# Visualizing diagram factorizations in Temperley–Lieb algebras

## by Dana Ernst

Published April 12, 2014 in Research

The Temperley-Lieb Algebra of type A, 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. Then in 1987, V. Jones showed that this algebra occurs naturally as a quotient of an algebra whose underlying structure is the symmetric group. Similarly, there is also a diagrammatic representation for the Temperley-Lieb algebra of type B involving decorated diagrams. Multiplying diagrams is easy to do. However, taking a given diagram and finding the corresponding reduced factorization is generally difficult. We have an efficient (and colorful) algorithm for obtaining a reduced factorization for Temperley-Lieb diagrams of types A and B.

This talk was given by my undergraduate research student Sarah Salmon (Northern Arizona University) on April 12, 2014 at the 2014 Arizona–Nevada Academy of Science Annual Meeting. This joint work with Michael Hastings.