Interval groups related to finite Coxeter groups. I
In the 1960s, \textit{F. Garside} introduced a new algebraic aspect to the study of the braid group \(B_{n}\) in his solution to the conjugacy problem in \(B_{n}\) [Q. J. Math., Oxf. II. Ser. 20, 235--254 (1969; Zbl 0194.03303)]: Garside defined a submonoid \(B^{+}_{n}\) of \(B_{n}\) and a distinguished element \(\Delta_{n} \in B^{+}_{n}\) with the property that every element \(g\in B_{n}\) can be written in the form \(g=\Delta_{n}^{m}g'\) with \(m \in \mathbb{Z}\) and \(g' \in B_{n}^{+}\). It was then realised that Garside's approach extends to all Artin groups of spherical type, independently in [\textit{E. Brieskorn} and \textit{K. Saito}, Invent. Math. 17, 245--271 (1972; Zbl 0243.20037)] and [\textit{P. Deligne}, Invent. Math. 17, 273--302 (1972; Zbl 0238.20034)]. In this paper, the authors 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 that they 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 cycle 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.
dual approach to Coxeter and Artin groups
scientific article; zbMATH DE number 7681942
scientific article