Suppose I have a finite group $G$. With this group, I can associate an ortho-normal Hilbert space spanned by elements of the group $$\mathcal{H} = \{|g\rangle: g \in G \}$$.

I could alternatively work in the representation basis, related to the group element basis by the Fourier transform: $$ |\mu,a,b\rangle = \sqrt{\frac{n_\mu}{|G|}} \sum_{g\in G}\Gamma^{ab}_\mu(g)|g\rangle $$ where $\mu$ labels irreps of the group, $n_\mu$ is the dimension of $\mu$, and $\Gamma^{ab}_\mu$ are the irrep matrices, with $a,b = 1,\dots,n_\mu$.

My question is the following: suppose I consider instead the group $G \times G$ with basis $\{|\mu i j\rangle|\nu k l\rangle\}$. Does there exist a projection operator which projects onto the subspace where irreps are tied together i.e. take the form $\{|\mu i j\rangle|\mu k l\rangle\}$?

In other words, I'd like to find an operator $P$ such that $$ \begin{equation} P |\mu i j\rangle |\nu k l\rangle = \delta_{\mu,\nu} \sum_{a,b}\sum_{c,d} A_{ab}^{ij}B_{cd}^{kl} |\mu a b\rangle|\mu cd\rangle \end{equation} $$ where I am allowing for the fact that there may be some mixing between states within the same irrep (if it has dimension >1).


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