**McKay Tantalizers**

Jeffrey M. Barnes, advised by Prof. Tom Halverson

Presented at the undergraduate poster session of the AMS/MAA Joint Mathematics Mettings, San Antonio, TX (January 2006)

**Abstract: **Each subgroup `G` of the special linear group `SL`_{2}(ℂ) acts
naturally as transformations on `V` = ℂ^{2}. In this research we study the centralizer algebra `C _{k}^{G}` =

`End`(

_{G}`V`

^{⊗k}) of endomorphisms that commute with

`G`on the

`k`-fold tensor product of

`V`. It is known that when

`G`=

`SL`

_{2}(ℂ), the centralizer

`C`is the Temperley-Lieb algebra, with Catalan dimension 1/(2

_{k}^{G}`k`+ 1)(2

`k`+ 1

`k`). We study the case of the the tetrahedral group

**T**, the octahedral group

**O**, and the icosahedral group

**I**. In each case, we construct a graph called the Bratteli diagram, which describes the structure of

`C`and yields a combinatorial recurrence for the dimension of

_{k}^{G}`C`. We explicitly compute

_{k}^{G}`C`in low dimensions and we make a number of conjectures about the recursive structure of

_{k}^{G}`C`in higher dimensions.

_{k}