Oriented Matroid Circuit Polytopes
arXiv:2501.00108
The paper studies polytopes defined from the signed circuits and cocircuits of oriented matroids, giving dimension formulas, face descriptions, Ehrhart series, and symmetric group actions, with special focus on type A root system constructions that relate to graphic zonotopes and symmetric edge polytopes.
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.
17 pages, 9 figures, 1 table