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. In the present work, we focus on fully commutative involutions, which are characterized in terms of Viennot's heaps. By encoding the latter by Dyck-type lattice walks, we enumerate fully commutative involutions according to their length, for all classical finite and affine Coxeter groups. In the finite cases, we also find explicit expressions for their generating functions with respect to the major index. Finally in affine type A, we connect our results to Fan-Green's cell structure of the corresponding Temperley-Lieb algebra. (C) 2015 Elsevier B.V. All rights reserved.
Biagioli R, Jouhet F, Nadeau P (2015). Combinatorics of fully commutative involutions in classical Coxeter groups. DISCRETE MATHEMATICS, 338, 2242-2259 [10.1016/j.disc.2015.05.023].
Combinatorics of fully commutative involutions in classical Coxeter groups
Biagioli R;
2015
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. In the present work, we focus on fully commutative involutions, which are characterized in terms of Viennot's heaps. By encoding the latter by Dyck-type lattice walks, we enumerate fully commutative involutions according to their length, for all classical finite and affine Coxeter groups. In the finite cases, we also find explicit expressions for their generating functions with respect to the major index. Finally in affine type A, we connect our results to Fan-Green's cell structure of the corresponding Temperley-Lieb algebra. (C) 2015 Elsevier B.V. All rights reserved.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
