On inequivalent representations of matroids over finite fields. (English) Zbl 0856.05021

Oxley, James; Vertigan, Dirk

J. Comb. Theory, Ser. B 67, No. 2, 325-343 (1996).

It was conjectured by J. Kahn [On the uniqueness of matroid representations over GF(4), Bull. Lond. Math. Soc. 20, No. 1, 5-10 (1988; Zbl 0609.05028)] that, for each prime power \(q\), there is an integer \(n(q)\) such that no 3–connected \(\text{GF} (q)\)-representable matroid has more than \(n(q)\) inequivalent \(\text{GF} (q)\)-representations. At the time, this conjecture was known to be true for \(q = 2\) and \(q = 3\), and Kahn had just proved it for \(q = 4\). This paper proves the conjecture for \(q = 5\) showing that 6 is a sharp value for \(n(5)\). Moreover, it is also shown that the conjecture is false for all larger values of \(q\).

Reviewer: J.G.Oxley (Baton Rouge)

MSC:

05B35

Combinatorial aspects of matroids and geometric lattices

Keywords:

inequivalent representations; matroid roundedness; matroid representations