On maximal dihedral reflection subgroups and generalized noncrossing partitions

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.