Lattices of flats for symplectic matroids

TL;DR

This paper explores the structure of symplectic matroids with lattices of flats, introduces ranked symplectic matroids, and establishes their properties and connections to ordinary matroids using geometric and combinatorial methods.

Contribution

It provides a new construction of lattices for symplectic matroids, introduces ranked symplectic matroids, and characterizes them via recursive atom orderings.

Findings

Constructed lattices for symplectic matroids from subdivisions of the cross polytope.

Established a correspondence between symplectic and ordinary matroids.

Proved properties like minors and shellability for ranked 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.