Lagrangian Matroids: Representations of Type Bn

Coxeter matroids are combinatorial objects associated with finite Coxeter groups; they can be viewed as subsets M of the factor set W/P of a Coxeter group W by a parabolic subgroup P which satisfy a certain maximality property with respect to a family of shifted Bruhat orders on W/P . The classical matroids of matroid theory are exactly the Coxeter matroids for the symmetric group Symn (which is a Coxeter group of type An−1) and a maximal parabolic subgroup, while the maximality property turns out to be Gale’s classical characterisation of matroids [13]. The theory of Coxeter matroids sheds new light on the classical matroid theory and brings into the consideration a wider class of combinatorial objects [7]. Of this, we can specifically mention Lagrangian matroids, which are Coxeter matroids for the hyperoctahedral group BCn and a particular maximal parabolic subgroup. Lagrangian matroids are cryptomorphically equivalent to symmetric matroids or 2-matroids of Bouchet’s papers [9] and [11]. They are also equivalent to ∆-matroids [9] and to Dress and Havel’s metroids, see [12]. Because of the natural embedding of Coxeter groups Dn < BCn, the even ∆-matroids of Wenzel [15] are in fact Coxeter matroids for Dn. The present paper belongs to a series of publications aimed at the development of the concept of orientation for Coxeter matroids which would generalise the classical oriented matroids [1]. The concept of orientation for even ∆-matroids was introduced by Wenzel [15, 16] and developed by Booth [2] in a form which better fits the general theory. However, as we shall soon see, this concept does not cover all natural orientation structures on Lagrangian matroids.