I am familiar with the interpretation of the irreducible representations (finite dimensional) of $SU(2)$ in terms of homogeneous polynomials of degree $n$.
If I take two of these irreducible representations, say $\rho_{i}$ and $\rho_{j}$ of dimensions $i+1$ and $j+1$ respectively, and take their product $\rho_{i}\rho_{j}$, the resulting representation will be the direct sum of other $\rho_{k}$'s. For a fixed $k$, how do I calculate the multiplicity of $\rho_{k}$ as a summand in the product representation in terms of $i$ and $j$?