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Fully commutative elements in finite and affine Coxeter groups

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Abstract

An element of a Coxeter group \(W\) is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. These elements were extensively studied by Stembridge, in particular in the finite case. They index naturally a basis of the generalized Temperley–Lieb algebra. In this work we deal with any finite or affine Coxeter group \(W\), and we give explicit descriptions of fully commutative elements. Using our characterizations we then enumerate these elements according to their Coxeter length, and find in particular that the corrresponding growth sequence is ultimately periodic in each type. When the sequence is infinite, this implies that the associated Temperley–Lieb algebra has linear growth.

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Authors and Affiliations

  1. Institut Camille Jordan, Université Claude Bernard Lyon 1, 69622, Villeurbanne Cedex, France

    Riccardo Biagioli & Frédéric Jouhet

  2. CNRS, Institut Camille Jordan, Université Claude Bernard Lyon 1, 69622, Villeurbanne Cedex, France

    Philippe Nadeau

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  1. Riccardo Biagioli
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  2. Frédéric Jouhet
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  3. Philippe Nadeau
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Correspondence to Riccardo Biagioli.

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Communicated by J. S. Wilson.

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Biagioli, R., Jouhet, F. & Nadeau, P. Fully commutative elements in finite and affine Coxeter groups. Monatsh Math 178, 1–37 (2015). https://doi.org/10.1007/s00605-014-0674-7

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