Sortable elements in infinite Coxeter groups

TL;DR

This paper extends the theory of sortable elements from finite Coxeter groups to general Coxeter groups, providing uniform proofs, new results, and refined descriptions of Cambrian fans and related structures.

Contribution

It generalizes finite-type results to infinite Coxeter groups using uniform arguments and introduces new tools like a skew-symmetric form and the projection \\pidown^c.

Findings

Unified proofs for properties of sortable elements in all Coxeter groups

Description of fibers of the projection \\pidown^c and their relation to Cambrian fans

Transformation rules of sortable elements under reflection functors

Abstract

In a series of previous papers, we studied sortable elements in finite Coxeter groups, and the related Cambrian fans. We applied sortable elements and Cambrian fans to the study of cluster algebras of finite type and the noncrossing partitions associated to Artin groups of finite type. In this paper, as the first step towards expanding these applications beyond finite type, we study sortable elements in a general Coxeter group W. We supply uniform arguments which transform all previous finite-type proofs into uniform proofs (rather than type by type proofs), generalize many of the finite-type results and prove new and more refined results. The key tools in our proofs include a skew-symmetric form related to (a generalization of) the Euler form of quiver theory and the projection \pidown^c mapping each element of W to the unique maximal c-sortable element below it in the weak order. The fibers of \pidown^c essentially define the c-Cambrian fan. The most fundamental results are, first, a precise statement of how sortable elements transform under (BGP) reflection functors and second, a precise description of the fibers of \pidown^c. These fundamental results and others lead to further results on the lattice theory and geometry of Cambrian (semi)lattices and Cambrian fans.