On exterior powers of reflection representations, II
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916681688809472 |
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| author | Hu, Hongsheng |
| author_facet | Hu, Hongsheng |
| contents | Let $W$ be a group endowed with a finite set $S$ of generators. A representation $(V,ρ)$ of $W$ is called a reflection representation of $(W,S)$ if $ρ(s)$ is a (generalized) reflection on $V$ for each generator $s \in S$. In this paper, we prove that for any irreducible reflection representation $V$, all the exterior powers $\bigwedge ^d V$, $d = 0, 1, \dots, \dim V$, are irreducible $W$-modules, and they are non-isomorphic to each other. This extends a theorem of R. Steinberg which is stated for Euclidean reflection groups. Moreover, we prove that the exterior powers (except for the 0th and the highest power) of two non-isomorphic reflection representations always give non-isomorphic $W$-modules. This allows us to construct numerous pairwise non-isomorphic irreducible representations for such groups, especially for Coxeter groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_08215 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On exterior powers of reflection representations, II Hu, Hongsheng Representation Theory Combinatorics Group Theory Rings and Algebras 15A75, 05E10, 20C15, 20F55, 51F15 Let $W$ be a group endowed with a finite set $S$ of generators. A representation $(V,ρ)$ of $W$ is called a reflection representation of $(W,S)$ if $ρ(s)$ is a (generalized) reflection on $V$ for each generator $s \in S$. In this paper, we prove that for any irreducible reflection representation $V$, all the exterior powers $\bigwedge ^d V$, $d = 0, 1, \dots, \dim V$, are irreducible $W$-modules, and they are non-isomorphic to each other. This extends a theorem of R. Steinberg which is stated for Euclidean reflection groups. Moreover, we prove that the exterior powers (except for the 0th and the highest power) of two non-isomorphic reflection representations always give non-isomorphic $W$-modules. This allows us to construct numerous pairwise non-isomorphic irreducible representations for such groups, especially for Coxeter groups. |
| title | On exterior powers of reflection representations, II |
| topic | Representation Theory Combinatorics Group Theory Rings and Algebras 15A75, 05E10, 20C15, 20F55, 51F15 |
| url | https://arxiv.org/abs/2401.08215 |