On maximal dihedral reflection subgroups and generalized noncrossing partitions

TL;DR

This paper provides a new combinatorial proof that in any Coxeter group, distinct reflections are contained in a unique maximal dihedral subgroup, and applies this to show certain intervals form lattices, revealing new algebraic structures.

Contribution

It offers a root-system-free combinatorial proof of maximal dihedral reflection subgroups and applies this to establish lattice properties of generalized noncrossing partitions in Coxeter groups.

Findings

Unique maximal dihedral reflection subgroups for any pair of reflections.

Intervals of length 3 in the absolute order are lattices.

Interval groups of length 3 are quasi-Garside groups.

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.