Cluster algebras IV: Coefficients

TL;DR

This paper explores how coefficients influence cluster algebras, providing formulas, graph relationships, parametrizations, and generalizations of Y-systems, with implications for duality conjectures and universal coefficient choices.

Contribution

It introduces general formulas for cluster variables, analyzes exchange graph coverings, investigates parametrizations linked to duality conjectures, and generalizes Y-systems with proven Laurent phenomena and periodicity.

Findings

Exchange graph of principal coefficients covers all with same exchange matrix

Established Laurent phenomenon for generalized Y-systems

Identified universal coefficients for finite type cluster algebras

Abstract

We study the dependence of a cluster algebra on the choice of coefficients. We write general formulas expressing the cluster variables in any cluster algebra in terms of the initial data; these formulas involve a family of polynomials associated with a particular choice of "principal" coefficients. We show that the exchange graph of a cluster algebra with principal coefficients covers the exchange graph of any cluster algebra with the same exchange matrix. We investigate two families of parametrizations of cluster monomials by lattice points, determined, respectively, by the denominators of their Laurent expansions and by certain multi-gradings in cluster algebras with principal coefficients. The properties of these parametrizations, some proven and some conjectural, suggest links to duality conjectures of V.Fock and A.Goncharov [math.AG/0311245]. The coefficient dynamics leads to a natural generalization of Al.Zamolodchikov's Y-systems. We establish a Laurent phenomenon for such Y-systems, previously known in finite type only, and sharpen the periodicity result from [hep-th/0111053]. For cluster algebras of finite type, we identify a canonical "universal" choice of coefficients such that an arbitrary cluster algebra can be obtained from the universal one (of the same type) by an appropriate specialization of coefficients.