Extended weak order for the affine symmetric group

TL;DR

This paper analyzes the extended weak order for the affine symmetric group by connecting it to translation-invariant and total orders via two key lattices, $\\WO(\\mathrm{Tot})$ and $\\WO(\\mathrm{TTot}_n)$. It establishes that these lattices, along with $\\Dyer(\\widetilde{S}_n)$, are profinite semidistributive lattices, and develops canonical join representations through (cyclic) arc diagrams, including a detailed treatment of infinitely many cases such as $S_{\\infty}$ and TITOs. A central theme is the correspondence between join-irreducibles and arc-based combinatorics, with wide-generated elements playing a pivotal role in canonical representations. The profinite viewpoint yields algebraicity and facilitates transfer of finite-lattice methods to the infinite setting, enabling a unifying treatment of extended weak order across finite, affine, and infinite symmetric groups. The paper also conjectures that the extended weak order is profinite semidistributive for any Coxeter group, suggesting broad structural uniformity in these posets.

Abstract

The extended weak order on a Coxeter group $W$ is the poset of biclosed sets in its root system. In (Barkley-Speyer 2024), it was shown that when $W=\widetilde{S}_n$ is the affine symmetric group, then the extended weak order is a quotient of the lattice $L_n$ of translation-invariant total orderings of the integers. In this article, we give a combinatorial introduction to $L_n$ and the extended weak order on $\widetilde{S}_n$. We show that $L_n$ is an algebraic completely semidistributive lattice. We describe its canonical join representations using a cyclic version of Reading's non-crossing arc diagrams. We also show analogous statements for the lattice of all total orders of the integers, which is the extended weak order on the symmetric group $S_\infty$. A key property of both of these lattices is that they are profinite; we also prove that a profinite lattice is join semidistributive if and only if its compact elements have canonical join representations. We conjecture that the extended weak order of any Coxeter group is a profinite semidistributive lattice.

Figures (9)

• Figure 1: The Hasse diagram of weak order on $S_3$.
• Figure 2: The arc diagrams for arcs in $S_4$ with initial value $1$ and terminal value $4$. Below each diagram, we have indicated the associated JI in $S_4$.
• Figure 3: The intersection of $\mathcal{B}_3$ with a two-dimensional subspace of $\mathbb{R}^3$. Two points are in the same region of $\mathcal{B}_3$ if their coordinates are in the same order.
• Figure 4: The four shards of $\mathcal{B}_3$ are the two hyperplanes $H_{12},H_{23}$ and the two halves of $H_{13}$. The regions are labeled by the lower arc diagram of their associated permutation.
• Figure 5: The Hasse diagram of weak order on $\widetilde{S}_2$. Each affine permutation is labeled by its window notation.

Theorems & Definitions (98)

• theorem 2
• definition 2
• definition 3
• proposition 2
• definition 4
• proposition 3
• proposition 4
• definition 5
• proposition 5
• definition 6