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Consider the unique (up to unitary equivalence) unitary irreducible representation $(V_{j},D_{j})$ of $\mathrm{SU}(2)$ with dimension $2j+1$. Then, one usually defines the "Wigner D-matrices" to be the representation matrices of $D_{j}$ in the standard basis $\{\vert j,m\rangle\}_{-j\leq m\leq j}$ of $V_{j}$, i.e.

$$D_{mn}^{j}(g):=\langle j,m\vert D^{j}(g)\vert j,n\rangle\in\mathbb{C}$$

Now, on wikipedia, it is claimed that there is the following formula:

$$D^{j}_{mn}(g)D^{j^{\prime}}_{m^{\prime}n^{\prime}}(g)=\sum_{J=\vert j-j^{\prime}\vert}^{j+j^{\prime}}C_{mm^{\prime}(m+m^{\prime})}^{jj^{\prime}J}C_{nn^{\prime}(n+n^{\prime})}^{jj^{\prime}J}D^{J}_{(m+m^{\prime})(n+n^{\prime})}(g)$$ where $C_{mm^{\prime}M}^{jj^{\prime J}}:=\langle j_{1},j_{2},m_{1},m_{2}\vert J,M\rangle$ denotes the Clebsch-Gordan coefficients. However, in this lecture notes, it is claimed that (formulae (396) and (397))

$$D^{j}_{mn}(g)D^{j^{\prime}}_{m^{\prime}n^{\prime}}(g)=\sum_{J=\vert j-j^{\prime}\vert}^{j+j^{\prime}}\sum_{M,M^{\prime}}C_{mm^{\prime}M}^{jj^{\prime}J}C_{nn^{\prime}M^{\prime}}^{jj^{\prime}J}D^{J}_{MM^{\prime}}(g)$$

I can't see how these two formula are the same. Also, what is the range of the sums over $M$ and $M^{\prime}$ in the second formula, if I have understood it correctly? I guess, it is just $-J\leq M,M^{\prime}\leq J$...

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    $\begingroup$ You are "adding" angular momenta, so your Clebsches vanish unless $M=m+m'$, $M'=n+n'$, etc... Have you tried a simple example? $\endgroup$ Nov 30, 2021 at 15:31

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I think the two formulas are not a priori the same, because in your book not all properties of the CG coefficients have been used yet. For example the sum over $M,M'$ (which the book calls $m',m''$ I think) goes over all values that $M,M'$ can take, which is determined by $J$, i.e. they go from $-J$ to $J$ if I remember correctly (or with the variable names of the book $m'$ goes from $-j$ to $j$). Remind yourself: The CG coefficients are a priori just the general expansion coefficients as defined in equation (391) so there is one for each basis element, at least before you show specific properties.

This is done later (in equation (399)), where it gets shown that the (products of the two) CG coefficients are only not zero if $M=m+m',M'=n+n'$ (in the variables of the book that is $m'=m_1'+m_2',m''=m_1''+m_2''$. And if you plug that back into equation (397) your sum over $M,M'$ vanishes (in the book the sum over $m',m''$) and you have the formula that you found on Wikipedia.

Hopefully that helps and I'm not missing the point.

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