# The product of three spherical harmonics in higher dimension

As I see, e.g., in this question, a closed formula for the following integral $$\begin{equation} \int_{\mathbb{S}^N} Y_{\ell_1\ldots \ell_N}\,Y_{\ell'_1\ldots\ell'_N}\,Y_{\ell''_1\ldots \ell''_N}\, d^N S \end{equation}$$ is well-known in the $$N=2$$ case. (Here $$\mathbb{S}^N=\{x_1^2+\dots+x_{N+1}^2=1\}$$ is the $$N$$-dimensional sphere sitting in $$N+1$$ dimensional space. The symbol $$d^N S$$ denotes the $$N$$-dimensional surface element and the spherical harmonics quantum numbers reduce to $$\ell_1=\ell, \ell_2=m$$ in the two-dimensional case).

My question is whether a similar formula is known in arbitrary dimension (I am especially interested in the cases $$N=3$$, $$N=5$$)?

Hints on how to compute such a formula, or even special cases of it are also warmly appreciated.

• It's ugly to actually do it, but you can consider that the spherical harmonics are a representation of the $\mathrm{SO}(N)$ rotation group, and then seek the connection between your definition of them and a representation in matrix form (there will be ugly factors between them) - the integral over the sphere then becomes agroup integral, and the integrals for products of such matrices are, at least for three matrices, done in various places as applications of the Peter-Weyl theorem. Mar 17, 2015 at 14:48