I am working on the Fourier transform over finite non-abelian groups, specifically following [Diaconis][1]. 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? [1]: https://jdc.math.uwo.ca/M9140a-2012-summer/Diaconis.pdf

I am working on the Fourier transform over finite non-abelian groups, specifically following Diaconis. 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?