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A factorization of Temperley--Lieb diagrams

by Dana Ernst

Published January 3, 2014 in Research

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. Then in 1987, V. Jones showed that the Temperley--Lieb Algebra occurs naturally as a quotient of the Hecke algebra arising from a Coxeter group of type A (whose underlying group is the symmetric group). This realization of the Temperley--Lieb Algebra as a Hecke algebra quotient was generalized to the case of an arbitrary Coxeter group. 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 efficient algorithm for obtaining a reduced factorization for a given diagram.

This poster was presented by my undergraduate research students Michael Hastings and Sarah Salmon (Northern Arizona University) on January 17, 2014 at the 2014 Joint Mathematics Meetings in Baltimore, MD and on April 25, 2014 at the 2014 NAU Undergraduate Symposium at Northern Arizona University in Flagstaff, AZ.