Abstract
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.
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Vince, A., White, N. Orthogonal Matroids. Journal of Algebraic Combinatorics 13, 295–315 (2001). https://doi.org/10.1023/A:1011212331779
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DOI: https://doi.org/10.1023/A:1011212331779