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.Comment: Version 3 has a new title and a few other minor corrections; 38
pages, including a 6-page Jupyter notebook that contains SageMath code and
that is also available in the ancillary file