Kevin Grace

Templates for Representable Matroids

The matroid structure theory of Geelen, Gerards, and Whittle has led to an announced result that a highly connected member of a minor-closed class of matroids representable over a finite field is a mild
modification (known as a perturbation) of a frame matroid, the dual of a frame matroid, or a matroid representable over a proper subfield. They introduced the notion of a template to describe these perturbations in more detail. In this talk, we will define templates and discuss how templates are related to each other. We define a preorder on the set of frame templates over a finite field, and we determine the minimal nontrivial templates with respect to this preorder.

We use templates to obtain results about representability, extremal functions, and excluded minors for various minor-closed classes of matroids, subject to the announced result of Geelen, Gerards, and Whittle. These
classes include the class of 1-flowing matroids and three closely related classes of quaternary matroids -- the golden-mean matroids, the matroids representable over all fields of size at least 4, and the quaternary matroids representable over fields of all characteristics. This leads to a determination of the extremal functions for these classes, verifying a conjecture of Archer for matroids of sufficiently large rank.

This talk will include a brief introduction to matroid theory.

Kevin Grace is a new Heilbronn Fellow.