The Schur polynomials in all primitive nth roots of unity

2024/03/16 by Masaki Hidaka, Hidaka, Masaki, Minoru Itoh +1
Mathematics · #05C50 #05E05 #11B83 #11C08 #11C20 #11R18 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2403.10817

openalex publication_date 2024/03/16 · openalex created_date 2024/03/21 · openalex updated_date 2026/07/28

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 ∪ \ -∑ B \ with some basis B).

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