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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: http://homepages.physik.uni-muenchen.de/~vondelft/Papers/ClebschGordan/2012-01-02
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    Isn't this just the Clebsch–Gordan theorem? See theorem 15.14 [here](http://tartarus.org/gareth/maths/notes/ii/Representation_Theory.pdf), or theorem 21.7 [here](http://math.berkeley.edu/~teleman/math/RepThry.pdf).2012-01-02

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