Cyclotomic and simplicial matroids

Abstract

Two naturally occurring matroids representable over Q are shown to be dual: the {\it cyclotomic matroid} $\mu_n$ represented by the $n^{th}$ roots of unity $1,\zeta,\zeta^2,...,\zeta^{n-1}$ inside the cyclotomic extension $Q(\zeta)$, 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 $Q(\zeta)$ 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 $\mu_n$ in the case that $n$ has two prime factors.