
Classification of positive definite graphs	p. 216
Chamber systems	p. 220
Chamber systems	p. 220
Coxeter complex	p. 220
Residues and parabolic subgroups	p. 220
The geometric realization	p. 221
Flag complex of a vector space	p. 222
W-metric	p. 223
W-metrics and associated chamber systems	p. 223
Order complex of a semimodular lattice admits a W-metric	p. 224
Buildings	p. 226
Definition of buildings	p. 226
Generalized m-gons	p. 226
Buildings of projective spaces	p. 228
Building associated with a BN-pair	p. 230
Strongly transitive automorphism groups	p. 231
Representing Coxeter matroids in buildings	p. 233
Retractions	p. 233
Apartments are convex	p. 234
Geodesic galleries and reduced words	p. 235
Retractions give matroid maps	p. 236
Vector-space representations and building representations	p. 237
A[subscript n], B[subscript n], C[subscript n] and D[subscript n]-representations	p. 237
Buildings from flags of subspaces	p. 238
Vector-space representations of W-matroids are building representations	p. 239
Residues in buildings	p. 240
Residues are convex	p. 240
Residues are buildings	p. 240
Intersection of residues	p. 241
Intersection of a residue and an apartment	p. 241
Buildings of type A[subscript n-1] = Sym[subscript n]	p. 241
Combinatorial flag varieties, revisited	p. 243
Gaussian schemes	p. 243
Retractions	p. 245
Representation morphism	p. 245
Partial metric on [Omega * subscript W]	p. 246
The case W = A[subscript n-1]	p. 248
Open Problems	p. 248
Exercises for Chapter 7	p. 250
References	p. 253
Index	p. 259
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