On symmetry groups of oriented matroids

TL;DR

This paper investigates the symmetries of oriented matroids, revealing differences from geometric symmetries, and classifies certain symmetry groups by analyzing fixed-point properties and introducing the FPA property.

Contribution

It introduces a construction showing non-realizable matroidal symmetries, studies fixed-point properties, and classifies symmetry groups of oriented matroids, proposing a conjecture on FPA property.

Findings

Constructed 3D point configurations with non-geometric matroidal symmetries.

Proved fixed-point properties for rotational symmetries in rank 3.

Classified symmetry groups of simple oriented matroids of rank 3 and 4.

Abstract

Symmetries of geometric structures such as hyperplane arrangements, point configurations and polytopes have been studied extensively for a long time. However, symmetries of oriented matroids, a common combinatorial abstraction of them, are not understood well. In this paper, we aim to obtain a better understanding of symmetries of oriented matroids. First, we put focus on symmetries of matroids, and give a general construction that generates a $3$-dimensional point configuration with a matroidal symmetry that cannot be realized as a geometric symmetry. The construction is based on the observation that every non-trivial rotation in the $2$-dimensional Euclidean space has a unique fixed point but that there is no corresponding property for matroids. The construction suggests that the lack of the fixed point theorem generates a big gap between matroidal symmetries and geometric symmetries of point configurations. Motivated by this insight, we study fixed-point properties for symmetry groups of oriented matroids. For rotational symmetries of oriented matroids, we prove a useful property, which corresponds to the uniqueness of fixed points (in the rank $3$ case). Using it, we classify rotational and full symmetry groups of simple oriented matroids of rank $3$. In addition, we define fixed-point-admitting (FPA) property for subgroups of symmetry groups of oriented matroids, and make classification of rotational symmetry groups with FPA property of simple acyclic oriented matroids of rank $4$. We conjecture that the symmetry group of every acyclic simple oriented matroid has FPA property.