To let $ mathcal {U} (N) $ to be the unified group.

It is well known that

$$ int _ { mathcal {U} (N)} U_ {ij} U ^ dagger_ {nm} dU = delta_ {im} delta_ {jn} frac {1} {N}, $$

from where $ dU $ is the hair measure.

It can also be calculated more complicated average values, such

$$ int _ { mathcal {U} (N)} | U_ {11} | ^ 2 | U_ {12} | ^ 2 dU = frac {1} {N (N + 1)}. $$

Now let it go $ R_ lambda $ to be an irreducible representation of $ mathcal {U} (N) $ that is different from the basic one. Then the main orthogonality still applies

$$ int _ { mathcal {U} (N)} [R_lambda(U)]_ {ij}[R_lambda(U^dagger)]_ {nm} dU = delta_ {im} delta_ {jn} frac {1} {d_ lambda (N)}, $$

where the denominator is replaced by the dimension of Irrep.

My question is: are there any computations of such integrals with more matrix elements? To like

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