Matrix representations of frame and lifted-graphic matroids correspond to gain functions

TL;DR

This paper establishes a precise correspondence between matrix representations of 3-connected matroids over a field and biased graph representations, settling conjectures and linking projective equivalence classes to gain graph switching classes.

Contribution

It characterizes the relationship between matrix and biased graph representations of matroids, confirming conjectures and describing the structure of their equivalence classes.

Findings

Matrix A is projectively equivalent to a canonical gain graph matrix.

Projective equivalence classes correspond to gain graph switching classes.

Results settle four conjectures of Zaslavsky.

Abstract

Let $M$ be a 3-connected matroid and let $\mathbb F$ be a field. Let $A$ be a matrix over $\mathbb F$ representing $M$ and let $(G,\mathcal B)$ be a biased graph representing $M$. We characterize the relationship between $A$ and $(G,\mathcal B)$, settling four conjectures of Zaslavsky. We show that for each matrix representation $A$ and each biased graph representation $(G,\mathcal{B})$ of $M$, $A$ is projectively equivalent to a canonical matrix representation arising from $G$ as a gain graph over $\mathbb F^+$ or $\mathbb F^\times$ realizing $\mathcal{B}$. Further, we show that the projective equivalence classes of matrix representations of $M$ are in one-to-one correspondence with the switching equivalence classes of gain graphs arising from $(G,\mathcal B)$, except in one degenerate case.