Abstract:In this paper, we study the principal specialization of monomial symmetric polynomials and investigate the special values of these polynomials at \[ \zeta_{(n,k)} := ( 1, \zeta_n, \zeta_n^2, \dots, \zeta_n^{kn-1} ), \] where $\zeta_n$ is a primitive $n$th root of unity. We give explicit formulas for several classes of special values. We also 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 formulas for the case $k = 1$. 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 $k = 1$.
From: Naoya Yamaguchi [view email]
[v1]
Mon, 28 Mar 2022 00:16:20 UTC (12 KB)
[v2]
Sat, 19 Apr 2025 12:29:59 UTC (16 KB)
[v3]
Wed, 27 May 2026 04:50:32 UTC (14 KB)
[v4]
Mon, 7 Sep 2026 13:43:27 UTC (13 KB)
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