Computing Young's Natural Representations for Generalized Symmetric Groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918105926598656 |
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| author | Paul, Koushik Pfeiffer, Götz |
| author_facet | Paul, Koushik Pfeiffer, Götz |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_11223 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Computing Young's Natural Representations for Generalized Symmetric Groups Paul, Koushik Pfeiffer, Götz Representation Theory Group Theory 20C30, 20C15 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. |
| title | Computing Young's Natural Representations for Generalized Symmetric Groups |
| topic | Representation Theory Group Theory 20C30, 20C15 |
| url | https://arxiv.org/abs/2412.11223 |