Combinatorics of fully commutative involutions in classical Coxeter groups

TL;DR

This paper studies fully commutative involutions in classical Coxeter groups, using combinatorial models to enumerate and analyze their properties, including generating functions and connections to algebraic structures.

Contribution

It introduces a combinatorial encoding of fully commutative involutions via Dyck-type lattice walks and provides enumeration formulas for all classical finite and affine Coxeter groups.

Findings

Enumerated fully commutative involutions by length in classical Coxeter groups.

Derived explicit generating functions with respect to the major index in finite cases.

Connected affine type A involutions to the cell structure of Temperley–Lieb algebra.

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.