On maximal dihedral reflection subgroups and generalized noncrossing partitions

Gobet, Thomas

doi:10.1090/proc/16972

On maximal dihedral reflection subgroups and generalized noncrossing partitions
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by Thomas Gobet;

Proc. Amer. Math. Soc. 152 (2024), 4095-4101

DOI: https://doi.org/10.1090/proc/16972

Published electronically: August 26, 2024

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Abstract:

In this note, we give a new proof of a result of Matthew Dyer stating that in an arbitrary Coxeter group $W$, every pair $t,t’$ of distinct reflections lie in a unique maximal dihedral reflection subgroup of $W$. Our proof only relies on the combinatorics of words, in particular we do not use root systems at all. As an application, we deduce a new proof of a recent result of Delucchi-Paolini-Salvetti, stating that the poset $[1,c]_T$ of generalized noncrossing partitions in any Coxeter group of rank $3$ is a lattice. We achieve this by showing the more general statement that any interval of length $3$ in the absolute order on an arbitrary Coxeter group is a lattice. This implies that the interval group attached to any interval $[1,w]_T$ where $w$ is an element of an arbitrary Coxeter group with $\ell _T(w)=3$ is a quasi-Garside group.

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Proceedings of the American Mathematical Society

On maximal dihedral reflection subgroups and generalized noncrossing partitions
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Abstract: