Modularity and Structure in Matroids

Abstract

This thesis concerns sufficient conditions for a matroid to admit one of two types of structural characterization: a representation over a finite field or a description as a frame matroid.

We call a restriction N of a matroid M modular if, for every flat F of M, r_M(F) + r(N) = r_M(F ∩ E(N)) + r_M(F ∪ E(N)).

A consequence of a theorem of Seymour is that any 3-connected matroid with a modular U_{2,3}-restriction is binary. We extend this fact to arbitrary finite fields, showing that if N is a modular rank-3 restriction of a vertically 4-connected matroid M, then any representation of N over a finite field extends to a representation of M.

We also look at a more general notion of modularity that applies to minors of a matroid, and use it to present conditions for a matroid with a large projective geometry minor to be representable over a finite field. In particular, we show that a 3-connected, representable matroid with a sufficiently large projective geometry over a finite field GF(q) as a minor is either representable over GF(q) or has a U_{2,q^2+1}-minor.

A second result of Seymour is that any vertically 4-connected matroid with a modular M(K_4)-restriction is graphic. Geelen, Gerards, and Whittle partially generalized this from M(K_4) to larger frame matroids, showing that any vertically 5-connected, representable matroid with a rank-4 Dowling geometry as a modular restriction is a frame matroid. As with projective geometries, we prove a version of this result for matroids with large Dowling geometries as minors, providing conditions which imply that they are frame matroids.