# Representation Theory – Mean of the product of matrix elements in irreducible representations of unit groups

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

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