Extended weak order for the affine symmetric group

TL;DR

This paper explores the structure of the extended weak order on affine symmetric groups, revealing its algebraic properties, canonical representations, and conjecturing its generalization to all Coxeter groups.

Contribution

It introduces a combinatorial framework for the extended weak order, proves its algebraic semidistributivity, and establishes canonical join representations using non-crossing arc diagrams.

Findings

$L_n$ is an algebraic completely semidistributive lattice

Canonical join representations are described via cyclic non-crossing arc diagrams

Profinite lattices are join semidistributive if their compact elements have canonical join representations

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.