Combinatorial frameworks for cluster algebras

Abstract

We develop a general approach to finding combinatorial models for cluster algebras. The approach is to construct a labeled graph called a framework. When a framework is constructed with certain properties, the result is a model incorporating information about exchange matrices, principal coefficients, g-vectors, and g-vector fans. The idea behind frameworks arises from Cambrian combinatorics and sortable elements, and in this paper, we use sortable elements to construct a framework for any cluster algebra with an acyclic initial exchange matrix. This Cambrian framework yields a model of the entire exchange graph when the cluster algebra is of finite type. Outside of finite type, the Cambrian framework models only part of the exchange graph. In a forthcoming paper, we extend the Cambrian construction to produce a complete framework for a cluster algebra whose associated Cartan matrix is of affine type.

50 pages, 2 figures. v4: Final pre-publication version + one post-publication typo-fix

References in corpus (11)

• Cluster algebras II: Finite type classification
• Clusters, Coxeter-sortable elements and noncrossing partitions
• Cambrian Lattices
• Cambrian fans
• Sortable elements and Cambrian lattices
• Lattice congruences, fans and Hopf algebras
• Cluster algebras via cluster categories with infinite-dimensional morphism spaces
• Cluster algebras of finite type and positive symmetrizable matrices
• Donaldson-Thomas theory and cluster algebras
• Sortable elements in infinite Coxeter groups
• Mutations of group species with potentials and their representations. Applications to cluster algebras

Cited by in corpus (13)

• The greedy basis equals the theta basis
• A conjecture on $C$-matrices of cluster algebras
• A combinatorial approach to scattering diagrams
• Cambrian frameworks for cluster algebras of affine type
• Some Consequences of Categorification
• Ungarian Markov Chains
• An affine almost positive roots model
• Cluster Algebras and Scattering Diagrams, Part II. Cluster Patterns and Scattering Diagrams
• Affine cluster monomials are generalized minors
• Dominance phenomena: mutation, scattering and cluster algebras
• A Cambrian framework for the oriented cycle
• Lattice homomorphisms between weak orders
• Torsion pairs for quivers and the Weyl groups