McKay Centralizer Algebras

For a finite subgroup G of the special unitary group SU2, we study the centralizer algebra Zk(G) = EndG(V ⊗k) of G acting on the k-fold tensor product of its defining representation V = C2. These subgroups are in bijection with the simply-laced affine Dynkin diagrams. The McKay correspondence relates the representation theory of these groups to the associ-ated Dynkin diagram, and we use this connection to show that the structure and representation theory of Zk(G) as a semisimple algebra is controlled by the combinatorics of the corresponding Dynkin diagram