#### Vol. 10, No. 1, 2017

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Factorization of Temperley–Lieb diagrams

### Dana C. Ernst, Michael G. Hastings and Sarah K. Salmon

Vol. 10 (2017), No. 1, 89–108
##### Abstract

The Temperley–Lieb algebra is a finite-dimensional associative algebra that arose in the context of statistical mechanics and occurs naturally as a quotient of the Hecke algebra arising from a Coxeter group of type $A$. It is often realized in terms of a certain diagram algebra, where every diagram can be written as a product of “simple diagrams”. These factorizations correspond precisely to factorizations of the so-called fully commutative elements of the Coxeter group that index a particular basis. Given a reduced factorization of a fully commutative element, it is straightforward to construct the corresponding diagram. On the other hand, it is generally difficult to reconstruct the factorization given an arbitrary diagram. We present an efficient algorithm for obtaining a reduced factorization for a given diagram.

##### Keywords
diagram algebra, Temperley–Lieb algebra, Coxeter group, heap
##### Mathematical Subject Classification 2010
Primary: 20C08, 20F55, 57M15
##### Milestones
Received: 5 September 2015
Revised: 10 January 2016
Accepted: 14 January 2016
Published: 11 October 2016

Communicated by Scott T. Chapman
##### Authors
 Dana C. Ernst Department of Mathematics and Statistics Northern Arizona University Flagstaff, AZ 86011 United States Michael G. Hastings Department of Mathematics and Statistics Northern Arizona University Flagstaff, AZ 86011 United States Sarah K. Salmon Department of Mathematics University of Colorado Boulder Boulder, CO 80309 United States