Cambrian Lattices

Abstract

For an arbitrary finite Coxeter group W we define the family of Cambrian lattices for W as quotients of the weak order on W with respect to certain lattice congruences. We associate to each Cambrian lattice a complete fan, which we conjecture is the normal fan of a polytope combinatorially isomorphic to the generalized associahedron for W. In types A and B we obtain, by means of a fiber-polytope construction, combinatorial realizations of the Cambrian lattices in terms of triangulations and in terms of permutations. Using this combinatorial information, we prove in types A and B that the Cambrian fans are combinatorially isomorphic to the normal fans of the generalized associahedra and that one of the Cambrian fans is linearly isomorphic to Fomin and Zelevinsky's construction of the normal fan as a "cluster fan." Our construction does not require a crystallographic Coxeter group and therefore suggests a definition, at least on the level of cellular spheres, of a generalized associahedron for any finite Coxeter group. The Tamari lattice is one of the Cambrian lattices of type A, and two "Tamari" lattices in type B are identified and characterized in terms of signed pattern avoidance. We also show that open intervals in Cambrian lattices are either contractible or homotopy equivalent to spheres.

Revisions in exposition (partly in response to the suggestions of an anonymous referee) including many new figures. Also, Conjecture 1.4 and Theorem 1.5 are replaced by slightly more detailed statements. To appear in Adv. Math. 37 pages, 8 figures

References in corpus (3)

• Cluster algebras II: Finite type classification
• Lattice congruences, fans and Hopf algebras
• Lattice congruences of the weak order

Cited by in corpus (56)

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• Lattice congruences of the weak order
• Universal geometric cluster algebras
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• The Hopf algebra of diagonal rectangulations
• Cambrian Hopf Algebras
• 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
• Lattice structure of torsion classes for path algebras
• Associahedra via spines
• Non-kissing complexes and tau-tilting for gentle algebras
• Noncrossing partitions and the shard intersection order
• Polytopal realizations of finite type $\mathbf{g}$-vector fans
• Brick polytopes, lattice quotients, and Hopf algebras
• Quotientopes
• Geometric realizations of the accordion complex of a dissection
• Geometric combinatorial algebras: cyclohedron and simplex
• Polyhedral models for generalized associahedra via Coxeter elements
• Lattice Properties of Oriented Exchange Graphs and Torsion Classes
• Compatibility fans for graphical nested complexes
• The diameter of type D associahedra and the non-leaving-face property
• Cambrian combinatorics on quiver representations (type A)
• Celebrating Loday's Associahedron
• Crosscut-simplicial Lattices
• Hopf dreams and diagonal harmonics
• On Maximal Green Sequences For Type A Quivers
• Hopf algebras on decorated noncrossing arc diagrams
• The Pop-stack-sorting Operator on Tamari Lattices
• Shard polytopes
• Minimal length maximal green sequences
• Acyclic reorientation lattices and their lattice quotients
• Noncrossing Arc Diagrams, Tamari Lattices, and Parabolic Quotients of the Symmetric Group
• Trimness of Closed Intervals in Cambrian Semilattices
• Fan realizations for some 2-associahedra
• Ungarian Markov Chains
• Noncrossing arc diagrams and canonical join representations
• The canonical complex of the weak order
• Hochschild polytopes
• The weak order on Weyl posets
• Coxeter-biCatalan combinatorics
• The $s$-weak order and $s$-permutahedra I: combinatorics and lattice structure
• Lattices of acyclic pipe dreams
• Geometric realizations of the $s$-weak order and its lattice quotients
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• Dominance phenomena: mutation, scattering and cluster algebras
• Separating trees and simple congruences of the weak order
• The $m$-Cover Posets and Their Applications
• Permutree sorting
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• Rowmotion Markov Chains
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• $c$-Birkhoff polytopes