Length enumeration of fully commutative elements in finite and affine Coxeter groups

TL;DR

This paper provides explicit formulas for counting fully commutative elements in finite and affine Coxeter groups, improving previous recursive methods with a new recursive approach that yields closed-form generating functions.

Contribution

It introduces an alternative recursive description that produces explicit expressions for the generating functions of fully commutative elements.

Findings

Explicit formulas for generating functions derived

Recursive description simplifies previous methods

Enhanced understanding of element enumeration in Coxeter groups

Abstract

An element w of a Coxeter group W is said to be fully commutative, if any reduced expression of w can be obtained from any other by transposing adjacent pairs of generators. These elements were described in 1996 by Stembridge in the case of finite irreducible groups, and more recently by Biagioli, Jouhet and Nadeau (BJN) in the affine cases. We focus here on the length enumeration of these elements. Using a recursive description, BJN established for the associated generating functions systems of non-linear q-equations. Here, we show that an alternative recursive description leads to explicit expressions for these generating functions.