Universal geometric cluster algebras

Abstract

We consider, for each exchange matrix B, a category of geometric cluster algebras over B and coefficient specializations between the cluster algebras. The category also depends on an underlying ring R, usually the integers, rationals, or reals. We broaden the definition of geometric cluster algebras slightly over the usual definition and adjust the definition of coefficient specializations accordingly. If the broader category admits a universal object, the universal object is called the cluster algebra over B with universal geometric coefficients, or the universal geometric cluster algebra over B. Constructing universal coefficients is equivalent to finding an R-basis for B (a "mutation-linear" analog of the usual linear-algebraic notion of a basis). Polyhedral geometry plays a key role, through the mutation fan F_B, which we suspect to be an important object beyond its role in constructing universal geometric coefficients. We make the connection between F_B and g-vectors. We construct universal geometric coefficients in rank 2 and in finite type and discuss the construction in affine type.

49 pages, 5 figures. Version 6: Small post-publication changes, including fixing a typo

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• Cambrian Lattices
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• Universal geometric cluster algebras from surfaces

Cited by in corpus (29)

• The greedy basis equals the theta basis
• Cluster Configuration Spaces of Finite Type
• Polytopal realizations of finite type $\mathbf{g}$-vector fans
• Compatibility fans for graphical nested complexes
• Universal geometric coefficients for the once-punctured torus
• Linearity of stability conditions
• Density of $g$-vector cones from triangulated surfaces
• Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras
• Cluster automorphism groups of cluster algebras of finite type
• Scattering fans
• A combinatorial approach to scattering diagrams
• ABHY Associahedra and Newton polytopes of $F$-polynomials for finite type cluster algebras
• Cluster automorphism groups of cluster algebras with coefficients
• $F$-matrices of cluster algebras from triangulated surfaces
• Cambrian frameworks for cluster algebras of affine type
• Universal geometric cluster algebras from surfaces
• An affine almost positive roots model
• The No Gap Conjecture for tame hereditary algebras
• Cluster Algebras and Scattering Diagrams, Part II. Cluster Patterns and Scattering Diagrams
• Acyclic cluster algebras with dense $g$-vector fans
• On Jacobian algebras associated with the once-punctured torus
• Universal geometric coefficients for the four-punctured sphere
• Discrete Dynamical Systems From Real Valued Mutation
• Deformation Theory for Finite Cluster Complexes
• Dominance phenomena: mutation, scattering and cluster algebras
• A Cambrian framework for the oriented cycle
• Infinitesimal semi-invariant pictures and co-amalgamation
• Denseness of $g$-vector cones from weighted orbifolds
• New Combinatorial Formulas for Cluster Monomials of Type A Quivers