The Schur polynomials in all primitive nth roots of unity

Abstract

We show that the Schur polynomials in all primitive nth roots of unity are 1, 0, or −1, if n has at most two distinct odd prime factors. This result can be regarded as a generalization of properties of the coefficients of the cyclotomic polynomial and its multiplicative inverse. The key to the proof is the concept of a unimodular system of vectors. Namely, this result can be reduced to the unimodularity of the tensor product of two maximal circuits (here we call a vector system a maximal circuit, if it can be expressed as $B\cup \{-\sum B\}$ with some basis B).