Oriented Matroid Circuit Polytopes

TL;DR

This paper explores polytopes derived from oriented matroids, providing dimension formulas, face structures, Ehrhart series, and symmetry actions, linking combinatorics, geometry, and algebraic properties.

Contribution

It offers explicit descriptions of polytopes from oriented matroids, including their face structures, Ehrhart series, and symmetry group actions, especially in type A root systems.

Findings

Dimensions of polytopes from graphical oriented matroids determined

Explicit face structures and Ehrhart series for type A polytopes provided

Symmetric group actions on polytopes fully characterized

Abstract

Matroids give rise to several natural constructions of polytopes. Inspired by this, we examine polytopes that arise from the signed circuits of an oriented matroid. We give the dimensions of these polytopes arising from graphical oriented matroids and their duals. Moreover, we consider polytopes constructed from cocircuits of oriented matroids generated by the positive roots in any type A root system. We give an explicit description of their face structure and determine the Ehrhart series. We also study an action of the symmetric group on these polytopes, giving a full description the subpolytopes fixed by each permutation. These type A polytopes are graphic zonotopes, are polar duals of symmetric edge polytopes, and also make an appearance in Stapledon's paper introducing Equivariant Ehrhart Theory.