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For (real) unitary irreducible representations $\{\rho\}$ of a compact group $\mathcal{G}$, the Schur orthogonality relations have that

$$\int_{\mathcal{G}} \rho^{k_{1}}_{a_{1} b_{1}}(g) \rho^{k_{2}}_{a_{2} b_{2}}(g)\, dg = \frac{1}{d^{k_{1}}}\delta_{k_{1} k_{2}} \delta_{a_{1} a_{2}} \delta_{b_{1} b_{2}}$$

where $d^{k_{1}}$ is the dimension of $\rho^{k_{1}}$. I'm looking for a simplification of the similar case

$$\int_{\mathcal{G}} \rho^{k_{1}}_{a_{1} b_{1}}(g) \rho^{k_{2}}_{a_{2} b_{2}}(g) \rho^{k_{3}}_{a_{3} b_{3}}(g) \, dg =\ \textbf{?}$$

Does such a general relationship exist? Or does it need to be evaluated on per group basis?

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    The case of three seems to be asking for an explicit decomposition of the tensor product of two irreducibles, and then application of the Schur orthogonality relations. So this would be difficult in general. – 2017-02-14

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