In "Another subexponential-time quantum algorithm for the dihedral hidden subgroup problem", Kuperberg writes that $\mathbb{C}[G]$ has a "Burnside decomposition" of
$$\mathbb{C}[G]\cong \bigoplus_{V} V^*\otimes V$$
where each $V$ is an irreducible representation.
He defines $\mathbb{C}[G]$ as a Hilbert space with an orthonormal basis identified by $G$, a finite group. So $\mathbb{C}[G]$ is the span of $\{\vert g\rangle :g\in G\}$ for orthonormal vectors $\vert g\rangle$.
This means $\mathbb{C}[G]$ works naturally as the representation space of either the left-regular or right-regular representation of $G$. Thus, it makes sense to me that, letting $V_i$ be the representation spaces of different representations, we would have
$$\mathbb{C}[G]=\bigoplus_{i} V_i.$$
But I don't see why the tensor product appears. Even if we consider $B(\mathbb{C}[G])$, the set of operators on $\mathbb{C}[G]$, to get the density matrices on this Hilbert space, we would end up with vectors of the form $v_j\otimes v_i$, where $v_j\in V_j$ and $v_i\in V_i$ for $i\neq j$.
Alternatively, he may be identifying $\mathbb{C}[G]$ with the left- or right-regular representation itself, and then decomposing it into the representations, not the representation spaces. But this doesn't make sense to me either; $V$ is finite-dimensional, so $V^*\otimes V\cong M_n$ for some $n$, so this decomposition would imply that every irreducible representation of $G$ is isomorphic to $M_n$ for some $n$, which I don't think is true.
So what is this decomposition?