  For a finite subgroup $G$ of the special unitary group $SU_2$, we study the
centralizer algebra $Z_k(G) = End_G(V^{\otimes k})$ of $G$ acting on the
$k$-fold tensor product of its defining representation $V= \mathbb{C}^2$. These
subgroups are in bijection with the simply-laced affine Dynkin diagrams. The
McKay correspondence relates the representation theory of these groups to the
associated Dynkin diagram, and we use this connection to show that the
structure and representation theory of $Z_k(G)$ as a semisimple algebra is
controlled by the combinatorics of the corresponding Dynkin diagram.
