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On exterior powers of reflection representations, II
- Part of
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- Journal:
- Canadian Journal of Mathematics , First View
- Published online by Cambridge University Press:
- 24 March 2025, pp. 1-23
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- Article
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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.
