Oriented Lagrangian Matroids

The aim of this paper is to develop, by analogy with the theory of oriented (ordinary) matroids [2], an oriented version of the theory of Lagrangian symplectic matroids [7]. Recall that the concept of an oriented matroid axiomatises, in combinatorial form, the properties of the ensemble of non-zero k× k minors of a real k× n matrix A, say. The k-subsets of indices of columns of A such that the corresponding k × k minors are non-zero form the collection of bases of an (ordinary) matroid [18]. When we consider also the signs of these minors we obtain a considerably richer combinatorial structure, an (ordinary) oriented matroid. These structures have important applications in combinatorics, geometry and topology. To give an idea of the concept of an oriented Lagrangian matroid, we consider one of the simplest situations in which they arise. Assume that we have a symmetric n×n matrix A over some field K. The collection of sets of row (equivalently, column) indices K ⊆{1, . . . ,n} corresponding to non-zero diagonal minors |ai j|i, j∈K of A forms what is known as a ∆-matroid [10] (we include for formal reasons the empty set K = / 0, and specify that a 0×0 minor takes the value 1). These are equivalent structures to Lagrangian symplectic matroids. When K = R, we may consider also the signs of these minors. What follows in this introduction is a very elementary description of the resulting combinatorial structure; all necessary proofs, and detailed explanations, may be found later in the paper. Let our symmetric n× n matrix A over K have columns indexed by I = 1, . . . ,n, and let e1, . . . ,en denote the standard basis in Rn. Given some B ⊆ I we define a point in Rn by B → ∑ i∈B ei −∑ i/ ∈B ei,