Affine extended weak order is a lattice

TL;DR

The paper proves Dyer's conjecture that the extended weak order is a complete lattice for affine Coxeter groups by introducing clean arrangements and a framework of suitable orderings that reduce the problem to rank-3 subsystems. It shows that root poset order ideals in finite or rank-3 untwisted affine root systems are clean, and develops folding techniques to extend the results to all crystallographic finite and affine types. It also analyzes difficulties in twisted affine and non-crystallographic cases, including explicit counterexamples, and outlines a roadmap toward a general verification via suitable orders. Overall, the work provides a structural toolkit for establishing lattice properties of biclosed sets and sets the stage for extending Dyer’s conjectures more broadly.

Abstract

Coxeter groups are equipped with a partial order known as the weak order, such that $u \leq v$ if the inversions of $u$ are a subset of the inversions of $v$. In finite Coxeter groups, weak order is a complete lattice, but in infinite Coxeter groups it is only a meet semi-lattice. Motivated by questions in Kazhdan-Lusztig theory, Matthew Dyer introduced a larger poset, now known as extended weak order, which contains the weak order as an order ideal and coincides with it for finite Coxeter groups. The extended weak order is the containment order on certain sets of positive roots: those which satisfy a geometric condition making them "biclosed". The finite biclosed sets are precisely the inversion sets of Coxeter group elements. Generalizing the result for finite Coxeter groups, Dyer conjectured that the extended weak order is always a complete lattice, even for infinite Coxeter groups. In this paper, we prove Dyer's conjecture for Coxeter groups of affine type. To do so, we introduce the notion of a clean arrangement, which is a hyperplane arrangement where the regions are in bijection with biclosed sets. We show that root poset order ideals in a finite or rank 3 untwisted affine root system are clean. We set up a general framework for reducing Dyer's conjecture to checking cleanliness of certain subarrangements. We conjecture this framework can be used to prove Dyer's conjecture for all Coxeter groups.

Figures (12)

• Figure 1: Implications between different notions of convexity are shown with solid arrows. If $X$ is clean, then the dashed implications also hold.
• Figure 2: The Dynkin diagrams associated to finite irreducible root systems.
• Figure 3: The extended Dynkin diagrams associated to untwisted affine root systems.
• Figure 4: The $A_3$ hyperplane arrangement.
• Figure 5: Case 1 in the proof of Lemma \ref{['lem:KeyCleanLemmaRephrase']} for $A_3$.

Theorems & Definitions (65)

• Conjecture 1.1: Dyer2019Dyer1994
• Conjecture 1.2: Conjecture A Dyer2019Dyerpreprint
• Conjecture 1.3
• Conjecture 1.4
• Remark 2.1
• Remark 2.2
• Lemma 2.3
• proof
• Remark 2.4
• Theorem 3.1