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?
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3$\begingroup$ crossposted math.stackexchange.com/questions/2877494/… --- don't do that, please. $\endgroup$– Carlo BeenakkerCommented Aug 10, 2018 at 14:33
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$\begingroup$ @CarloBeenakker Sorry, I didn't realize that was bad form; I wasn't getting an answer there so posted it here. $\endgroup$– onamoonlessnightCommented Aug 10, 2018 at 14:52
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1$\begingroup$ it's not simply a matter of bad form: you run the risk that someone spends time answering your question on one site, not knowing that it has already been answered or partially answered on the other site; at the very least you should disclose on each site that you are asking this elsewhere as well $\endgroup$– Carlo BeenakkerCommented Aug 10, 2018 at 15:02
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4$\begingroup$ Understood. I will indicate the crossposting in this instance and avoid this in future. $\endgroup$– onamoonlessnightCommented Aug 10, 2018 at 15:05
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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
answered Aug 10, 2018 at 14:54
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$\begingroup$ It took me a while to parse the notation but this seems to be what I wanted. Are the Clebsch--Gordan coefficients $\langle j_1 m_1 j_2 m_2 | j_3 m_3 \rangle $ there the same as the ones one would call the CG coefficients $c(\mu, \nu; \lambda)$ in the expansion $ \pi_\mu \otimes \pi_\nu = \bigoplus_\lambda c(\mu, \nu; \lambda) \pi_\lambda$, in that I would have $c = 0$ or $1$ for $\mathrm{SU}(2)$? $\endgroup$ Commented Aug 12, 2018 at 10:36
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2$\begingroup$ No there are not. The physicist's CG coefficients $\langle j_1 m_1 j_2 m_2|j_3 m_3\rangle$ are matrix elements of explicit projectors (or injections given by the transpose) of $\pi_{\mu}\otimes\pi_{\nu}$ onto some irreducible $\pi_{\lambda}$. The mathematician's CG coefficient $c(\mu,\nu;\lambda)$ is the multiplicity of the irreducible in the tensor product. $\endgroup$ Commented Aug 14, 2018 at 14:04