On Perturbations of Highly Connected Dyadic Matroids

Abstract

Geelen, Gerards, and Whittle [3] announced the following result: let q=pk be a prime power, and let M be a proper minor-closed class of GF(q)-representable matroids, which does not contain PG(r − 1, p) for sufficiently high r. There exist integers k, t such that every vertically k-connected matroid in M is a rank-(≤t) perturbation of a frame matroid or the dual of a frame matroid over 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.</p