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