[Paper Review] Generic Caldero-Chapoton functions with coefficients and applications to surface cluster algebras

Christof Geiß, Daniel Labardini-Fragoso+1 more|arXiv (Cornell University)|Jul 10, 2020

Algebraic structures and combinatorial models35 references4 citations

TL;DR

This paper establishes a geometric framework for generic Caldero-Chapoton functions with coefficients in cluster algebras arising from surfaces, using regular maps and irreducible components of representation varieties. It proves that for any surface with at least two marked boundary points, the generic Caldero-Chapoton functions form a basis over the Laurent polynomial ring of coefficients, even when coefficients are not of full rank, generalizing earlier results via elementary algebraic geometry without requiring Jacobi-finiteness.

ABSTRACT

We realize Derksen-Weyman-Zelevinsky's mutations of representations as densely-defined regular maps on representation spaces, and study the generic values of Caldero-Chapoton functions with coefficients, giving, for instance, a sufficient combinatorial condition for their linear independence. For a quiver with potential $(Q,S)$, we show that if $k$ is a vertex not incident to any oriented 2-cycle, then every generically $τ$-reduced irreducible component $Z$ of any affine variety of (decorated) representations has a dense open subset $U$ on which Derksen-Weyman-Zelevinsky's mutation of representations $μ_k$ can be defined consistently as a regular map to an affine variety of (decorated) representations of the Jacobian algebra of the mutated QP $μ_k(Q,S)$. Our techniques involve only basic linear algebra and elementary algebraic geometry, and do not require to assume Jacobi-finiteness. Thus, the paper yields a new and more general proof of the mutation invariance of generic Caldero-Chapoton functions, generalizing and providing a new natural geometric perspective on results of Derksen-Weyman-Zelevinsky and Plamondon. (For Jacobi-finite non-degenerate quivers with potential, this invariance was shown by Plamondon using the machinery of Ginzburg dg-algebras and Hom-finite generalized cluster categories.) We apply our results, together with results of Mills, Muller and Qin, to prove that for any choice of geometric coefficient systems, not necessarily of full rank, the cluster algebra associated to a possibly punctured surface with at least two marked points on the boundary has the generic Caldero-Chapoton functions as a basis over the Laurent polynomial ring of coefficients.

Motivation & Objective

• To provide a geometric, mutation-invariant framework for generic Caldero-Chapoton functions with coefficients in cluster algebras.
• To generalize Plamondon’s mutation invariance result beyond Jacobi-finite settings using only basic algebraic geometry.
• To establish that generic Caldero-Chapoton functions form a basis for surface cluster algebras with arbitrary geometric coefficients, even when not of full rank.
• To prove that the upper cluster algebra coincides with the cluster algebra for surface quivers with at least two boundary marked points.
• To unify results from Mills, Muller, and Qin with a new geometric perspective on cluster basis constructions.

Proposed method

• Derksen-Weyman-Zelevinsky’s representation mutations are realized as densely-defined regular maps on representation varieties of quivers with potential.
• Upper semicontinuity of g-vectors, E-invariants, and Hom-dimensions is established for arbitrary associative algebras.
• Generically τ-reduced irreducible components are shown to admit dense open subsets where mutation is consistently defined as a regular map to the mutated Jacobian algebra.
• The Zariski closure of mutated orbits is proven to yield generically τ-reduced components in the mutated setting, establishing a mutation rule at the level of components.
• The construction is applied to triangulated surfaces, using results from Mills, Muller, and Qin to prove basis properties over arbitrary coefficient systems.
• The method avoids Ginzburg dg-algebras and Hom-finiteness assumptions, relying only on linear algebra and elementary algebraic geometry of minors.

Experimental results

Research questions

• RQ1 Can the mutation invariance of generic Caldero-Chapoton functions be proven without assuming Jacobi-finiteness or using advanced homological tools?
• RQ2 Under what combinatorial conditions on the exchange matrix is the set of generic Caldero-Chapoton functions linearly independent over the coefficient ring?
• RQ3 Does the generic Caldero-Chapoton basis coincide with the upper cluster algebra for surface cluster algebras with non-full-rank coefficient systems?
• RQ4 Can the mutation of irreducible components of representation varieties be consistently defined via regular maps in the absence of Jacobi-finiteness?
• RQ5 Is the generic basis of a surface cluster algebra independent of the choice of geometric coefficient system, including non-full-rank ones?

Key findings

• The paper proves that for any surface with at least two marked points on the boundary, the generic Caldero-Chapoton functions form a basis of the upper cluster algebra over the Laurent polynomial ring of coefficients.
• The mutation of representations is shown to define a densely-defined regular map on a dense open subset of the representation variety, even without Jacobi-finiteness.
• The Zariski closure of the image of a generic component under mutation yields a generically τ-reduced component in the mutated setting, establishing a mutation rule at the level of components.
• The set of generic Caldero-Chapoton functions spans the cluster algebra for any geometric coefficient system, including non-full-rank ones, for surface quivers with at least two boundary marked points.
• The generic basis coincides with the bangle basis of Musiker-Schiffler-Williams in the coefficient-free and principal coefficient cases for unpunctured surfaces with at least two marked points.
• The result generalizes Plamondon’s invariance result by avoiding Ginzburg dg-algebras and Hom-finiteness, using only elementary algebraic geometry.