Principal specialization of monomial symmetric polynomials and group determinants of cyclic groups

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Abstract In this paper, we study the principal specialization of monomial symmetric polynomials and investigate the special values of these polynomials at $$\begin{aligned} \zeta _{(n,k)} := ( 1, \zeta _n, \zeta _n^2, \dots , \zeta _n^{kn-1} ), \end{aligned}$$ ζ ( n , k ) : = ( 1 , ζ n , ζ n 2 , ⋯ , ζ n k n - 1 ) , where $$\zeta _n$$ ζ n is a primitive $$n$$ 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 $$k = 1$$ 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 $$k = 1$$ k = 1 .