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

TL;DR

This paper explores the principal specialization of monomial symmetric polynomials at roots of unity, deriving explicit formulas, connecting these to circulant determinants, and extending classical results for cyclic groups.

Contribution

It provides explicit formulas for special values of monomial symmetric polynomials at roots of unity and links these to group determinants, extending prior results for cyclic groups.

Findings

Explicit formulas for special values at roots of unity

Connection between polynomial values and circulant determinants

Extended classical results for cyclic groups

Abstract

In this paper, we consider the principal specialization of monomial symmetric polynomials and investigate the special values of these polynomials at the point $$ \zeta_{(n,k)} := ( 1, \zeta_n, \zeta_n^2, \dots, \zeta_n^{kn-1} ), $$ where \(\zeta_n\) is a primitive \(n\)th root of unity. We give explicit formulas for several special values. Also, we show that these special values naturally appear as the coefficients in the expansion of the $k$th power of the circulant determinant of order $n$ (the group determinant of the cyclic group of order $n$). These results extend Ore's results for $k = 1$. Furthermore, we determine the number of terms in the $k$th power of the group permanent of the cyclic group of order $n$. This extends Brualdi and Newman's result for $k = 1$.