On exterior powers of reflection representations

TL;DR

This paper generalizes Steinberg's 1968 theorem by showing that exterior powers of irreducible reflection representations remain irreducible and non-isomorphic even without the assumption of an invariant inner product.

Contribution

The authors extend Steinberg's theorem to broader contexts lacking an invariant inner product, broadening the understanding of reflection representations.

Findings

Exterior powers of irreducible reflection representations are irreducible.

Exterior powers are pairwise non-isomorphic.

The result holds without requiring an invariant inner product.

Abstract

In 1968, R. Steinberg proved a theorem stating that the exterior powers of an irreducible reflection representation of a Euclidean reflection group are again irreducible and pairwise non-isomorphic. We extend this result to a more general context where the inner product invariant under the group action may not necessarily exist.