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 ⟨ σ ⟩ of order r, and an absolutely irreducible representation: H GL (n, L) ρ: H → 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. ∙ ρ is equivalent to a representation ': H GL (n, L) ρ ′: H → GL (n, L) such that the action of σ on the entries of the matrices corresponds to the action of τ on H, and ∙ the induced representation ind ₇, ₇ () 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 A₅ A 5 over Q5 Q 5 and a four dimensional irreducible representation of the double cover of A₇ A 7 over Q-7 Q - 7.