On perturbations of highly connected dyadic matroids

TL;DR

This paper challenges a recent theorem on highly connected matroids over finite fields by providing counterexamples and proposing weaker conjectures, impacting the understanding of matroid perturbations and structure.

Contribution

It presents counterexamples to a known result on dyadic matroids and introduces weaker conjectures to refine the structural theory of matroid perturbations.

Findings

Counterexamples to the existing theorem on dyadic matroids.

Implications for the structure of frame templates.

Discussion on weaker conjectures and their consequences.

Abstract

Geelen, Gerards, and Whittle [3] announced the following result: let $q = p^k$ be a prime power, and let $\mathcal{M}$ be a proper minor-closed class of $\mathrm{GF}(q)$-representable matroids, which does not contain $\mathrm{PG}(r-1,p)$ for sufficiently high $r$. There exist integers $k, t$ such that every vertically $k$-connected matroid in $\mathcal{M}$ is a rank-$(\leq t)$ perturbation of a frame matroid or the dual of a frame matroid over $\mathrm{GF}(q)$. They further announced a characterization of the perturbations through the introduction of subfield templates and frame templates. We show a family of dyadic matroids that form a counterexample to this result. We offer several weaker conjectures to replace the ones in [3], discuss consequences for some published papers, and discuss the impact of these new conjectures on the structure of frame templates.