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On inequivalent representations of matroids over non-prime fields. (English) Zbl 1230.05087
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 \(q \geqslant m^m\).

05B35 Combinatorial aspects of matroids and geometric lattices
Full Text: DOI
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