Computing Young's Natural Representations for Generalized Symmetric Groups

TL;DR

This paper introduces an algorithmic framework to compute explicit matrices for all irreducible representations of generalized symmetric groups, extending classical Young's representations to more complex group structures.

Contribution

It develops a unified method to construct Young's natural representations for generalized symmetric groups, including new approaches for the case when r=2, related to Weyl groups of type B.

Findings

Provides explicit Specht matrices with entries 0 and ±1

Extends classical Young representations to wreath products of cyclic groups and symmetric groups

Specialized construction for the case r=2, Weyl group of type B

Abstract

We provide an algorithmic framework for the computation of explicit representing matrices for all irreducible representations of a generalized symmetric group $\Grin_n$, i.e., a wreath product of cyclic group of order $r$ with the symmetric group $\Symm_n$. The basic building block for this framework is the Specht matrix, a matrix with entries $0$ and $\pm1$, 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 $\Grin_n$ is the Weyl group of type $B_n$, a different construction is based on a notion of pairs of biwords.