Dillon Mayhew, Towards Rota’s conjecture for gain-graphic matroids

In some sense, matroids are generalisations of graphs. The idea of graph minors extends to matroids, and so does the idea of a minor-closed class. We can think of a minor-closed class of matroids as being an analogue to the class of graphs embeddable on a surface. Any such class of graphs has a corresponding class of minimal forbidden minors, and these forbidden minors characterise the class. A minor-closed class of matroids is characterised by its minimal forbidden minors in the same way.

Rota’s conjecture is the most famous problem in matroid theory. It says that when F is a finite field, there is a finite number of minimal forbidden minors for the class of matroids that can be represented by vectors over the field of scalars F. A proof has been announced by Geelen, Gerards, and Whittle.

Gain-graphic matroids are analogues to matroids represented by vectors: instead of representing the matroid using numbers from a field, we use elements from a group. So we can ask for an analogue of Rota’s conjecture, except for gain-graphic matroids.

In this talk I will outline our intended path towards Rota’s conjecture for gain-graphic matroids. This is joint work with Daryl Funk.