Involve: A Journal of Mathematics

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  • Volume 10, Number 1 (2017), 89-108.

Factorization of Temperley–Lieb diagrams

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

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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.

Article information

Involve, Volume 10, Number 1 (2017), 89-108.

Received: 5 September 2015
Revised: 10 January 2016
Accepted: 14 January 2016
First available in Project Euclid: 22 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20C08: Hecke algebras and their representations 20F55: Reflection and Coxeter groups [See also 22E40, 51F15] 57M15: Relations with graph theory [See also 05Cxx]

diagram algebra Temperley–Lieb algebra Coxeter group heap


Ernst, Dana C.; Hastings, Michael G.; Salmon, Sarah K. Factorization of Temperley–Lieb diagrams. Involve 10 (2017), no. 1, 89--108. doi:10.2140/involve.2017.10.89.

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