Lattice homomorphisms between weak orders

TL;DR

This paper classifies all surjective lattice homomorphisms between weak orders of finite Coxeter groups, revealing they correspond to diagram modifications and providing insights into Cambrian lattices and fans.

Contribution

It provides a complete classification of surjective lattice homomorphisms between weak orders on finite Coxeter groups, linking diagram modifications to homomorphisms.

Findings

Surjective homomorphisms exist if and only if diagrams are obtained by deleting vertices, edges, or decreasing labels.

Homomorphisms are determined by restrictions to rank-two standard parabolic subgroups.

Classification applies to Cambrian lattices and fans, enabling new refinement constructions.

Abstract

We classify surjective lattice homomorphisms $W\to W'$ between the weak orders on finite Coxeter groups. Equivalently, we classify lattice congruences $\Theta$ on $W$ such that the quotient $W/\Theta$ is isomorphic to $W'$. Surprisingly, surjective homomorphisms exist quite generally: They exist if and only if the diagram of $W'$ is obtained from the diagram of $W$ by deleting vertices, deleting edges, and/or decreasing edge labels. A surjective homomorphism $W\to W'$ is determined by its restrictions to rank-two standard parabolic subgroups of $W$. Despite seeming natural in the setting of Coxeter groups, this determination in rank two is nontrivial. Indeed, from the combinatorial lattice theory point of view, all of these classification results should appear unlikely a priori. As an application of the classification of surjective homomorphisms between weak orders, we also obtain a classification of surjective homomorphisms between Cambrian lattices and a general construction of refinement relations between Cambrian fans.