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Oriented Matroid Circuit Polytopes

Laura Escobar, Jodi McWhirter

TL;DR

The paper introduces oriented matroid circuit (OMC) polytopes $P_{\, ho{M}}$ derived from signed circuits, proving circuit-to-vertex correspondence and central symmetry. It develops a complete theory for graphical and cocircuit OMC polytopes, giving exact dimension formulas and detailing how bridges, loops, and dualities affect the polytopes. Focusing on the type A family with the complete graph, the authors identify $ ext{P}_{n-1}$ with the cycle graphic zonotope, the polar dual of the symmetric edge polytope of $K_n$, and the tropical unit ball, and they fully describe its face structure and Ehrhart theory. They further compute the equivariant Ehrhart theory under the $S_{n+1}$ action, including fixed polytopes $ ext{P}_n^{\sigma}$ and the corresponding $H^*$-series, connecting to Stapledon’s framework and providing a new, direct derivation via fixed polytopes. The work reveals rich connections between oriented matroids, zonotopes, toric geometry, and equivariant combinatorics, offering a unified approach to studying these polytopes across several Coxeter-type contexts.

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.

Oriented Matroid Circuit Polytopes

TL;DR

The paper introduces oriented matroid circuit (OMC) polytopes derived from signed circuits, proving circuit-to-vertex correspondence and central symmetry. It develops a complete theory for graphical and cocircuit OMC polytopes, giving exact dimension formulas and detailing how bridges, loops, and dualities affect the polytopes. Focusing on the type A family with the complete graph, the authors identify with the cycle graphic zonotope, the polar dual of the symmetric edge polytope of , and the tropical unit ball, and they fully describe its face structure and Ehrhart theory. They further compute the equivariant Ehrhart theory under the action, including fixed polytopes and the corresponding -series, connecting to Stapledon’s framework and providing a new, direct derivation via fixed polytopes. The work reveals rich connections between oriented matroids, zonotopes, toric geometry, and equivariant combinatorics, offering a unified approach to studying these polytopes across several Coxeter-type contexts.

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.
Paper Structure (13 sections, 16 theorems, 51 equations, 9 figures, 1 table)

This paper contains 13 sections, 16 theorems, 51 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Oriented matroid circuit polytopes
  3. Oriented matroid circuit polytopes
  4. Graphical oriented matroid circuit polytopes
  5. Graphical oriented matroid cocircuit polytopes
  6. Bridges and loops
  7. A closer look: The complete graph
  8. The polytope $\mathcal{P}_{n-1}$
  9. Other appearances
  10. Face structure of $\mathcal{P}_{n-1}$
  11. Ehrhart theory of $\mathcal{P}_{n-1}$
  12. Equivariant Ehrhart theory of $\mathcal{P}_{n-1}$
  13. Two equivariant Ehrhart theories

Key Result

Proposition 2.4

The vertices of $P_\mathcal{M}$ are precisely those $\textbf{v}_{\widetilde{X}}$ such that $\widetilde{X}$ is a circuit of $\mathcal{M}$.

Figures (9)

  • Figure 1: The complete graph $\widetilde{K_3}$ on 3 nodes, oriented so that the arrow points at the larger number.
  • Figure 2: A cut of $\widetilde{K_3}$ with $V_1 = \{1\}$ and $V_2 = \{2,3\}$. This depicts the cocircuits $(\{\vec{12},\vec{13}\},\varnothing)$ and $(\varnothing,\{\vec{12},\vec{13}\})$.
  • Figure 3: $\mathcal{P}_{2}$ with aff$(\mathcal{P}_{2})$ pictured as generated by $(1,1,0)$ and $(-1,0,1)$.
  • Figure 4: $\mathcal{P}_{2}$ (left) and $\mathcal{P}_{3}$ (right).
  • Figure 5: The polar dual of the symmetric edge polytope $\mathcal{S}_{\widetilde{G}}$ (left) and the OMC polytope $P_{\mathcal{M}^*(\widetilde{G})}$ (right).
  • ...and 4 more figures

Theorems & Definitions (33)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • Remark 2.6
  • Proposition 2.7
  • proof
  • Proposition 2.8
  • ...and 23 more