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?



Clebsch–Gordan decomposition for $\mathrm{SU}(2)$, in indices
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>


