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The classical Schur-Weyl duality deals with the decomposition of $V^{\otimes k}$ into irreps of $S_k\times GL(n)$, where $V=\mathbb{C}^n$ is the defining irrep of $GL(n)$.

Is there a version of the duality if I take $V$ to be a more general irrep of $GL(n)$?

The motivation for this question is the irrep decomposition of generalized symmetric tensor products in physics context. I.e. when I have a quantity of the form $$ A^{\mu\nu\lambda}A^{\rho\sigma\gamma}B^{\alpha\beta}B^{\epsilon\delta}, $$ where $A$ and $B$ have some definite Young symmetry, and I want to know the irrep content of it. More generally, I can have more $A$'s, but only allow a subgroup of permutations for them (if I had Schur-Weyl duality for the irrep of $A$, I would look for singlets in the restrictions of $S_k$ representations to the smaller subgroup). I can in principle figure the answer quite generally by using character arguments, and in some simple cases like this one by other techniques, but it would be nice to know a more efficient genear way. If anyone knows an efficient method to tackle this kind of problems without regard to Schur-Weyl duality, that's also great.

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I guess that what I was looking for is the following. If $V$ is a representation of $GL(n)$, then we can consider $GL(n)$ to be a subgroup of $GL(\dim V)$, use the Schur-Weyl duality for $S_k\times GL(\dim V)$ to identify the irreps with respect to $S_k$, and then restrict to $GL(n)$. That is, $$ V\otimes V\otimes \ldots \otimes V=\sum_\lambda \rho^\lambda_{S_k}\otimes Res^{GL(\dim V)}_{GL(n)} \rho^\lambda_{GL(\dim V)}, $$ where the sum is over all appropriate Young tableax.

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