A generalization in affine type A of Coxeter sortable elements and Reading's bijection with noncrossing partitions

TL;DR

This paper extends the concept of Coxeter sortable elements and their bijection with noncrossing partitions from finite to affine symmetric groups, using biclosed sets and pattern avoidance criteria.

Contribution

It introduces a new characterization of affine c-sortable elements via pattern avoidance and establishes a bijection with affine c-noncrossing partitions through combinatorial objects.

Findings

Characterization of affine c-sortable elements by pattern avoidance.

Definition of c-sortable biclosed sets using TITOs on ℤ.

Establishment of a bijection with affine c-noncrossing partitions.

Abstract

This paper generalizes in the affine symmetric group the notion of Coxeter sortable (or c-sortable for short) elements, as well as the classical bijection between c-sortable elements and c-noncrossing partitions defined by Reading in finite Coxeter groups. The generalization to the affine symmetric group of the c-sortable elements is achieved by using biclosed sets of reflections. Using recent works from Barkley and Speyer, these biclosed sets admit a sort of "one-line notation" called a TITO on $\mathbb{Z}$ (translation-invariant total order on $\mathbb{Z}$) that coincides with the usual one-line notation in the case of an affine permutation. We characterize the c-sortable elements of the affine symmetric group by pattern avoidance on their one-line notation, mirroring the well-known characterizations of c-sortable elements in the classical finite types. Based on this criterion, we then define the c-sortable biclosed sets, generalizing the c-sortable elements, as biclosed sets such that their TITOs on $\mathbb{Z}$ avoid certain patterns. We also build a bijection from our set of c-sortable biclosed sets to the set of c-noncrossing partitions using various combinatorial objects and their one-to-one correspondences. First, the TITOs, in bijection with the biclosed sets of the affine symmetric group using results from Barkley and Speyer. Second, the c-noncrossing partitions of an annulus, in bijection with the c-noncrossing partitions of the affine symmetric group, using results from Digne and Reading. Finally, the cyclic noncrossing arc diagrams, defined by Barkley, for which we exhibit a subset in bijection with both the set of c-sortable biclosed sets and the set of c-noncrossing partitions.