Inequivalent representations of certain classes of matroids

Abstract

Geelen, Oxley, Vertigan and Whittle conjectured that, for any integer r exceeding two, and any prime power q, there exists an integer n(q, r), such that any 3-connected GF(q)-representable matroid that has no minor isomorphic to the rank-r free-swirl or the rank-r free-spike has at most n(q, r) inequivalent GF(q)-representations. We prove this conjecture for the class of matroids that have no U3,6-minor, the class of Dowling matroids, and the class produced by applying the truncation operator to the family of bicircular matroids.