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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 $$q \geqslant m^m$$.

##### MSC:
 05B35 Combinatorial aspects of matroids and geometric lattices
##### Keywords:
matroids; inequivalent representations; connectivity
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##### 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
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