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

In this paper, we study the principal specialization of monomial symmetric polynomials and investigate the special values of these polynomials at \[ ζ_{(n,k)} := ( 1, ζ_n, ζ_n^2, \dots, ζ_n^{kn-1} ), \] where $ζ_n$ is a primitive $n$th root of unity. We give explicit formulas for several classes of