Matroid lifts and representability

TL;DR

This paper investigates matroid lifts, proving a conjecture for representable matroids, providing a new method to identify non-representable matroids, and analyzing a construction related to gain graphs over finite fields.

Contribution

It simplifies Walsh's construction of matroid lifts, proves the conjecture for representable matroids, and refutes it in general, offering new tools for matroid theory and non-representability certification.

Findings

Conjecture holds for representable matroids.

Construction fails for non-representable cases.

Counterexamples for three-vertex gain graph constructions.

Abstract

A 1965 result of Crapo shows that every elementary lift of a matroid $M$ can be constructed from a linear class of circuits of $M$. In a recent paper, Walsh generalized this construction by defining a rank-$k$ lift of a matroid $M$ given a rank-$k$ matroid $N$ on the set of circuits of $M$, and conjectured that all matroid lifts can be obtained in this way. In this sequel paper we simplify Walsh's construction and show that this conjecture is true for representable matroids but is false in general. This gives a new way to certify that a particular matroid is non-representable, which we use to construct new classes of non-representable matroids. Walsh also applied the new matroid lift construction to gain graphs over the additive group of a non-prime finite field, generalizing a construction of Zaslavsky for these special groups. He conjectured that this construction is possible on three or more vertices only for the additive group of a non-prime finite field. We show that this conjecture holds for four or more vertices, but fails for exactly three.