Inequivalent Representations of Bias Matroids

Abstract

Suppose that $q$ is a prime power exceeding five. For every integer $N$ there exists a 3-connected GF($q$)-representable matroid that has at least $N$ inequivalent GF($q$)-representations. In contrast to this, Geelen, Oxley, Vertigan and Whittle have conjectured that, for any integer $r > 2$, there exists an integer $n(q,\, r)$ such that if $M$ is a 3-connected GF($q$)-representable matroid and $M$ has no rank-$r$ free-swirl or rank-$r$ free-spike minor, then $M$ has at most $n(q,\, r)$ inequivalent GF($q$)-representations. The main result of this paper is a proof of this conjecture for Zaslavsky's class of bias matroids.