To what extent does the relation between the diagonal representation of $SU(n)$ in $(\mathbb{C}^n)^{\otimes k}$ and representations of the symmetric group $S\_k$ remain valid when instead of the group $SU(n)$ we take the general unitary group $U(\mathcal{H})$ ($\mathcal{H}$ is some separable Hilbert space) and its diagonal representation in $\mathcal{H}^{\otimes k}$? In particular is it true that irreducible representations of of $U(\mathcal{H})$ in $\mathcal{H}^{\otimes k}$ correspond to young diagrams of $S\_k$ just like in the finite dimensional case? Is it possible to get any irreducible representation of $U(\mathcal{H})$ by considering decompositions of $\mathcal{H}^{\otimes k}$ for sufficiently big $k$?



Is there  "Schur-Weyl duality" for infinite dimensional unitary group?
The answer to all of your questions is yes. This is a theorem announced by Kirillov in


Kirillov, A. A. Representations of the infinite-dimensional unitary group. *Dokl. Akad. Nauk. SSSR*, 1973, 212, 288-290


and proved by Olshanski in


Olshanski, G. I. Unitary representations of the infinite-dimensional classical groups $U(p,\infty)$, $SO\_0(p,\infty )$, $Sp(p,\infty)$, and of the corresponding motion groups. *Funktsional. Anal. i Prilozhen.*, 1978, 12, 32-44, 96


**Edit**: This answer applies to continuous unitary representations of $U(\mathcal{H})$, where the latter group is equipped with the strong operator topology (which is the usual topology on this group). However, if you are only interested in representations on a separable Hilbert space, the continuity assumption can be dropped, as is shown in arXiv:1109.1200.


