Matrices for finite group representations that respect Galois automorphisms

2023/06/09 by Benson, David J.
#20C15 #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2306.06280

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 ρ\colon H→\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→\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 \sf ind_{H,H\rtimes⟨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 over ℚ[√5] and a four dimensional irreducible representation of the double cover of A₇ over ℚ[√(-7)].

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