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On the length of fully commutative elements

Author: Philippe Nadeau
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 05E15, 16Z05; Secondary 05A15
Published electronically: February 8, 2018
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Abstract: In a Coxeter group $ W$, an element is fully commutative if any two of its reduced expressions can be linked by a series of commutations of adjacent letters. These elements have particularly nice combinatorial properties, and index a basis of the generalized Temperley-Lieb algebra attached to $ W$.

We give two results about the sequence counting these elements with respect to their Coxeter length. First we prove that this sequence always satisfies a linear recurrence with constant coefficients, by showing that reduced expressions of fully commutative elements form a regular language. Then we classify those groups $ W$ for which the sequence is ultimately periodic, extending a result of Stembridge. These results are applied to the growth of generalized Temperley-Lieb algebras.

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Additional Information

Philippe Nadeau
Affiliation: CNRS, Institut Camille Jordan, Université Claude Bernard Lyon 1, 69622 Villeurbanne Cedex, France

Keywords: Coxeter groups, fully commutative elements, Coxeter length, rational functions, finite automata, periodic sequences, reduced expressions, Temperley--Lieb algebra
Received by editor(s): June 27, 2016
Received by editor(s) in revised form: December 12, 2016
Published electronically: February 8, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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