Finitary affine oriented matroids

TL;DR

This paper develops the foundational theory of affine oriented matroids, introducing new axioms, studying their properties, and exploring their applications to arrangements and group actions, including shellability and topological classifications.

Contribution

It provides the first axiomatic framework for affine oriented matroids on arbitrary sets, including finitary cases, and connects them to complexes, arrangements, and group actions.

Findings

Proved shellability of finitary affine oriented matroids.

Characterized those affinely homeomorphic to Euclidean space.

Extended the multiplicity Tutte polynomial to group actions on semimatroids.

Abstract

We initiate the axiomatic study of affine oriented matroids (AOMs) on arbitrary ground sets, obtaining fundamental notions such as minors, reorientations and a natural embedding into the frame work of Complexes of Oriented Matroids. The restriction to the finitary case (FAOMs) allows us to study tope graphs and covector posets, as well as to view FAOMs as oriented finitary semimatroids. We show shellability of FAOMs and single out the FAOMs that are affinely homeomorphic to $\mathbb{R}^n$. Finally, we study group actions on AOMs, whose quotients in the case of FAOMs are a stepping stone towards a general theory of affine and toric pseudoarrangements. Our results include applications of the multiplicity Tutte polynomial of group actions of semimatroids, generalizing enumerative properties of toric arrangements to a combinatorially defined class of arrangements of submanifolds. This answers partially a question by Ehrenborg and Readdy.