The geometry of ranked symplectic matroids

TL;DR

This paper investigates the geometric structures of ranked symplectic matroids, showing their relation to ordinary matroids, computing their order complex dimension, and exploring their polytopes and toric varieties.

Contribution

It introduces geometric constructions for ranked symplectic matroids, demonstrating their properties and relationships to ordinary matroids, and provides partial proof of Mason's conjecture.

Findings

Ranked symplectic matroids sit between two ordinary matroids.

Their order complex dimension is computed via the Möbius function.

Their matroid polytope relates to flats and the Bergman fan.

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 M\"obius 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.