Absolute order and involutions

TL;DR

This paper investigates the structure of the absolute order on Coxeter groups when restricted to involutions, characterizing those that form lattices using involutive parabolic subgroups.

Contribution

It provides a classification of involutions in Coxeter groups for which the associated order interval is a lattice, introducing the concept of involutive parabolic subgroups.

Findings

Identifies involutions with lattice intervals in the absolute order

Characterizes these involutions via involutive parabolic subgroups

Advances understanding of order structures in Coxeter groups

Abstract

We study the restriction of the absolute order on a Coxeter group $W$ to an interval $[1,w]_T$, where $w\in W$ is an involution. We characterize and classify those involutions $w$ for which $[1,w]_T$ is a lattice, using the notion of involutive parabolic subgroups.