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arXiv · 2403.10817

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

Masaki Hidaka·Minoru Itoh
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Abstract

We show that the Schur polynomials in all primitive $n$th 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 $B$).

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Masaki Hidaka, Minoru Itoh. 2024-03-16. The Schur polynomials in all primitive $n$th roots of unity. https://arxiv.org/abs/2403.10817

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