×

zbMATH — the first resource for mathematics

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 \(q \geqslant m^m\).

MSC:
05B35 Combinatorial aspects of matroids and geometric lattices
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] J. Geelen, G. Whittle, Inequivalent representations of matroids over prime fields, in preparation. · Zbl 1281.05041
[2] Kahn, J., On the uniqueness of matroid representations over \(\mathit{GF}(4)\), Bull. lond. math. soc., 20, 5-10, (1988) · Zbl 0609.05028
[3] Olsen, J.E., A combinatorial problem on finite abelian groups, I, J. number theory, 1, 8-10, (1961)
[4] Oxley, J.G., Matroid theory, (1992), Oxford University Press New York · Zbl 0784.05002
[5] Oxley, J.; Vertigan, D.; Whittle, G., On inequivalent representations of matroids over finite fields, J. combin. theory ser. B, 67, 325-343, (1996) · Zbl 0856.05021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.