Lattices of flats for symplectic matroids

Abstract

We are interested in expanding our understanding of symplectic matroids by exploring the properties of a class of symplectic matroids with a "lattice of flats". Taking a well-behaved family of subdivisions of the cross polytope we obtain a construction of lattices, resembling a known definition for the geometric lattice corresponding to ordinary matroid. We construct a correspondence to a set of enveloped symplectic matroids, we denote ranked symplectic matroids. As a by-product of our construction, we also obtain a new way of finding symplectic matroids from ordinary ones and an embedding Theorem into geometric lattices. The second part of this paper is dedicated to the properties of ranked symplectic matroids and their enveloping ordinary matroids. We focus on establishing a geometric approach to the study of ranked symplectic matroids, demonstrating the ability to take minors, and proving shellability. We finish with a characterization of ranked symplectic matroids using recursive atom orderings.