Orthogonal matroids

Vince, Andrew ; White, Neil

Journal of Algebraic Combinatorics, Tome 13 (2001) no. 3, pp. 295-315.

Résumé

Summary: The notion of matroid has been generalized to Coxeter matroid by Gelfand and Serganova. To each pair (W, P) consisting of a finite irreducible Coxeter group $W$ and parabolic subgroup $P$ is associated a collection of objects called Coxeter matroids. The (ordinary) matroids are the special case where $W$ is the symmetric group (the $A _{n}$ case) and $P$ is a maximal parabolic subgroup.This generalization of matroid introduces interesting combinatorial structures corresponding to each of the finite Coxeter groups. Borovik, Gelfand and White began an investigation of the $B _{n}$ case, called symplectic matroids. This paper initiates the study of the $D _{n}$ case, called orthogonal matroids. The main result (Theorem 2) gives three characterizations of orthogonal matroid: algebraic, geometric, and combinatorial. This result relies on a combinatorial description of the Bruhat order on $D _{n}$ (Theorem 1). The representation of orthogonal matroids by way of totally isotropic subspaces of a classical orthogonal space (Theorem 5) justifies the terminology orthogonal matroid.

Détail

Keywords: orthogonal matroid, Coxeter matroid, Coxeter group, Bruhat order