The geometry of ranked symplectic matroids

Abstract

This paper is a continuation of my paper "Lattices of flats for symplectic matroids". We explore geometric constructions originating from the lattice of flats of ranked symplectic matroids. We observe that a ranked symplectic matroid always sits between two ordinary matroids and use this fact to prove that it has many of the same properties of ordinary matroids. We compute the dimension of its order complex using its Möbius function, We show that its matroid polytope is geometrically defined using its flats and connected to its Bergman fan. We finish by highlighting differences between its toric variety and the toric variety of an ordinary matroid, and give a partial proof of Mason's conjecture for ranked symplectic matroids.