On Inequivalent Representations of Matroids over Finite Fields*
It was conjectured by Kahn in 1988 that, for each prime power $q$, there is an
integer $n(q)$ such that no 3--connected $GF(q)$--representable matroid has
more than $n(q)$ inequivalent $GF(q)$--representations. At the time, this
conjecture was known to be true for $q=2$ and $q=3$, and Kahn had just proved
it for $q=4$. In this paper, we prove the conjecture for $q=5$ showing that
6 is a sharp value for $n(5)$. Moreover, we also show using two different classes of examples
that the conjecture is false for all larger values of $q$.
This paper is dedicated to Don Row who introduced all three author
s to matroids.