Coxeter Matroids

Neil White, Journalist und ehemaliger Verleger von Hochglanzmagazinen (New Orleans Magazine, Coast Magazine, Coast Business Journal), lebt heute als Autor von Theaterstücken und Essays in Oxford, Mississippi, wo er einen kleinen Verlag besitzt.

Matroids appear in diverse areas of mathematics, from combinatorics to algebraic topology and geometry, and "Coxeter Matroids" provides an intuitive and interdisciplinary treatment of their theory. In this text, matroids are examined in terms of symmetric and finite reflection groups; also, symplectic matroids and the more general coxeter matroids are carefully developed. The Gelfand-Serganova theorem, which allows for the geometric interpretation of matroids as convex polytopes with certain symmetry properties, is presented, and in the final chapter, matroid representations and combinatorial flag varieties are discussed. With its excellent bibliography and index and ample references to current research, this work will be useful for graduate students and research mathematicians.

1 Matroids and Flag Matroids.- 1.1 Matroids.- 1.2 Representable matroids.- 1.3 Maximality Property.- 1.4 Increasing Exchange Property.- 1.5 Sufficient systems of exchanges.- 1.6 Matroids as maps.- 1.7 Flag matroids.- 1.8 Flag matroids as maps.- 1.9 Exchange properties for flag matroids.- 1.10 Root system.- 1.11 Polytopes associated with flag matroids.- 1.12 Properties of matroid polytopes.- 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.- 2.5 Representations of matroids.- 2.6 Representation of flag matroids.- 2.7 Every flag matroid is representable.- 2.8 Exercises for Chapter 2.- 3 Symplectic Matroids.- 3.1 Definition of symplectic matroids.- 3.2 Root systems of type Cn.- 3.3 Polytopes associated with symplectic matroids.- 3.4 Representable symplectic matroids.- 3.5 Homogeneous symplectic matroids.- 3.6 Symplectic flag matroids.- 3.7 Greedy Algorithm.- 3.8 Independent sets.- 3.9 Symplectic matroid constructions.- 3.10 Orthogonal matroids.- 3.11 Open problems.- 3.12 Exercises for Chapter 3.- 4 Lagrangian Matroids.- 4.1 Lagrangian matroids.- 4.2 Circuits and strong exchange.- 4.3 Maps on orientable surfaces.- 4.4 Exercises for Chapter 4.- 5 Reflection Groups and Coxeter Groups.- 5.1 Hyperplane arrangements.- 5.2 Polyhedra and polytopes.- 5.3 Mirrors and reflections.- 5.4 Root systems.- 5.5 Isotropy groups.- 5.6 Parabolic subgroups.- 5.7 Coxeter complex.- 5.8 Labeling of the Coxeter complex.- 5.9 Galleries.- 5.10 Generators and relations.- 5.11 Convexity.- 5.12 Residues.- 5.13 Foldings.- 5.14 Bruhat order.- 5.15 Splitting the Bruhat order.- 5.16 Generalized permutahedra.- 5.17 Symmetricgroup as a Coxeter group.- 5.18 Exercises for Chapter 5.- 6 Coxeter Matroids.- 6.1 Coxeter matroids.- 6.2 Root systems.- 6.3 The Gelfand-Serganova Theorem.- 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.- 6.12 Flag matroids and concordance.- 6.13 Combinatorial flag variety.- 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.- 7.3 Deletion Property.- 7.4 Deletion property and Coxeter groups.- 7.5 Reflection representation of W.- 7.6 Classification of finite Coxeter groups.- 7.7 Chamber systems.- 7.8 W-metric.- 7.9 Buildings.- 7.10 Representing Coxeter matroids in buildings.- 7.11 Vector-space representations and building representations.- 7.12 Residues in buildings.- 7.13 Buildings of type An-1 = Symn.- 7.14 Combinatorial flag varieties, revisited.- 7.15 Open Problems.- 7.16 Exercises for Chapter 7.- References.