The Temperley--Lieb Algebra, invented by Temperley and Lieb in 1971, is a finite dimensional associative algebra that arose in the context of statistical mechanics. Later in 1971, R. Penrose showed that this algebra can be realized in terms of certain diagrams. In the cases when diagrammatic representations are known to exist, it turns out that every diagram can be written as a product of "simple diagrams." These factorizations correspond precisely to factorizations in the underlying group. Given a diagrammatic representation and a reduced factorization of a group element, it is easy to construct the corresponding diagram. However, given a diagram, it is generally difficult to reconstruct the factorization of the corresponding group element. In cases that include Temperley--Lieb algebras of types A and B, we have devised an algorithm for counting the number of times each simple diagram appears in a reduced factorization for a given diagram.
This talk was given by my undergraduate research student Sarah Salmon (Northern Arizona University) on February 1, 2014 at the Nebraska Conference for Undergraduate Women in Mathematics. This joint work with Michael Hastings.