Let $V=\mathbb C^d$, and let $A$ be the subalgebra of $\operatorname{End}_{\mathbb C}(V^{\otimes n})$ generated by $\rho(GL_d(\mathbb C))$. Let $B$ be the image of $\mathbb C[S_n]$ under the place-permutation action. The algebras $A$ and $B$ centralize each other. When $d \ge n$, each is the full centralizer of the other.