Sortable elements in infinite Coxeter groups

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.

This is essentially the final version, which will appear in Transactions of the AMS. Minor changes have been made in response to comments by referees

References in corpus (3)

• Cambrian fans
• Sortable elements and Cambrian lattices
• Cluster characters for triangulated 2-Calabi--Yau categories

Cited by in corpus (17)

• Universal geometric cluster algebras
• Brick polytopes of spherical subword complexes and generalized associahedra
• Many non-equivalent realizations of the associahedron
• Subword complexes, cluster complexes, and generalized multi-associahedra
• A Hopf algebra of subword complexes
• Combinatorial frameworks for cluster algebras
• Noncrossing partitions and the shard intersection order
• Sortable Elements for Quivers with Cycles
• A combinatorial approach to scattering diagrams
• Noncrossing Arc Diagrams, Tamari Lattices, and Parabolic Quotients of the Symmetric Group
• Trimness of Closed Intervals in Cambrian Semilattices
• An affine almost positive roots model
• Noncrossing partitions, clusters and the Coxeter plane
• Parabolic Tamari Lattices in Linear Type B
• Coxeter-biCatalan combinatorics
• Dominance phenomena: mutation, scattering and cluster algebras
• Torsion pairs for quivers and the Weyl groups