Matroids from gain graphs over quotient groups

TL;DR

This paper introduces a new method to construct matroids from gain graphs over Frobenius groups, generalizing previous models and establishing the necessity of Frobenius group structure for certain matroid properties.

Contribution

It presents a novel construction of matroids from gain graphs over Frobenius groups, extending existing frameworks and proving the essential role of Frobenius groups in this context.

Findings

The construction generalizes several existing matroid constructions.

Frobenius group structure is necessary for the described matroid properties.

Characterization of when a gain graph yields an elementary lift of a frame matroid.

Abstract

We present a new construction for matroids from gain graphs that simultaneously generalizes several existing constructions. The construction takes as input a gain graph over a Frobenius group $\Gamma$ with Frobenius kernel $\Gamma_1$ and outputs an elementary lift of the frame matroid of the underlying gain graph over the quotient group $\Gamma/\Gamma_1$. While the hypothesis that $\Gamma$ is a Frobenius group may seem unusual, we prove that it is in some sense necessary: if $\Gamma$ is any finite group with a nontrivial proper normal subgroup $\Gamma_1$ and there is a construction that takes in a complete $\Gamma$-gain graph and outputs an elementary lift $M$ of the frame matroid of the underlying $(\Gamma/\Gamma_1)$-gain graph so that a cycle of the graph is a circuit of $M$ if and only if it is $\Gamma$-balanced, then $\Gamma$ is a Frobenius group with Frobenius kernel $\Gamma_1$.