内容

目次

1 Matroids and Flag Matroids.- 1.1 Matroids.- 1.1.1 Definition in terms of bases.- 1.1.2 Examples.- 1.1.3 Circuits.- 1.2 Representable matroids.- 1.3 Maximality Property.- 1.4 Increasing Exchange Property.- 1.5 Sufficient systems of exchanges.- 1.5.1 Strong Exchange Property.- 1.6 Matroids as maps.- 1.7 Flag matroids.- 1.7.1 Flags.- 1.7.2 Flag matroids.- 1.7.3 Matroid quotients.- 1.7.4 Equivalence of Maximality Property and concordance of constituents.- 1.7.5 Representable flag matroids.- 1.7.6 Higgs lift.- 1.8 Flag matroids as maps.- 1.9 Exchange properties for flag matroids.- 1.9.1 Increasing Exchange Property for flag matroids.- 1.9.2 Failure of the Strong Exchange Property for flag matroids.- 1.10 Root system.- 1.10.1 Roots.- 1.10.2 Transpositions and reflections.- 1.10.3 Geometric representation of flags.- 1.10.4 Orderings associated with the root system.- 1.11 Polytopes associated with flag matroids.- 1.11.1 Polytopes associated with flag matroids.- 1.11.2 Main Theorem.- 1.12 Properties of matroid polytopes.- 1.12.1 Adjacency in matroids.- 1.12.2 Groups generated by transpositions.- 1.12.3 Components of matroids and the transposition graph.- 1.12.4 2-dimensional faces of matroid polytopes.- 1.12.5 Dimension of the matroid polytope.- 1.13 Minkowski sums.- 1.14 Exercises for Chapter 1.- 2 Matroids and Semimodular Lattices.- 2.1 Lattices as generalizations of projective geometry.- 2.2 Semimodular lattices.- 2.3 Jordan—Hölder permutation.- 2.4 Geometric lattices.- 2.4.1 Bases of lattices.- 2.4.2 Closure operators.- 2.4.3 Geometric lattice determined by a matroid.- 2.5 Representations of matroids.- 2.6 Representation of flag matroids.- 2.6.1 Retractions.- 2.6.2 Matroid maps from chains.- 2.7 Every flag matroid is representable.- 2.8 Exercises for Chapter 2.- 3 Symplectic Matroids.- 3.1 Definition of symplectic matroids.- 3.1.1 Hyperoctahedral group and admissible permutations.- 3.1.2 Admissible orderings.- 3.1.3 Symplectic matroids.- 3.2 Root systems of type Cn.- 3.2.1 Roots.- 3.2.2 Simple systems of roots.- 3.2.3 Correspondences.- 3.3 Polytopes associated with symplectic matroids.- 3.3.1 Geometric representation of admissible sets.- 3.3.2 Gelfand—Serganova Theorem for symplectic matroids.- 3.4 Representable symplectic matroids.- 3.4.1 Isotropic subspaces.- 3.4.2 Symplectic matroids from isotropic subspaces.- 3.4.3 Examples.- 3.4.4 Operations on representations.- 3.5 Homogeneous symplectic matroids.- 3.6 Symplectic flag matroids.- 3.6.1 Examples.- 3.6.2 Representable symplectic flag matroids.- 3.7 Greedy Algorithm.- 3.8 Independent sets.- 3.9 Symplectic matroid constructions.- 3.10 Orthogonal matroids.- 3.10.1 Dn-admissible orderings.- 3.10.2 Orthogonal matroids.- 3.10.3 Representable orthogonal matroids.- 3.10.4 Orthogonal flag matroids.- 3.11 Open problems.- 3.12 Exercises for Chapter 3.- 4 Lagrangian Matroids.- 4.1 Lagrangian matroids.- 4.1.1 Transversals.- 4.1.2 Symmetric Exchange Axiom.- 4.1.3 Represented Lagrangian matroids.- 4.1.4 Homogeneous Lagrangian matroids.- 4.2 Circuits and strong exchange.- 4.2.1 Dual matroid.- 4.2.2 Circuits.- 4.2.3 Circuits and cocircuits.- 4.2.4 Strong Exchange Property.- 4.2.5 Circuit characterizations of Lagrangian matroids.- 4.3 Maps on orientable surfaces.- 4.3.1 Maps on compact surfaces.- 4.3.2 Matroids, representations and maps.- 4.4 Exercises for Chapter 4.- 5 Reflection Groups and Coxeter Groups.- 5.1 Hyperplane arrangements.- 5.1.1 Chambers of a hyperplane arrangement.- 5.1.2 Galleries.- 5.2 Polyhedra and polytopes.- 5.3 Mirrors and reflections.- 5.3.1 Systems of mirrors and of reflections.- 5.3.2 Finite reflection groups.- 5.4 Root systems.- 5.4.1 Mirrors and their normal vectors.- 5.4.2 Root systems.- 5.4.3 Positive and simple systems.- 5.4.4 Classification of root systems.- 5.5 Isotropy groups.- 5.6 Parabolic subgroups.- 5.7 Coxeter complex.- 5.7.1 Chambers.- 5.7.2 Generation by simple reflections.- 5.7.3 Action of W on W.- 5.8 Labeling of the Coxeter complex.- 5.9 Galleries.- 5.9.1 Bending.- 5.10 Generators and relations.- 5.10.1 Coxeter group.- 5.11 Convexity.- 5.12 Residues.- 5.12.1 The mirror system of a residue.- 5.12.2 Residues are convex.- 5.12.3 Gate property of residues.- 5.12.4 Opposite chamber in a residue.- 5.13 Foldings.- 5.14 Bruhat order.- 5.14.1 Characterization of the Bruhat order.- 5.14.2 Bruhat ordering on W / WJ.- 5.15 Splitting the Bruhat order.- 5.15.1 Some properties of the length function l(w).- 5.15.2 The property Z.- 5.16 Generalized permutahedra.- 5.17 Symmetric group as a Coxeter group.- 5.17.1 Coxeter complex of the symmetric group.- 5.17.2 Permutahedron.- 5.17.3 Length in Symn.- 5.17.4 Bruhat order in Symn.- 5.18 Exercises for Chapter 5.- 6 Coxeter Matroids.- 6.1 Coxeter matroids.- 6.1.1 The Maximality Property.- 6.1.2 Matroid maps.- 6.1.3 Flag matroids are Coxeter matroids.- 6.1.4 The Strong Exchange Property.- 6.1.5 The Increasing Exchange Property.- 6.2 Root systems.- 6.2.1 Orbits of W on V.- 6.2.2 Orderings of W · ?J.- 6.3 The Gelfand—Serganova Theorem.- 6.3.1 A Useful reformulation of the Gelfand—Serganova Theorem.- 6.3.2 A corollary.- 6.4 Coxeter matroids and polytopes.- 6.5 Examples.- 6.6 W-matroids.- 6.7 Characterization of matroid maps.- 6.8 Adjacency in matroid polytopes.- 6.9 Combinatorial adjacency.- 6.10 The matroid polytope.- 6.11 Exchange groups of Coxeter matroids.- 6.11.1 Dimension of the matroid polytope.- 6.12 Flag matroids and concordance.- 6.12.1 Shifts.- 6.12.2 Concordance.- 6.12.3 Constituents of a flag matroid.- 6.13 Combinatorial flag variety.- 6.13.1 Definition of the combinatorial flag variety.- 6.13.2 Weak map ordering.- 6.13.3 Expansion.- 6.14 Shellable simplicial complexes.- 6.15 Shellability of the combinatorial flag variety.- 6.16 Open problems.- 6.17 Exercises for Chapter 6.- 7 Buildings.- 7.1 Gaussian decomposition.- 7.2 BN-pairs.- 7.2.1 Definition of a BN-pair.- 7.2.2 Standard generators are involutions.- 7.2.3 Length function.- 7.2.4 Bruhat decomposition.- 7.2.5 Refinement of Axiom BN1.- 7.3 Deletion Property.- 7.4 Deletion property and Coxeter groups.- 7.5 Reflection representation of W.- 7.5.1 Construction.- 7.5.2 The Coxeter graph.- 7.5.3 Irreducibility of the reflection representation.- 7.5.4 Finite Coxeter groups are Euclidean reflection groups.- 7.5.5 Positive and negative roots.- 7.5.6 The reflection representation is faithful.- 7.6 Classification of finite Coxeter groups.- 7.6.1 Labeled graphs and associated bilinear forms.- 7.6.2 Classification of positive definite graphs.- 7.7 Chamber systems.- 7.7.1 Chamber systems.- 7.7.2 Coxeter complex.- 7.7.3 Residues and parabolic subgroups.- 7.7.4 The geometric realization.- 7.7.5 Flag complex of a vector space.- 7.8 W-metric.- 7.8.1 W-metrics and associated chamber systems.- 7.8.2 Order complex of a sermmodular lattice admits a W-metric.- 7.9 Buildings.- 7.9.1 Definition of buildings.- 7.9.2 Generalized m-gons.- 7.9.3 Buildings of projective spaces.- 7.9.4 Building associated with a BN-pair.- 7.9.5 Strongly transitive automorphism groups.- 7.10 Representing Coxeter matroids in buildings.- 7.10.1 Retractions.- 7.10.2 Apartments are convex.- 7.10.3 Geodesic galleries and reduced words.- 7.10.4 Retractions give matroid maps.- 7.11 Vector-space representations and building representations.- 7.11.1 An, Bn, Cn and Dn-representations.- 7.11.2 Buildings from flags of subspaces.- 7.11.3 Vector-space representations of W-matroids are building representations.- 7.12 Residues in buildings.- 7.12.1 Residues are convex.- 7.12.2 Residues are buildings.- 7.12.3 Intersection of residues.- 7.12.4 Intersection of a residue and an apartment.- 7.13 Buildings of type An-1 = Symn.- 7.14 Combinatorial flag varieties, revisited.- 7.14.1 Gaussian schemes.- 7.14.2 Retractions.- 7.14.3 Representation morphism.- 7.14.4 Partial metric on ?W*.- 7.14.5 The case W = An-1.- 7.15 Open Problems.- 7.16 Exercises for Chapter 7.- References.