On the weak order of Coxeter groups

TL;DR

This paper explores conjectural relationships between weak order extensions, closure operators, and Bruhat order in Coxeter groups, proposing new structures and connections that deepen understanding of their algebraic and combinatorial properties.

Contribution

It introduces a refined conjecture linking weak order, closure operators, and Bruhat order, and defines an analogue of weak order for parabolic subsets of root systems.

Findings

Meet and join in weak order described via closure operators

Galois connections relate subgroup structures to weak order lattices

Proposed weak order analogue for parabolic subsets with conjectural properties

Abstract

This paper provides some evidence for conjectural relations between extensions of (right) weak order on Coxeter groups, closure operators on root systems, and Bruhat order. The conjecture focused upon here refines an earlier question as to whether the set of initial sections of reflection orders, ordered by inclusion, forms a complete lattice. Meet and join in weak order are described in terms of a suitable closure operator. Galois connections are defined from the power set of W to itself, under which maximal subgroups of certain groupoids correspond to certain complete meet subsemilattices of weak order. An analogue of weak order for standard parabolic subsets of any rank of the root system is defined, reducing to the usual weak order in rank zero, and having some analogous properties in rank one (and conjecturally in general).