McKay Tantalizers
Jeffrey M. Barnes, advised by Prof. Tom Halverson
Presented at the undergraduate poster session of the AMS/MAA Joint Mathematics Mettings, New Orleans, LA (January 2007)
Abstract: Via a classical construction, we associate a family of algebras {CkG} to each finite subgroup G of the special unitary group SU2. These algebras have beautiful combinatorics vis-à-vis walks on their associated Dynkin diagram or McKay quiver. The construction works like this: Let V = ℂ2 be the natural two-dimensional representation of SU2 given by usual matrix-vector multiplication and let V⊗k be its k-fold tensor product. Then for any subgroup G ⊆ SU2, we let CkG be the algebra of transformations on V⊗k that commute with G — that is, the tensor power centralizer algebra (or tantalizer) of G. We prove that the dimension of G equals the number of closed walks of length 2k on the Dynkin diagram. The finite subgroups of SU2 reduce (in a concrete way) to two infinite classes — the cyclic groups of order m and the dicyclic groups of order 4m — and three exceptional cases, the binary polyhedral groups. In all but the exceptional cases, we have a bijection between a basis of CkG and these walks, and we have strong conjectures about the general structure of these algebras.