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

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 $\mathfrak S_n$.