Generating functions for monomial characters of wreath products $\mathbb Z/d \mathbb Z \wr \mathfrak S_n$

TL;DR

This paper develops generating functions for monomial characters of wreath products of cyclic groups with symmetric groups, extending classical results and expressing these functions via determinants and permanents.

Contribution

It introduces a new framework for generating functions of monomial characters of wreath products, generalizing previous work on symmetric groups.

Findings

Derived explicit formulas for generating functions in terms of determinants and permanents.

Extended classical results from symmetric groups to wreath products.

Provided a mathematical foundation for analyzing monomial characters in complex group structures.

Abstract

Let $\mathbb Z/d\mathbb Z \wr \mathfrak S_n$ denote the wreath product of the cyclic group $\mathbb Z/d\mathbb Z$ with the symmetric group $\mathfrak S_n$. We define generating functions for monomial (induced one-dimensional) characters of $\mathbb Z/d\mathbb Z \wr \mathfrak S_n$ and express these in terms of determinants and permanents. This extends work of Littlewood ({\em The Theory of Group Characters and Representations of Groups}, 1940) and Merris and Watkins ({\em Linear Algebra Appl.}, {\bf 64}, 1985) on generating functions for the monomial characters of $\mathfrak S_n$.