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Matroids and Flag Matroids
Matroids and Semimodular Lattices
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adjacent admissible ordering arbitrary Assume Axiom basis Bruhat order building of type called Chapter circuit coedge collection of bases columns common panel concordant constituents contains convex hull Corollary corresponding cosets Coxeter complex Coxeter group Coxeter matroids defined Deletion denote dimension edge elements equivalent example Exercise finite Coxeter group flag matroid flag variety Gale order Gaussian scheme geometric lattice graph hence hyperplane Increasing Exchange Property intersection isotropic subspaces Lagrangian matroid Lagrangian orthogonal matroid Lemma linear matrix matroid map matroid of rank maximal chains maximal element Maximality Property mirrors Notice obviously ordinary matroids parabolic subgroup permutation Prove quotient reflection group representation residues root system satisfies scalar semimodular lattice simplicial complex spanned Strong Exchange Property subsets symmetric group symplectic flag matroid symplectic matroid Theorem totally isotropic transposition vector space vertex vertices w e Symn W-matroid wall H