# Projecting GxG onto subspace with tied irreducible representations

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