The Schur polynomials in all $n$th primitive roots of unity

Masaki Hidaka, Minoru Itoh

Published 2024-03-16Version 1

We show that the Schur polynomials in all $n$th primitive 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. This result is reduced in turn to four propositions on the unimodularity of vector systems, and the last proposition is proved by using graph theory.