Matroid lifts and representability

TL;DR

The paper investigates lifts of matroids beyond elementary lifts via a simplified rank-$k$ lift construction using a matroid on circuits. It proves that a local closure condition yields a well-defined rank-$r(N)$ lift $M^N$ of $M$ and that this subsumes Walsh's original construction, with representable matroids satisfying a universal 'every lift is $M^N$' property, while non-representable cases disprove universality. The work provides a concrete non-representable counterexample, $K(r,t)$, showing the converse fails in general and yielding new non-representability certificates with potential for infinite antichains. It then applies the framework to gain graphs, establishing a classification: for $n \ge 4$ non-elementary lifts exist precisely when the group $\Gamma$ has a nontrivial partition (and in the abelian case this forces $\Gamma \cong \mathbb{Z}_p^j$), while for $n=3$ a nontrivial partition yields rank-2 lifts, with an all-groups converse linking the existence of higher-rank lifts to group partitions. Overall, the results extend Zaslavsky's gain-graph approach, reveal limits of Walsh's construction, and furnish a robust certificate for non-representability with wide-reaching implications for matroid theory.

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.

Theorems & Definitions (35)

• Theorem 1: brylawski1986Crapo
• Theorem 2
• Theorem 3
• Theorem 4: walsh2022new
• Theorem 5: walsh2022new
• Theorem 6
• Theorem 7: walsh2022new
• Proposition 8
• proof
• Proposition 9