On inequivalent representations of matroids over non-prime fields. (English) Zbl 1230.05087

Geelen, Jim; Gerards, Bert; Whittle, Geoff

J. Comb. Theory, Ser. B 100, No. 6, 740-743 (2010).

Summary: For each finite field \(\mathbb F\) of prime order there is a constant \(c\) such that every 4-connected matroid has at most \(c\) inequivalent representations over \(\mathbb F\). We had hoped that this would extend to all finite fields, however, it was not to be. The \((m,n)\)-mace is the matroid obtained by adding a point freely to \(M(K_{m,n})\). For all \(n \geqslant 3\), the \((3,n)\)-mace is 4-connected and has at least \(2^n\) representations over any field \(\mathbb F\) of non-prime order \(q \geqslant 9\). More generally, for \(n \geqslant m\), the \((m,n)\)-mace is vertically \((m+1)\)-connected and has at least \(2^n\) inequivalent representations over any finite field of non-prime order \(q \geqslant m^m\).

MSC:

05B35

Combinatorial aspects of matroids and geometric lattices

Keywords:

matroids; inequivalent representations; connectivity

References:

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