Let $G$ be a finite group, $R = \mathbb{C}G$ the regular representation of $G$, and $\rho : G \to \text{GL}(V)$ a finite dimensional representation of $G$. Write $\sigma: G \to \text{GL}(V \otimes R)$ for the tensor product of $\rho$ and the regular representation. How do I construct a vector space isomorphism $f: V \otimes \mathbb{C}G \overset{\sim}{\to} V \otimes \mathbb{C}G$ such that$$f(v \otimes gh) = \sigma(g)(f(v \otimes h)),\text{ }\forall v \in V,\text{ }g,\,h \in G.$$The above isomorphism means that the map $f^{-1}$ yields an isomorphism of representations $V \otimes R \cong R \oplus R \oplus \dots \oplus R$ ($\text{dim}\,V$ summands).



Construction of vector space isomorphism where $f(v \otimes gh) = \sigma(g)(f(v \otimes h)),\text{ }\forall v \in V,\text{ }g,\,h \in G$
The map $\phi: v\otimes g\mapsto g^{-1}v\otimes g$ gives an isomorphism from $V\otimes R$ to $V\_0\otimes R$, where $G$ acts trivially on $V\_0$. Taking a basis of $V\_0$ gives you an isomorphism $V\_0\otimes R\simeq R^{\dim V}$. I don't think there is a canonical isomorphism, though.


