On the weak order of Coxeter groups

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).

37 pages, submitted

References in corpus (2)

• On rigidity of abstract root systems of Coxeter systems
• Geometric Combinatorics of Weyl Groupoids

Cited by in corpus (19)

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• A new family of posets generalizing the weak order on some Coxeter groups
• On inversion sets and the weak order in Coxeter groups
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• Garside families in Artin-Tits monoids and low elements in Coxeter groups
• Shi arrangements and low elements in Coxeter groups
• Lattice homomorphisms between weak orders
• Parabolic subgroup orbits on finite root systems
• Biclosed sets in real hyperplane arrangements
• A characterization of simplicial oriented geometries as groupoids with root systems
• Local Finiteness of the Twisted Bruhat Orders on Affine Weyl Groups
• Twisted Weak Orders of Coxeter Groups
• Oriented Matroid Structures From Realized Root Systems
• Shi arrangements restricted to Weyl cones