Interval groups related to finite Coxeter groups I

TL;DR

This paper presents new presentations for interval groups associated with quasi-Coxeter elements in type D_n Coxeter groups, revealing that most are not isomorphic to Artin groups and introducing novel combinatorial techniques.

Contribution

It provides explicit presentations for interval groups in type D_n Coxeter groups and introduces combinatorial methods for analyzing quasi-Coxeter elements.

Findings

Derived presentations over Carter generating sets

Most interval groups are not isomorphic to Artin groups

Established properties related to Coxeter and Artin groups

Abstract

We derive presentations of the interval groups related to all quasi-Coxeter elements in the Coxeter group of type $D_n$. Type $D_n$ is the only infinite family of finite Coxeter groups that admits proper quasi-Coxeter elements. The presentations we obtain are over a set of generators in bijection with what we call a Carter generating set, and the relations are those defined by the related Carter diagram together with a twisted or a cycle commutator relator, depending on whether the quasi-Coxeter element is a Coxeter element or not. The proof is based on the description of two combinatorial techniques related to the intervals of quasi-Coxeter elements. In a subsequent work [4], we complete our analysis to cover all the exceptional cases of finite Coxeter groups, and establish that almost all the interval groups related to proper quasi-Coxeter elements are not isomorphic to the related Artin groups, hence establishing a new family of interval groups with nice presentations. Alongside the proof of the main results, we establish important properties related to the dual approach to Coxeter and Artin groups.