Oriented Lagrangian Orthogonal Matroid Representations

Several attempts have been made to extend the theory of matroids (here referred to as ordinary or classical matroids) to theories of more general objects, in particular the Coxeter matroids of Borovik, Gelfand and White ([7], first introduced as WP-matroids in [10]), and the ∆-matroids and (equivalent but for notation) symmetric matroids of Bouchet (see, for example, [8]). The special cases of Coxeter matroids for the Coxeter groups BCn and Dn and a maximal parabolic subgroup are called symplectic and orthogonal matroids respectively, and may be viewed as collections of k-element subsets of the 2n-element set {1, . . . ,n,1∗, . . . ,n∗} with maximality conditions, where k is between 1 and n. In the case where k = n, these structures are called Lagrangian matroids and are isomorphic in a natural way to Bouchet’s symmetric matroids [6, 11], with orthogonal matroids giving even symmetric matroids. Classical matroids now appear as a special case of even Lagrangian matroids. A concept of representation of even ∆and symmetric matroids by skew-symmetric n× n matrices was developed in [9]. In turn, symplectic and orthogonal matroids may be represented by k-dimensional totally isotropic subspaces of 2n-dimensional symplectic and orthogonal vector spaces [6, 11]; it is from this that the names of these structures arise. Attempts have also been made to extend the (classical) theory of oriented matroids to this larger concept. A theory of orientation of Lagrangian symplectic matroids was presented in [4]. However, in the case when the matroid is even (as all orthogonal matroids are), this theory is both uninteresting and trivial; in particular, it is uninteresting for classical matroids. In [12], Wenzel presents an orientation concept for even ∆-matroids, and their representations, which includes classical oriented matroids as a special case. In this paper we extend this theory to Lagrangian orthogonal matroids and their representations, and give a completely natural transformation from a representation of a classical oriented matroid to a representation of the same oriented matroid embedded as a Lagrangian orthogonal matroid. We are interested in representations of Lagrangian matroids as isotropic subspaces because such representations arise in the study of maps on surfaces [2, 3], and also because of their natural connections with Schubert cells. Since classical represented matroids correspond to thin Schubert cells in the Grassmannian [5], oriented matroids provide a stratification of the Grassmannian finer than thin Schubert cells but coarser than their connected components. Similarly, these other concepts of orientation provide stratifications of Lagrangian varieties which split thin Schubert cells into unions of connected components.