Computing Young's Natural Representations for Generalized Symmetric Groups

2024/12/15 by Paul, Koushik, Pfeiffer, Götz
#20C15 #20C30 #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2412.11223

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 ±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 \Grin_n is the Weyl group of type B_n, a different construction is based on a notion of pairs of biwords.

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