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On exterior powers of reflection representations, II

Part of: Special aspects of infinite or finite groups Representation theory of groups Algebraic combinatorics Metric geometry Basic linear algebra

Published online by Cambridge University Press:  24 March 2025

Hongsheng Hu Open the ORCID record for Hongsheng Hu [Opens in a new window]
Hongsheng Hu*
Affiliation:
School of Mathematics, Hunan University, Changsha 410082, China
*
e-mail: huhongsheng@hnu.edu.cn
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Abstract

Let W be a group endowed with a finite set S of generators. A representation $(V,\rho )$ of W is called a reflection representation of $(W,S)$ if $\rho (s)$ is a (generalized) reflection on V for each generator $s \in S$. In this article, 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.

Keywords

Exterior powersgeneralized reflectionsreflection representationsCoxeter groups

MSC classification

Primary: 15A75: Exterior algebra, Grassmann algebras
Secondary: 05E10: Combinatorial aspects of representation theory 20C15: Ordinary representations and characters 20F55: Reflection and Coxeter groups 51F15: Reflection groups, reflection geometries

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Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 23
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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