We know the Wigner-Eckart theorem generalizes to say $\mathrm{SU}(3)$ (see for example this answer). In a different direction, I am curious if/how it generalizes to finite groups of $\mathrm{SU}(2)$.
Let $G \subset \mathrm{SU}(2)$ be a finite group and let $\lambda_1, \lambda_2, \lambda_3$ be irreps of $G$. The Wigner-Eckart Theorem in this case seems to be something along the lines of $$ \langle \lambda_1 ,p_1; \alpha_1| T_{\lambda_2, p_2;\alpha_2} | \lambda_3,p_3; \alpha_3 \rangle = \sum_{i} \langle \lambda_1,p_1|| T_{\lambda_2,p_2} || \lambda_3,p_3 \rangle_i \times (\text{CG}). $$ Here $\alpha_1,\alpha_2,\alpha_3$ are internal labels (within the irrep), CG is something like a Clebsch-Gordan coefficient, and $\langle \lambda_1,p_1|| T_{\lambda_2,p_2} || \lambda_3,p_3 \rangle$ is the reduced matrix element that doesn't depend on the internal labels. We need to sum over $i$ because in general $\lambda_2 \otimes \lambda_3$ might contain more than 1 copy of $\lambda_1$ (unlike the case $G = \mathrm{SU}(2)$). Lastly, $p_1,p_2,p_3$ are extra labels one might need in order to identity which irrep we have (since in general, there might be more than 1 copy of $\lambda_1$ and $\lambda_2$ in spin $j$).
Can anyone confirm if this is correct?
