Cluster algebras IV: Coefficients
Sergey Fomin, Andrei Zelevinsky
TL;DR
This paper analyzes how the coefficient data influence cluster algebras, introducing principal coefficients as a universal reference and deriving separation formulas that express any cluster variable via $F$-polynomials and initial data. It shows that the exchange graph with principal coefficients covers all graphs sharing the same exchange matrix and develops two parameterizations of cluster monomials through denominator vectors and $\mathbf{g}$-vectors, closely tied to generalized $Y$-systems. In finite type, the authors identify a universal coefficient framework, prove linear independence of cluster monomials, and connect $F$-polynomials to Fibonacci polynomials, culminating in the universal coefficient construction for finite types. Overall, the work illuminates coefficient dynamics, connects to dualities with Fock–Goncharov theory, and establishes a rich set of conjectures guiding future exploration of canonical bases and Langlands-type dualities in 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.