On exterior powers of reflection representations, II

Abstract

Let $W$ be a group endowed with a finite set $S$ of generators. A representation $(V,\ddot{I})$ of $W$ is called a reflection representation of $(W,S)$ if $\ddot{I}(s)$ 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.

22 pages. Published version. Comments welcome!