My naive understanding of taking the tensor product of two functions $f(x)$ and $g(x)$ is that the resulting function should act on two variables, i.e. $f \otimes g = f(x) g(y)$. This is discussed in several Mathematica posts (here and here). However, consider two spherical harmonic functions, $Y_{\ell m}$ and $Y_{\ell' m'}$. In the Wikipedia page on Clebsch-Gordan coefficients, the product $Y_{\ell m}(\Omega) Y_{\ell' m'}(\Omega)$ is given in terms of Clebsh-Gordan coefficients, as though the above expression should be considered as the tensor product of the two different representations, even though they act on the same variable. I'm not sure how to reconcile these two different statements - does anyone have a good explanation for this?
I'm also not quite sure how to understand the coefficients on the Wikipedia page other than the $\langle\ell_1 m_1 \ell_2 m_2|LM\rangle$ term. If we're thinking of the product $Y_{\ell m}(\Omega) Y_{\ell' m'}(\Omega)$ as a tensor product, shouldn't the decomposition into the $|LM\rangle$ basis be simply given by the Clebsh-Gordan coefficients?
