# integral of product of three basis functions and Clebsh-Gordan coefficients

Suppose I have an orthonormal basis $\{b_i\}_{i=1}^\infty$ for an $L_2$ space (for example, the $b_i$ could be spherical harmonics on the round sphere with the Euclidean $L_2$ inner product). I want to calculate the coefficients $$C_{i,j,k} = \int b_ib_jb_k;$$ are these the same thing as the "Clebsh-Gordan coefficients" for the basis $\{b_i\}$? If not, are they related, and how? My reading suggests they are closely related, but I am having trouble untangling the mathematics from the quantum mechanics.

• Related – Giuseppe Negro Jun 7 '16 at 21:47
• I have found some more understandable information in the lecture notes on "Quantum Mathematics" by Peter Olver: see here (scroll to "Lecture Notes: Quantum Mathematics"). I am writing an answer below with more information. – Giuseppe Negro Aug 24 '17 at 9:55
• Here's a recent related question with a good answer. – Giuseppe Negro Mar 12 '18 at 15:50

The key is that, if $\{b_i\ :\ i=0, 1,2\ldots \}$ is an orthonormal and real basis of $L^2(d\mu)$, then $$\tag{1} \int b_ib_j b_k\, d\mu= c_{i,j}^k \quad \iff \quad b_i b_j = \sum_k c_{i,j}^k b_k.$$ And it is the latter equation that is related to Clebsch-Gordan stuff.
Indeed, the Clebsch-Gordan $c_{i,j}^k$ appear in formulas such as $$\rho_i\otimes \rho_j = \bigoplus_k c_{i,j}^k \rho_k,$$ where the $\rho_i$ are irreducible representations of some group $G$. This already bears some resemblance with (1). Taking the characters $\chi_i(g)=\mathrm{trace}\,\rho_i(g)$, the previous equation becomes $$\chi_i\cdot \chi_j = \sum_k c_{i,j}^k \chi_k,$$ and the resemblance is even greater.
• For the relationship between "matrix elements" and special functions, from which the $c_{i, j}^k$ symbols come out, see Koornwinder's tutorial here: arxiv.org/abs/1606.08189 – Giuseppe Negro Jul 6 '18 at 19:13