Clusters, Coxeter-sortable elements and noncrossing partitions

Abstract

We introduce Coxeter-sortable elements of a Coxeter group W. For finite W, we give bijective proofs that Coxeter-sortable elements are equinumerous with clusters and with noncrossing partitions. We characterize Coxeter-sortable elements in terms of their inversion sets and, in the classical cases, in terms of permutations.

Minor changes in exposition, including: More precise statement in Remark 6.8; Added Remark 6.9, an observation which is helpful in the sequel (math.CO/0512339); Updated textual references to the sequel and to a paper in preparation (with D. Speyer). 28 pages, 8 figures

References in corpus (7)

• Cluster algebras II: Finite type classification
• Cambrian Lattices
• Cambrian fans
• Sortable elements and Cambrian lattices
• Lattice congruences, fans and Hopf algebras
• Lattice congruences of the weak order
• Lattices in finite real reflection groups

Cited by in corpus (64)

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• Cambrian fans
• Sortable elements and Cambrian lattices
• Universal geometric cluster algebras
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• 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
• Denominator vectors and compatibility degrees in cluster algebras of finite type
• Noncrossing partitions and the shard intersection order
• Chains in the noncrossing partition lattice
• The facial weak order and its lattice quotients
• Rowmotion in slow motion
• The Jordan-Hölder property and Grothendieck monoids of exact categories
• C-sortable words as green mutation sequences
• Isometry classes of generalized associahedra
• Quotient closed subcategories of quiver representations
• Sortable Elements for Quivers with Cycles
• The Core Label Order of a Congruence-Uniform Lattice
• Permutahedra and Associahedra: Generalized associahedra from the geometry of finite reflection groups
• The Shard Intersection Order on Permutations
• On cycle decompositions in Coxeter groups
• Cluster algebras of type D: pseudotriangulations approach
• Consequences of the Lakshmibai-Sandhya Theorem: the ubiquity of permutation patterns in Schubert calculus and related geometry
• Noncrossing Arc Diagrams, Tamari Lattices, and Parabolic Quotients of the Symmetric Group
• Admissible sequences, preprojective modules, and reduced words in the Weyl group of a quiver
• Cambrian frameworks for cluster algebras of affine type
• Preprojective representations of valued quivers and reduced words in the Weyl group of a Kac-Moody algebra
• W-Associahedra are In-Your-Face
• Tilting theory of preprojective algebras and $c$-sortable elements
• Noncrossing partitions, clusters and the Coxeter plane
• Ungarian Markov Chains
• Permutahedra and generalized associahedra
• An affine almost positive roots model
• The local $h$-vector of the cluster subdivision of a simplex
• Classifying τ-tilting modules over preprojective algebras of Dynkin type
• Hom-configurations and noncrossing partitions
• The weak order on Weyl posets
• Koszulity of dual braid monoid algebras via cluster complexes
• The sorting order on a Coxeter group
• On inversion sets and the weak order in Coxeter groups
• Two-term tilting complexes and simple-minded systems of self-injective Nakayama algebras
• Affine cluster monomials are generalized minors
• Dominance phenomena: mutation, scattering and cluster algebras
• Bender--Knuth Billiards in Coxeter Groups
• Refined enumeration of noncrossing chains and hook formulas
• A Cambrian framework for the oriented cycle
• Coxeter-biCatalan combinatorics
• Parabolic Tamari Lattices in Linear Type B
• Lattice homomorphisms between weak orders
• SB-Labelings, Distributivity, and Bruhat Order on Sortable Elements
• The greedy flip tree of a subword complex
• The extended permutohedron on a transitive binary relation
• The $m$-Cover Posets and Their Applications
• Facets of the $m$-generalized cluster complex and regions in the $m$-extended Catalan arrangement of type $A_n$
• Torsion pairs for quivers and the Weyl groups
• Permutree sorting
• Sublattices of associahedra and permutohedra
• Bubble Lattices I: Structure
• Rowmotion Markov Chains
• The excedance quotient of the Bruhat order, Quasisymmetric Varieties and Temperley-Lieb algebras
• $c$-Birkhoff polytopes