  In geometric representation theory, it is common to compute equivariant $K$
theory of schemes like $Hilb^n ( \mathbb{A}^2 )$ or $Hilb^n (X)$ for an ALE
resolution $X \to \mathbb{A}^2 / \Gamma$. If we abandon the algebraic nature
and just look at this homotopically we see close relatives of $BS_n$ and
$B(\Gamma \wr S_n)$. Therefore we compute the topological K theory of these
classifying spaces to fill in a small gap in the literature.
