Matrices for finite group representations that respect Galois automorphisms

We are given a finite group H , an automorphism τ τ of H of order r , a Galois extension L / K of fields of characteristic zero with cyclic Galois group σ ⟨ σ ⟩ of order r , and an absolutely irreducible representation ρ :H→ GL (n,L) ρ : H → GL ( n , L ) such that the action of τ τ on the character of ρ ρ is the same as the action of σ σ . Then the following are equivalent. • ∙ ρ ρ is equivalent to a representation ρ ':H→ GL (n,L) ρ ′ : H → GL ( n , L ) such that the action of σ σ on the entries of the matrices corresponds to the action of τ τ on H , and • ∙ the induced representation ind H,H τ(ρ ) 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 Q[√5] Q [ 5 ] and a four dimensional irreducible representation of the double cover of A₇ A 7 over Q[√-7] Q [ - 7 ] .