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 ${\mathfrak{S}}_n$.

10 pages