[Paper Review] Clusters and seeds in acyclic cluster algebras

Aslak Bakke Buan, Bethany Marsh+2 more|arXiv (Cornell University)|Oct 17, 2005

Algebraic structures and combinatorial models4 references4 citations

TL;DR

This paper proves that in acyclic cluster algebras without coefficients, a seed is uniquely determined by its cluster, confirming a conjecture by Fomin and Zelevinsky. Using a representation-theoretic approach via cluster categories and a new positivity condition, the authors establish a surjective map from cluster variables to indecomposable exceptional objects in the cluster category, and interpret the denominator of non-polynomial cluster variables as composition factors of these objects.

ABSTRACT

We show that for cluster algebras associated with finite quivers without oriented cycles (with no coefficients), a seed is determined by its cluster, as conjectured by Fomin and Zelevinsky.We also obtain an interpretation of the monomial in the denominator of a non-polynomial cluster variable in terms of the composition factors of an indecomposable exceptional module over an associated hereditary algebra.

Motivation & Objective

• To prove Fomin and Zelevinsky's conjecture that in acyclic cluster algebras without coefficients, a seed is uniquely determined by its cluster.
• To provide a representation-theoretic interpretation of the denominator of non-polynomial cluster variables in terms of composition factors of indecomposable exceptional modules over a hereditary algebra.
• To establish a surjective map from cluster variables to indecomposable exceptional objects in the cluster category, offering a new proof of the cluster-tilting correspondence in finite type.
• To extend the understanding of cluster algebra structure beyond finite type using categorical and homological tools.

Proposed method

• Introduce a new positivity condition to determine when a rational expression is in reduced form, enabling control over denominators in cluster variables.
• Construct a surjective map α from cluster variables to indecomposable exceptional objects in the cluster category C_H associated with a hereditary algebra H = kQ.
• Use the cluster-tilting graph's connectivity to show that all tilting objects arise from mutations of the initial seed, ensuring surjectivity of α and its induced maps.
• Leverage the equivalence between seeds and tilting seeds in the cluster category to translate cluster algebra problems into module-theoretic questions.
• Apply results from cluster category theory, including almost split triangles and sequences, to build cluster variables from exceptional objects.
• Use induction on mutation sequences to show that all seeds satisfy a property (⋆̃) linking clusters to tilting objects via the map α.

Experimental results

Research questions

• RQ1 Is a seed in an acyclic cluster algebra with no coefficients uniquely determined by its cluster?
• RQ2 Can the denominator of a non-polynomial cluster variable be interpreted in terms of composition factors of an indecomposable exceptional module over a hereditary algebra?
• RQ3 Does a surjective map from cluster variables to indecomposable exceptional objects in the cluster category exist, and can it be constructed using elementary positivity?
• RQ4 How does the representation-theoretic framework of cluster categories help in proving structural results about cluster algebras beyond finite type?
• RQ5 What is the relationship between the cluster-tilting graph and the mutation graph of seeds in the acyclic case?

Key findings

• The paper proves that in acyclic cluster algebras without coefficients, a seed is uniquely determined by its cluster, confirming the Fomin–Zelevinsky conjecture.
• The denominator of any non-polynomial cluster variable is shown to correspond exactly to the composition factors of an indecomposable exceptional module in the cluster category.
• A surjective map α from cluster variables to indecomposable exceptional objects in the cluster category is constructed using a new positivity condition, providing a direct link between cluster algebra elements and representation theory.
• The proof yields a new, independent proof of the bijection between cluster variables and indecomposable exceptional objects in the cluster category for finite-type cluster algebras.
• The method extends previous results from finite type to the acyclic case, offering a uniform framework for understanding cluster algebra structure.
• The authors establish that the cluster-tilting graph is connected, which ensures that all tilting objects arise from mutations of the initial seed, supporting the surjectivity of the map α.