Matroidal representations of low rank
Jaiung Jun, Kalina Mincheva, Jeffrey Tolliver
TL;DR
The paper develops a framework to study tropical subrepresentations of the Boolean regular representation $\mathbb{B}[G]$ as matroidal representations, proving a complete classification for rank-2 and rank-3 cases. It shows that rank-2 $G$-invariant matroids on a ground set correspond to $G$-invariant nontrivial equivalence relations, while rank-3 matroids correspond to data $(H,\sim)$ with compatibility conditions on $G/H$, with simplicity tied to $H$ being trivial. In the abelian setting, it introduces distinct difference systems generalizing modular Golomb rulers, enabling constructive families of simple rank-$k$ matroids; in particular, the prime-cyclic case yields a full classification for simple rank-3 $G$-invariant matroids. The results connect matroid theory, tropical geometry, and number theoretic constructions, enhancing understanding of how group structure constrains matroidal representations and offering new avenues for applications in combinatorial and tropical contexts.
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.