# Visualizing diagram factorizations in Temperley-Lieb algebras, Part 1

## by Dana Ernst

Published March 1, 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. Given an expression of a symmetric group element, it is easy to construct the corresponding diagram. However, given a diagram, it is generally difficult to reconstruct a factorization of the corresponding permutation. We have devised an efficient and visually appealing algorithm for obtaining a reduced factorization for a given diagram. This talk will be an introduction to Temperley-Lieb diagrams and an explanation of our algorithm. An extension involving diagrams of type B will be addressed in "Visualizing diagram factorizations in Temperley-Lieb algebras, Part 2” by Sarah Salmon (NAU).

This talk was given by my undergraduate research student Michael Hastings (Northern Arizona University) on March 1, 2014 at the Southwestern Undergraduate Mathematics Research Conference (SUnMaRC) at Mesa Community College, Mesa, AZ. This joint work with Sarah Salmon.