论文标题
关于广义对称组表示的决定因素
On the determinant of representations of generalized symmetric groups
论文作者
论文摘要
在本文中，我们研究了广义对称群体$ \ mathbb {z} _r \ wr s_n $的不可约说明的决定因素。我们给出一个明确的公式，以计算$ \ mathbb {z} _r \ wr s_n $的不可约表示的决定因素。最近，几位作者表征并计算了具有非平凡决定因素的给定有限群体的不可还原表示的数量。受这些结果的动机，对于给定的整数$ n $，$ r $ a奇数和$ζ$ $ \ mathbb {z} _r \ wr s_n $带有$ n <r $的非平凡乘法性，我们获得了一个明显的公式，以计算$ n_qu的$ n _ $ $ n $ $ \ n $ \ n $ \ n n ymath的数量。决定因素为$ζ$。
In this paper we study the determinant of irreducible representations of the generalized symmetric groups $\mathbb{Z}_r \wr S_n$. We give an explicit formula to compute the determinant of an irreducible representation of $\mathbb{Z}_r \wr S_n$. Recently, several authors have characterized and counted the number of irreducible representations of a given finite group with nontrivial determinant. Motivated by these results, for given integer $n$, $r$ an odd prime and $ζ$ a nontrivial multiplicative character of $\mathbb{Z}_r \wr S_n$ with $n<r$, we obtain an explicit formula to compute $N_ζ(n)$, the number of irreducible representations of $\mathbb{Z}_r \wr S_n$ whose determinant is $ζ$.
