Efficient computation of Fourier inversion for finite groups

Abstract

Let G be a finite group and let S be a complex-valued function on G. If ρ is a matrix representation of G (of dimension d ρ ) then the Fourier transform of f at G. If ρ is defined to be the matrix f(ρ) = Σ s ∈ G f(s)ρ(s). Let R be a complete set of inequivalent irreducible matrix representations of G. The Fourier transform of f (with respect to R) is defined as the set of matrices {f(ρ)} ρ ∈ R . Recovery of f from its Fourier transform may be accomplished via the Fourier inversion formula, f(s) = |G| Σ d ρ trace(f(ρ)ρ(s) −1 )