**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 {`C _{k}^{G}`} to each finite subgroup

`G`of the special unitary group

`SU`

_{2}. These algebras have beautiful combinatorics vis-à-vis walks on their associated Dynkin diagram or McKay quiver. The construction works like this: Let

`V`= ℂ

^{2}be the natural two-dimensional representation of

`SU`

_{2}given by usual matrix-vector multiplication and let

`V`

^{⊗k}be its

`k`-fold tensor product. Then for any subgroup

`G`⊆

`SU`

_{2}, we let

`C`be the algebra of transformations on

_{k}^{G}`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 2

`k`on the Dynkin diagram. The finite subgroups of

`SU`

_{2}reduce (in a concrete way) to two infinite classes — the cyclic groups of order

`m`and the dicyclic groups of order 4

`m`— and three exceptional cases, the binary polyhedral groups. In all but the exceptional cases, we have a bijection between a basis of

`C`and these walks, and we have strong conjectures about the general structure of these algebras.

_{k}^{G}