Vol. 10, No. 1, 2017

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ISSN: 1944-4184 (e-only)
ISSN: 1944-4176 (print)
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