  Let $\mathbf{S}_k$ denote a maximal torus in the complex Lie group
$\mathbf{G} = \mathrm{SL}_n(\mathbb{C})/C_k$ and let $T_k$ denote a maximal
torus in its compact real form $\mathrm{SU}_n(\mathbb{C})/C_k$, where $k$
divides $n$. Let $W$ denote the Weyl group of $\mathbf{G}$, namely the
symmetric group $\mathfrak{S}_n$. We elucidate the structure of the extended
quotient $\mathbf{S}_k // W$ as an algebraic variety and of $T_k // W$ as a
topological space, in both cases describing them as bundles over unions of
tori. Corresponding to the invariance of $K$-theory under Langlands duality,
this calculation provides a homotopy equivalence between $T_k // W$ and its
dual $T_{\frac{n}{k}} // W$. Hence there is an isomorphism in cohomology for
the extended quotients which is stratified as a direct sum over conjugacy
classes of the Weyl group. We use our formula to compute a number of examples.
