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It is well-known that tensor products of irreducible representations of a finite group decompose into direct sums of irreducible representations according to fusion rules $$\Gamma_i \otimes \Gamma_j=\bigoplus_k \Gamma_k^{\oplus N_{\Gamma_i,\Gamma_j}^{\Gamma_k}}$$ where the $N_{\Gamma_i,\Gamma_j}^{\Gamma_k}$ are multiplicities. In particular, this means that these rules hold at each group element:

$$\Gamma_i(g) \otimes \Gamma_j(g)=\bigoplus_k \Gamma_k(g)^{\oplus N_{\Gamma_i,\Gamma_j}^{\Gamma_k}}$$ for all $g\in G$. Is there a generalization of this formula when the two irreps on the left hand side are evaluated at different group elements:

$$\Gamma_i(g) \otimes \Gamma_j(h)=?$$

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There are no such rules. You are asking about the character $\Gamma_i \otimes \Gamma_j$ of the representation of $G\times G$. It is irreducible itself, so there is no other writing in terms of irreducibles.

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