Let $\mathcal{O}_n$ be the Cuntz algebra with generators $a_1,...,a_n$. We can define an action of $U(n,\mathbb{C})$ (the group of $n\times n$ unitary operators) on $\mathcal{O}_n$ in a very natural way: just replace each generator $a_i$ with $\sum_j b_{ij} a_j $, where $(b_{ij}) \in U(n,\mathbb{C})$.
If we calculate the fixed points of such action we will see that it is $C^*(S_\infty)$ (the univsersal $C^*$-algebra of the group of bijections of $\mathbb{N}$ which is identity almost everywhere). So we have some sort of "correspondence" between $U(n,\mathbb{C})$ and $S_\infty$. Is it connected with Schur-Weyl duality? If yes, how?