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