On a generalization of symmetric edge polytopes to regular matroids

TL;DR

This paper reveals that symmetric edge polytopes are fundamentally linked to graphical matroids and extends their construction to all regular matroids, providing combinatorial descriptions and triangulations.

Contribution

It establishes the matroidal nature of symmetric edge polytopes and introduces a generalized construction for all regular matroids with explicit combinatorial and geometric properties.

Findings

Symmetric edge polytopes are unimodularly equivalent iff they share the same graphical matroid.

A generalized symmetric edge polytope can be constructed from any regular matroid.

The Ehrhart theory of the polar of these polytopes relates to the lattice of flows of the dual matroid.

Abstract

Starting from any finite simple graph, one can build a reflexive polytope known as a symmetric edge polytope. The first goal of this paper is to show that symmetric edge polytopes are intrinsically matroidal objects: more precisely, we prove that two symmetric edge polytopes are unimodularly equivalent precisely when they share the same graphical matroid. The second goal is to show that one can construct a generalized symmetric edge polytope starting from every regular matroid. Just like in the usual case, we are able to find combinatorial ways to describe the facets and an explicit regular unimodular triangulation of any such polytope. Finally, we show that the Ehrhart theory of the polar of a given generalized symmetric edge polytope is tightly linked to the structure of the lattice of flows of the dual regular matroid.