Matrices for finite group representations that respect Galois automorphisms

Abstract

We are given a finite group $H$, an automorphism $τ$ of $H$ of order $r$, a Galois extension $L/K$ of fields of characteristic zero with cyclic Galois group $\langleσ\rangle$ of order $r$, and an absolutely irreducible representation $ρ\colon H\to\operatorname{\sf GL}(n,L)$ such that the action of $τ$ on the character of $ρ$ is the same as the action of $σ$. Then the following are equivalent. $\bullet$ $ρ$ is equivalent to a representation $ρ'\colon H\to\operatorname{\sf GL}(n,L)$ such that the action of $σ$ on the entries of the matrices corresponds to the action of $τ$ on $H$, and $\bullet$ the induced representation $\operatorname{\sf ind}_{H,H\rtimes\langleτ\rangle}(ρ)$ has Schur index one; that is, it is similar to a representation over $K$. As examples, we discuss a three dimensional irreducible representation of $A_5$ over $\mathbb{Q}[\sqrt5]$ and a four dimensional irreducible representation of the double cover of $A_7$ over $\mathbb{Q}[\sqrt{-7}]$.