ON INEQUIVALENT REPRESENTATIONS OF MATROIDS OVER FINITE-FIELDS

Kahn conjectured in 1988 that, for each prime power q, there is an int eger n(q) such that no 3-connected GF(q)-representable matroid has mor e than n(q) inequivalent GF(q)-representations. At the time, this conj ecture 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 th at 6 is a sharp value for n(5). Moreover, we also show that the conjec ture is false for all larger values of q. (C) 1996 Academic Press, Inc .