On exterior powers of reflection representations, II

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.