Principal Specialization of Monomial Symmetric Polynomials and Group Determinants of Cyclic Groups

2022/03/28 by Naoya Yamaguchi, Yamaguchi, Naoya, Yuka Yamaguchi +3
Chemistry · Mathematics · #11P05 #20C15 #Axial and Atropisomeric Chirality Synthesis #Combinatorics (math.CO) #FOS: Mathematics #Functional Equations Stability Results #Molecular spectroscopy and chirality #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2203.14422

openalex publication_date 2022/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

In this paper, we study the principal specialization of monomial symmetric polynomials and investigate the special values of these polynomials at ζ_(n,k) := ( 1, ζ_n, ζ_n², …, ζ_n^kn-1 ), where ζ_n is a primitive nth root of unity. We give explicit formulas for several classes of special values. We also show that these special values naturally appear as the coefficients in the expansion of the kth power of the circulant determinant of order n (the group determinant of the cyclic group of order n). These results extend Ore's formulas for the case k = 1. Furthermore, we determine the number of terms in the kth power of the group permanent of the cyclic group of order n. This extends Brualdi and Newman's result for k = 1.

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