# rt.representation theory – Multiplicities of irreducible \$U(n)\$-modules in the tensor product \$V_{lambda}otimes V_{mu}\$

It is known that every irreducible representation of the unitary group $$U(n)$$ can be uniquely described by the non-increasing sequence $$lambda=(lambda_1,ldots,lambda_n)$$ of integers (denote the corresponding representation as $$V_{lambda}$$). Does there exist a formula (like Littlewood-Richardson rule for $$GL(n)$$) for the decomposition of the tensor product $$V_{lambda}otimes V_{mu}$$?

In terms of characters it means that we need to decompose the product of ‘Schur functions’ $$s_{lambda}s_{mu}$$ into the sum of some $$s_{nu}$$ (but here $$lambda,mu,nu$$ might have negative ‘parts’).

The case when $$lambda_1geldotsgelambda_nge 0gemu_1geldotsgemu_n$$ is especially interesting for me.

Any references would be appreciated.