Isomorphism and non-isomorphism for interval groups of type Dn

Barbara Baumeister, Derek Holt, 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 J. Algebra 629 (2023), 399–423.

Abstract

We consider presentations that were derived in \cite{BaumeisterNeaimeRees} for the interval groups associated with proper quasi-Coxeter elements of the Coxeter group $W(D_n)$. We use combinatorial methods to derive alternative presentations for the groups, and use these new presentations to show that the interval group associated with a proper quasi-Coxeter element of $W(D_n)$ cannot be isomorphic to the Artin group of type $D_n$. While the specific problems we solve arise from the study of interval groups, their solution provides an illustration of how techniques indicated by computational observation can be used to derive properties of all groups within an infinite family.

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