On a generalization of symmetric edge polytopes to regular matroids

(This talk does NOT assume any prior knowledge of matroids!)

Symmetric edge polytopes are a class of reflexive lattice polytopes depending on the combinatorial data of a graph. Such objects arise in many different contexts, including finite metric space theory, physics and optimal transport, and have been studied extensively in the last few years.

The aim of this talk is to show that symmetric edge polytopes are special instances of a more general construction that associates a reflexive lattice polytope with every regular matroid. A matroid is called regular if it can be represented over every field; by work of Tutte, a matroid is regular if and only if it can be represented by a totally unimodular matrix, i.e. a matrix whose square submatrices of any size all have determinant equal to -1, 0 or 1.

We will show that regular matroids are the right framework for studying symmetric edge polytopes, as two (classical) symmetric edge polytopes turn out to be unimodularly equivalent precisely when the two associated graphs give rise to the same graphic matroid up to isomorphism.

This is joint work with Martina Juhnke-Kubitzke and Melissa Koch.