Orthogonal matroids (Q5939550)

In [Usp. Mat. Nauk 42, No. 2, 107-134 (1987; Zbl 0629.14035), Sov. Math., Dokl. 35, 63-66 (1987); translation from Dokl. Akad. Nauk SSSR 292, 524-528 (1987; Zbl 0645.22005), and Russ. Math. Surv. 42, No. 2, 133-168 (1987; Zbl 0639.14031)] \textit{I. M. Gelfand} and \textit{V. V. Serganova} generalize the notion of matroid to Coxeter matroids. More precisely, to every Coxeter group \(W\) and parabolic subgroup \(P\) these authors associated a family of objects called Coxeter matroids. The present paper initiates the study of the \(W=D_n\) case, called orthogonal matroids because of examples constructed from orthogonal geometries. The main result gives an algebraic, geometric and combinatorial characterization of orthogonal matroids. The proofs make full use of a description of the Bruhat order on \(D_n.\)