Document Type

Honors Project - Open Access

Abstract

The finite subgroups of the special unitary group SU2 have been classified to be isomorphic to one of the following groups: cyclic, binary dihedral, binary tetrahedral, binary octahedral, and binary icosahedral, of order n, 4n, 24, 48, and 120, respectively. Associated to each group is a representation graph, which by the McKay correspondence is a Dynkin diagram of type Aˆ n−1, Dˆ n+2, Eˆ 6, Eˆ 7, or Eˆ 8. The centralizer algebra Zk(G) = EndG(V ⊗k ) is the algebra of transformations that commute with G acting on the k-fold tensor product of the defining representation V = C 2 . The dimension of the centralizer algebra equals the number of walks on the corresponding Dynkin diagram, beginning and ending at the root node. These dimensions are generalizations of Catalan numbers, with formulas which include binomial coefficients and the Lucas numbers. In uniform ways, we find two bases for these algebras which are each in bijection with the walks on the Dynkin diagram, the first of which works for the binary tetrahedral, octahedral, and icosahedral groups, and the second of which we conjecture to work for all groups, but has only been shown to work for low values of k. This result allows us to give a presentation of generators and relations for the centralizer algebra Zk(G). These results answer an open question in combinatorial representation theory.

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