Classification of Coxeter groups with finitely many elements of $\mathbf{a}$-value 2

TL;DR

This paper classifies Coxeter groups with finitely many elements having an a-value of 2, using Coxeter diagrams, providing a clear structural understanding of these groups.

Contribution

It offers a complete classification of Coxeter groups with finitely many a-value 2 elements based on their diagrams, advancing the understanding of Lusztig's a-function.

Findings

Identification of all Coxeter groups with finitely many a-value 2 elements

Explicit classification in terms of Coxeter diagrams

Enhanced understanding of Lusztig's a-function in Coxeter groups

Abstract

We consider Lusztig's $\mathbf{a}$-function on Coxeter groups (in the equal parameter case) and classify all Coxeter groups with finitely many elements of $\mathbf{a}$-value 2 in terms of Coxeter diagrams.