 Fourier transform on a finite group

For a scalar function on a finite group, one normalized transform convention assigns the matrix ${\mathbb{E}}_{x}f(x)\rho (x)$ to each chosen unitary irreducible representation. This map is a weighted Hilbert space isomorphism by the Parseval identity on a finite group. Another common convention uses $\rho (x{)}^{\ast }$; the corresponding convolution theorem on a finite group then reverses the matrix product order for $(f\ast g)(x)={\mathbb{E}}_{y}f(y)g(y^{-1}x)$.