I am working on the Fourier transform over finite non-abelian groups, specifically following [Diaconis](https://jdc.math.uwo.ca/M9140a-2012-summer/Diaconis.pdf). He defines it as follows (p.7):


Let $P$ be a probability on a finite group $G$. The Fourier transform of $P$ at the representation $\rho$ is the matrix
\begin{equation\*}
\widehat{P}(\rho) = \sum\_{s\in G}P(s)\rho(s)
\end{equation\*}


My understanding is that the Fourier transform is a ring isomorphism between the set of functions $L(G) = \{f: G \rightarrow \mathbb{C}\}$ (with the operations of pointwise addition and convolution) and some other set (with pointwise addition and pointwise multiplication) via the convolution theorem. What is this other set?



Image of Fourier transform for finite non-abelian groups
The transform, which is a ring isomorphism,
$$
P \mapsto \widehat{P} = \sum\_{g\in G} P(g)g
$$
goes from your $L(G)$ to the group algebra ${\mathbb C}G$. The convolution corresponds to the multiplication in the group algebra:
<http://mathworld.wolfram.com/GroupAlgebra.html>


If you apply an irreducible representation $\rho$ to the righthand side, this corresponds to the projection to the matrix direct summand in the Wedderburn decomposition:
$$
{\mathbb C}G = \oplus\_{\mu\in Irr(G)} M\_{d(\mu)} \rightarrow M\_{d(\rho)}
$$
where $d(\rho)$ is the degree of your representation. Thus, you end up in the matrix algebra, while the convolution corresponds to the matrix multiplication. It is no longer an isomorphism, just a homomorphism.


