Computing Young's Natural Representations for Generalized Symmetric Groups

We provide an algorithmic framework for the computation of explicit representing matrices for all irreducible representations of a generalized symmetric group ${\text{Grin}}_n$, i.e., a wreath product of cyclic group of order $r$ with the symmetric group ${\text{Symm}}_n$. The basic building block for this framework is the Specht matrix, a matrix with entries $0$ and $\pm 1$, defined in terms of pairs of certain words. Combinatorial objects like Young diagrams and Young tableaus arise naturally from this setup. In the case $r=1$, we recover Young’s natural representations of the symmetric group. For general $r$, a suitable notion of pairs of $r$-words is used to extend the construction to generalized symmetric groups. Separately, for $r=2$, where ${\text{Grin}}_n$ is the Weyl group of type $B_n$, a different construction is based on a notion of pairs of biwords.