Jeffrey M. Barnes

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)

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Abstract: Each subgroup G of the special linear group SL2(ℂ) acts naturally as transformations on V = ℂ2. In this research we study the centralizer algebra CkG = EndG(Vk) of endomorphisms that commute with G on the k-fold tensor product of V. It is known that when G = SL2(ℂ), the centralizer CkG is the Temperley-Lieb algebra, with Catalan dimension 1/(2k + 1)(2k + 1k). 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 CkG and yields a combinatorial recurrence for the dimension of CkG. We explicitly compute CkG in low dimensions and we make a number of conjectures about the recursive structure of Ck in higher dimensions.