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Alright, let's go on a thrilling tour through the theory of representations. Notation in the following is the same as in the OP, except that we call the trivial representation $\boldsymbol{1}$, as is canon. We start from my expression in the question $$ I := \int \alpha(g)^i_{i'}\beta(g)^j_{j'}\gamma(g)^k_{k'} = \sum_\rho \sum_{\mu = ...


For SU(N), the adjoint representation can be obtained from a tensor product of the fundamental representation with its dual and projecting out the scalar. Thus we can replace the adjoint representation indices $a, ...$ by double indices $i \bar{j}$ and write this relation as: $$W^{i \bar{j}}_{k \bar{l}} = U^i_k U^{\dagger j}_l - \frac{1}{N} \delta^i_k ...

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