Weak order on groups generated by involutions

TL;DR

This paper explores the properties of weak orders in involution-generated groups, extending known results from Coxeter systems to a broader class, and provides classifications and new characterizations.

Contribution

It introduces the concept of involution systems, characterizes when their weak order forms a complete meet-semilattice, and extends Coxeter system results to larger classes including Cactus groups.

Findings

Weak order is a complete meet-semilattice for certain involution systems.

Provided a finite presentation for involution systems with sign character.

Classified involution systems in rank 3 and characterized Coxeter systems via weak order.

Abstract

In this article, we propose to initiate the general study of involution systems. An {\em involution system}, that is, a group $W$ generated by a set of involutions $S$, is naturally endowed with a {\em weak order} arising from orienting the Cayley graph of $(W,S)$. In the case of a Coxeter system $(W,S)$, Bj\"orner showed that the weak order is a complete meet-semilattice. This fact has many important consequences for Coxeter systems and their related structures. In this article, we discuss the following question: For which involution systems is the weak order a complete meet-semilattice? The class of involution systems that satisfies this condition is larger than the class of Coxeter systems (it contains, for instance, Cactus groups). In the case of an involution system with sign character, we provide a finite presentation by generators and relations and a classification in rank 3. We also obtain new characterizations of Coxeter systems in terms of the weak order, and prove a number of results on certain subclasses of these involution systems. Finally, we discuss further works and open problems in relation to biautomatic structures, geometric representations, mediangle graphs, and more.