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
$$\int\_{\mathcal{U}(N)} \Bigl|[R\_\lambda(U)]\_{11}\Bigr|^2\Bigl|[R\_\lambda(U)]\_{12}\Bigr|^2 dU=\text{?}$$



Average of product of matrix elements in irreducible representations of unitary groups
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](https://aip.scitation.org/doi/abs/10.1063/1.1832753)
or the multivolume series on "Representation of Lie Groups and Special Functions" by 
Vilenkin and Klimyk.


