TITLE: Clebsch–Gordan decomposition for $\mathrm{SU}(2)$, in indices
QUESTION [5 upvotes]: Let $\pi_m$, $m \geq 0$, be the unitary irreps of $\mathrm{SU}(2)$. The Clebsch–Gordan decomposition then gives that
$$ \pi_m \otimes \pi_n = \bigoplus_{k=0}^{\min(m,n)}\pi_{m+n-2k}.$$
But suppose I want to think of this decomposition as matrices. Evaluating at a point $x \in \mathrm{SU}(2)$, on the left I have
$$ (\pi_m(x))_{ij} (\pi_n(x))_{pq}.$$
How do the indices $i$, $j$ and $p$, $q$ correspond to the indices on the big matrix on the right?

REPLY [3 votes]: You can find such a formula with indices here:
https://en.wikipedia.org/wiki/Wigner_D-matrix#Kronecker_product_of_Wigner_D-matrices,_Clebsch-Gordan_series