On the determinant of representations of generalized symmetric groups

TL;DR

This paper derives explicit formulas for the determinants of irreducible representations of generalized symmetric groups and counts those with a specific nontrivial determinant, advancing understanding of their representation theory.

Contribution

It provides explicit formulas for determinants of irreducible representations of $ ext{Z}_r times S_n$ and counts representations with a given nontrivial determinant, extending prior theoretical results.

Findings

Explicit determinant formulas for irreducible representations.

Counting formula for representations with a specific nontrivial determinant.

Results applicable when $n<r$ and $r$ is an odd prime.

Abstract

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 $\zeta$ a nontrivial multiplicative character of $\mathbb{Z}_r \wr S_n$ with $n<r$, we obtain an explicit formula to compute $N_{\zeta}(n)$, the number of irreducible representations of $\mathbb{Z}_r \wr S_n$ whose determinant is $\zeta$.