Matrices for finite group representations that respect Galois automorphisms

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2024年3月2日

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David J. Benson

摘要

We are given a finite group H, an automorphism $\tau $ $τ$ of H of order r, a Galois extension L/K of fields of characteristic zero with cyclic Galois group $\langle \sigma \rangle $ $⟨ σ ⟩$ of order r, and an absolutely irreducible representation $\rho :H\rightarrow \textsf {GL} (n,L)$ $ρ : H \to GL (n, L)$ such that the action of $\tau $ $τ$ on the character of $\rho $ $ρ$ is the same as the action of $\sigma $ $σ$ . Then the following are equivalent.

$\bullet $ $∙$ $\rho $ $ρ$ is equivalent to a representation $\rho ':H\rightarrow \textsf {GL} (n,L)$ $ρ^{'} : H \to GL (n, L)$ such that the action of $\sigma $ $σ$ on the entries of the matrices corresponds to the action of $\tau $ $τ$ on H, and

$\bullet $ $∙$ the induced representation $\textsf {ind} _{H,H\rtimes \langle \tau \rangle }(\rho )$ ${ind}_{H, H ⋊ ⟨ τ ⟩} (ρ)$ has Schur index one; that is, it is similar to a representation over K.

As examples, we discuss a three dimensional irreducible representation of $A_5$ $A_{5}$ over $\mathbb {Q}[\sqrt{5}]$ $Q [\sqrt{5}]$ and a four dimensional irreducible representation of the double cover of $A_7$ $A_{7}$ over $\mathbb {Q}[\sqrt{-7}]$ $Q [\sqrt{- 7}]$ .