Interval groups related to finite Coxeter groups I

Barbara Baumeister, George Neaime and Sarah Rees

Keywords

Coxeter groups, Quasi-Coxeter elements, Carter diagrams, Artin(--Tits) groups, dual approach to Coxeter and Artin groups, generalised non-crossing partitions, Garside structures, Interval (Garside) structures.

Status

Published in Algebr. Comb. 6 (2023), no. 2, 471–506.

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 \cite{BaumNeaReesPart2}, 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.

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