Jeffrey M. Barnes

McKay Centralizer Algebras

Jeffrey M. Barnes, Georgia Benkart, and Tom Halverson (2016)

Proceedings of the London Mathematical Society 112: 375–414

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Abstract: For a finite subgroup G of the special unitary group SU2, we study the centralizer algebra Zk(G) = EndG(Vk) 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 associated 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.