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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 $j$?

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This might be helpful: homepages.physik.uni-muenchen.de/~vondelft/Papers/ClebschGordan –  Rasmus Jan 2 '12 at 13:45
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Isn't this just the Clebsch–Gordan theorem? See theorem 15.14 here, or theorem 21.7 here. –  Zhen Lin Jan 2 '12 at 13:48

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As others have already pointed out the answer is known as the Clebsch-Gordan formula. This reads as follows. W.l.o.g we may assume that $i\ge j$. The multiplicity of $\rho_k$ in the tensor product $\rho_i\rho_j$ is equal to 1, if $k$ is one of the integers in the arithmetic progression $i+j$, $i+j-2$, $i+j-4,\ldots$, $i-j+2$, $i-j$, and is zero otherwise.

One of the simplest argument showing this depends on the fact that all f.d. representations of $SU(2)$ split into a direct sum of 1-dimensional representations of the subgroup of diagonal matrices. These are classified by integers $\ell$ (aka weights) and correspond to the 1-dimensional representation, where the matrix $\pmatrix{e^{ix}&0\cr0&e^{-ix}\cr}$ acts with eigenvalue $e^{i\ell x}$ for any real $x$. In the representation $\rho_k$ of bivariate homogeneous polynomials of degree $k$ the monomial $x_1^{k-t}x_2^t$ has weight $\ell=k-2t$. Here $t$ ranges over the integers in the interval $0\le t\le k$, so the weights $k, k-2,\ldots,-k+2,-k$ appear in $\rho_k$ each with multiplicity one.

When we study the tensor product of two representations such $\rho_i\rho_j$ the weights are added up in the sense that if $u\in\rho_i$ has weight $s$ and $v\in\rho_j$ has weight $t$, then the elementary tensor $u\otimes v\in \rho_i\rho_j$ has weight $s+t$. Keeping a tally of the multiplicities of the weights of the tensor products of monomials coming from the two representations then quickly leads to the Clebsch-Gordan formula.

Students of quantum mechanics often encounter this calculation when quantum mechanical addition of angular momenta is explained.

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