Cambrian fans

Abstract

For a finite Coxeter group W and a Coxeter element c of W, the c-Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of W. Its maximal cones are naturally indexed by the c-sortable elements of W. The main result of this paper is that the known bijection cl_c between c-sortable elements and c-clusters induces a combinatorial isomorphism of fans. In particular, the c-Cambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for W. The rays of the c-Cambrian fan are generated by certain vectors in the W-orbit of the fundamental weights, while the rays of the c-cluster fan are generated by certain roots. For particular ("bipartite") choices of c, we show that the c-Cambrian fan is linearly isomorphic to the c-cluster fan. We characterize, in terms of the combinatorics of clusters, the partial order induced, via the map cl_c, on c-clusters by the c-Cambrian lattice. We give a simple bijection from c-clusters to c-noncrossing partitions that respects the refined (Narayana) enumeration. We relate the Cambrian fan to well known objects in the theory of cluster algebras, providing a geometric context for g-vectors and quasi-Cartan companions.

Substantial revisions, mostly of an expository nature, in response to suggestions of the referees. This is the final version which will appear in the Journal of the European Mathematical Society (JEMS). 38 pages, 7 figures

References in corpus (1)

• Sortable elements and Cambrian lattices

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• Cambrian Hopf Algebras
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• Subword complexes, cluster complexes, and generalized multi-associahedra
• A Hopf algebra of subword complexes
• Combinatorial frameworks for cluster 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
• Geometric realizations of the accordion complex of a dissection
• Polyhedral models for generalized associahedra via Coxeter elements
• The facial weak order and its lattice quotients
• Associahedra for finite type cluster algebras and minimal relations between $\mathbf{g}$-vectors
• Compatibility fans for graphical nested complexes
• The diameter of type D associahedra and the non-leaving-face property
• Isometry classes of generalized associahedra
• Vertex barycenter of generalized associahedra
• Cambrian combinatorics on quiver representations (type A)
• Celebrating Loday's Associahedron
• Minkowski Decomposition of Associahedra and Related Combinatorics
• Sortable Elements for Quivers with Cycles
• Permutahedra and Associahedra: Generalized associahedra from the geometry of finite reflection groups
• Enumerative properties of Grid-Associahedra
• Shard polytopes
• Signed tree associahedra
• Stokes posets and serpent nests
• Cambrian frameworks for cluster algebras of affine type
• An affine almost positive roots model
• Cluster algebras of finite type via Coxeter elements and principal minors
• Ungarian Markov Chains
• Permutahedra and generalized associahedra
• Hochschild polytopes
• The weak order on Weyl posets
• Coxeter-biCatalan combinatorics
• Dominance phenomena: mutation, scattering and cluster algebras
• Lattice homomorphisms between weak orders
• A Cambrian framework for the oriented cycle
• A note on gamma triangles and local gamma vectors
• One-skeleton posets of Bruhat interval polytopes
• Torsion pairs for quivers and the Weyl groups
• Wigglyhedra
• Separating trees and simple congruences of the weak order
• $c$-Birkhoff polytopes
• Brick manifolds and toric varieties of brick polytopes