Let $\rho_R$ be the regular representation of a finite group $G$. I want to understand the tensor product of $\rho_R$ with itself, i.e., $\rho_R\otimes \rho_R$. Is the following correct? The class function inner product gives:
$$\langle\chi_{R\otimes R},\chi_i\rangle=\frac{1}{|G|}\chi_{R\otimes R}(1)\chi_i(1)=|G|d_i,$$
so that $\rho_R\otimes \rho_R=|G|\rho_R$.