Automorphisms of complex reflection groups

TL;DR

This paper characterizes automorphisms of finite complex reflection groups, showing they are mostly composed of reflection-preserving automorphisms combined with Galois automorphisms, with detailed structure and exceptions.

Contribution

It provides a detailed description of automorphisms of complex reflection groups, including their relation to Galois actions and invariants, extending understanding beyond known cases.

Findings

Automorphisms are products of reflection-preserving automorphisms and Galois automorphisms.

The Galois group injects into the outer automorphism group, matching Galois action on characters.

Fundamental invariants of the groups can be chosen to be rational.

Abstract

Let $G\subset\GL(\BC^r)$ be a finite complex reflection group. We show that when $G$ is irreducible, apart from the exception $G=\Sgot_6$, as well as for a large class of non-irreducible groups, any automorphism of $G$ is the product of a central automorphism and of an automorphism which preserves the reflections. We show further that an automorphism which preserves the reflections is the product of an element of $N_{\GL(\BC^r)}(G)$ and of a "Galois" automorphism: we show that $\Gal(K/\BQ)$, where $K$ is the field of definition of $G$, injects into the group of outer automorphisms of $G$, and that this injection can be chosen such that it induces the usual Galois action on characters of $G$, apart from a few exceptional characters; further, replacing if needed $K$ by an extension of degree 2, the injection can be lifted to $\Aut(G)$, and every irreducible representation admits a model which is equivariant with respect to this lifting. Along the way we show that the fundamental invariants of $G$ can be chosen rational.