CU Algebraic Lie Theory Seminar

CU Algebraic Lie Theory Seminar
SL(2) centralizer algebras Tom Halverson, Macalester College
February 15, 2011 Math 220 2-3pm	Abstract Tensor powers of the defining matrix representation of the special linear group SL(2,C) lead naturally to the study of the combinatorics of Catalan numbers and the Temperley-Lieb algebra. Tensor space decomposes into irreducible modules whose multiplicities are Catalan numbers, and the Temperley-Lieb algebra is the centralizer algebra of transformations which commute with SL(2,C). We will tell this story and see the Temperley-Lieb algebra appear as a diagram algebra which models these commuting transformations. Then we will generalize this story in two ways: We replace the defining matrix of SL(2,C) with a natural 3-dimensional representation. Then the tensor powers lead to the study of Motzkin numbers, which are less-famous cousins of the Catalan numbers, and the "Motzkin Algebra," a diagram centralizer algebra that generalizes Temperley-Lieb. We look at finite subgroups G of SL(2,C), and we show that the centralizer algebra, in this case, reflects the beautiful structure of the McKay correspondence. In particular, it has dimension and a basis labeled by walks on the Dynkin diagram of G.

SL(2) centralizer algebras

Tom Halverson, Macalester College

February 15, 2011

Math 220

2-3pm

Abstract

Tensor powers of the defining matrix representation of the special linear group SL(2,C) lead naturally to the study of the combinatorics of Catalan numbers and the Temperley-Lieb algebra. Tensor space decomposes into irreducible modules whose multiplicities are Catalan numbers, and the Temperley-Lieb algebra is the centralizer algebra of transformations which commute with SL(2,C). We will tell this story and see the Temperley-Lieb algebra appear as a diagram algebra which models these commuting transformations. Then we will generalize this story in two ways:

We replace the defining matrix of SL(2,C) with a natural 3-dimensional representation. Then the tensor powers lead to the study of Motzkin numbers, which are less-famous cousins of the Catalan numbers, and the "Motzkin Algebra," a diagram centralizer algebra that generalizes Temperley-Lieb.
We look at finite subgroups G of SL(2,C), and we show that the centralizer algebra, in this case, reflects the beautiful structure of the McKay correspondence. In particular, it has dimension and a basis labeled by walks on the Dynkin diagram of G.