CYCLOTOMIC AND SIMPLICIAL MATROIDS

Abstract. We show that two naturally occurring matroids representable over Q are equal: the cyclotomic matroid µn represented by the n th roots of unity 1, ζ, ζ 2,..., ζ n−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 the n th roots of unity, which is tight if and only if n has at most two odd prime factors. In addition, we study the Tutte polynomial of µn in the case that n has two prime factors. 1