Matroidal representations of low rank

TL;DR

This paper classifies tropical subrepresentations of the Boolean regular representation of finite groups, focusing on rank 3, and introduces new matroids related to equivalence relations, with connections to Golomb rulers.

Contribution

It provides a complete classification of rank 3 tropical subrepresentations of Boolean regular representations and generalizes Golomb rulers for abelian groups.

Findings

Classified all rank 3 tropical subrepresentations of $B[G]$

Introduced a new class of matroids from equivalence relations

Connected matroid theory with group actions and combinatorial structures

Abstract

We study tropical subrepresentations of the Boolean regular representation $\mathbb{B}[G]$ of a finite group $G$. These are equivalent to the matroids on ground set $G$ for which left-multiplication by each element of $G$ is a matroid automorphism. We completely classify the tropical subrepresentations of $\mathbb{B}[G]$ for rank 3. When $G$ is an abelian group, our approach can be seen as a generalization of Golomb rulers. In doing so, we also introduce an interesting class of matroids obtained from equivalence relations on finite sets.