Cyclotomic and simplicial matroids
We show that two naturally occurring matroids representable over ℚ are equal: thecyclotomic matroid µn represented by then th roots of unity 1, ζ, ζ2, …, ζn-1 inside the cyclotomic extension ℚ(ζ), and a direct sum of copies of a certain simplicial matroid, considered originally by Bolker in the context of transportation polytopes. A result of Adin leads to an upper bound for the number of ℚ-bases for ℚ(ζ) among then th roots of unity, which is tight if and only ifn has at most two odd prime factors. In addition, we study the Tutte polynomial of µn in the case thatn has two prime factors.
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