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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