Orthogonal Matroids

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 An 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 Bn case, called symplectic matroids. This paper initiates the study of the Dn 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 Dn (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.