Principal Specialization of Monomial Symmetric Polynomials and Group Determinants of Cyclic Groups
Naoya Yamaguchi, Yuka Yamaguchi, Genki Shibukawa
TL;DR
The paper addresses explicit principal specializations of monomial symmetric polynomials at $\zeta_{(n,k)}$ and links the resulting values $m_\lambda(\zeta_{(n,k)})$ to coefficients in the $k$th power of the circulant determinant, i.e., the group determinant of the cyclic group $\mathbb{Z}/n\mathbb{Z}$. It develops a framework relating MSP values to the expansion $C(x)^k=\sum_{\lambda\in\Lambda_n^k} m_\lambda(\zeta_{(n,k)}) x_\lambda$, and proves that the number of terms in $\Theta(\mathbb{Z}/n\mathbb{Z})^k$ and $P(\mathbb{Z}/n\mathbb{Z})^k$ equals $|\tilde{\Lambda}_n^k|=|S(n,kn)|$. The main contributions are explicit evaluations of $m_\lambda(\zeta_{(n,k)})$ for several partition types, proofs that the coefficients align with MSP values, and a complete counting result for terms in the $k$th powers of group determinants, together with generating-function techniques and symmetry relations. These results extend classical findings of Ore and Brualdi–Newman, sharpening the connection between symmetric function theory, circulant determinants, and the combinatorics of 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 $$ ζ_{(n,k)} := ( 1, ζ_n, ζ_n^2, \dots, ζ_n^{kn-1} ), $$ where \(ζ_n\) 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 $k = 1$.