论文标题
循环组的单一对称多项式和组决定因素的主专业化
Principal Specialization of Monomial Symmetric Polynomials and Group Determinants of Cyclic Groups
论文作者
论文摘要
在本文中，我们考虑了单一对称多项式的主要专业化，并在点$$ζ_ {（n，k）}：=（1，ζ_n，ζ_n，ζ_n^2，\ dots，qualliand pririent friant frient frient frient frient）的特殊值。我们为几个特殊值提供明确的公式。另外，我们表明这些特殊值自然地显示为扩展订单$ n $循环决定因素的$ k $ th功率的系数（订单$ n $的订单订单组的组决定因素）。这些结果将矿石的结果扩展到$ k = 1 $。此外，我们确定了订单$ n $的$ k $ th功率的$ k $ th功率中的条款数量。这扩展了Brualdi和Newman的结果，价格为$ K = 1 $。
In this paper, we consider the principal specialization of monomial symmetric polynomials and investigate the special values of these polynomials at the point $$ 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 $k = 1$.
