Average of product of matrix elements in irreducible representations of unitary groups

Question

Let $\mathcal{U}(N)$ be the unitary group.

It is well known that $$ \int_{\mathcal{U}(N)} U_{ij} U^\dagger_{nm} \,dU=\delta_{im}\delta_{jn}\frac{1}{N},$$ where $dU$ is the Haar measure.

More complicated averages can also be computed, such as $$\int_{\mathcal{U}(N)} |U_{11}|^2|U_{12}|^2 \,dU = \frac{1}{N(N+1)}.$$

Now, let $R_\lambda$ be an irreducible representation of $\mathcal{U}(N)$ which is different from the fundamental one. Then, the main orthogonality still holds, $$ \int_{\mathcal{U}(N)} [R_\lambda(U)]_{ij}[R_\lambda(U^\dagger)]_{nm} \, dU = \delta_{im} \delta_{jn} \frac{1}{d_\lambda(N)},$$ with the denominator replaced by the dimension of the irrep.

My question is: are there calculations of such integrals involving more matrix elements? Like $$\int_{\mathcal{U}(N)} \Bigl|[R_\lambda(U)]_{11}\Bigr|^2\Bigl|[R_\lambda(U)]_{12}\Bigr|^2 dU=\text{?}$$

Answer 1

One can reduce your more complicated example which is of the form $$ \int_{U(n)} \langle a, R_{\lambda}(U)b\rangle \langle c, R_{\lambda}(U)d\rangle \times ({\rm complex\ conjugate}) \ dU $$ for some vectors $a,b,c,d$ to the simpler case with two matrix elements as follows.

Use $$ \langle a, R_{\lambda}(U)b\rangle \langle c, R_{\lambda}(U)d\rangle = \langle a\otimes c, (R_{\lambda}(U)\otimes R_{\lambda}(U)) b\otimes d\rangle\ , $$ and then the decomposition into irreducibles of $R_{\lambda}\otimes R_{\lambda}$.

The catch though is that this requires not only the Littlewood-Richardson rule (which says what irreducibles occur and with what multiplicity), but a very explicit form of it which involves the Clebsch-Gordan coefficients (or analogues of Wigner's $3jm$ symbols) for $U(n)$. As far as I know, there are no good formulas for them in full generality. Some references: this article by Elvang, Cvitanović and Kennedy or the multivolume series on "Representation of Lie Groups and Special Functions" by Vilenkin and Klimyk.