Cluster algebras IV: Coefficients

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 parameterizations 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 parameterizations, some proven and some conjectural, suggest links to duality conjectures of Fock and Goncharov. The coefficient dynamics leads to a natural generalization of 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 an earlier paper. 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.

Some integrable maps and their Hirota bilinear forms

“…Due to the homogeneity of (10), the degree growth of the iterates can be determined from that of the denominators. If we further assume that there are no cancellations between numerators and denominators on the right-hand side, then (as is well known in the context of cluster algebras [6,7]), the denominator vector d n satisfies the max-plus tropical (or ultradiscrete) analogue of (10), which takes the form…”

“…There is another cluster algebra that arises, namely the one associated with the bilinear recurrence (6). The variables v n defined by (26) provide symplectic coordinates for the corresponding U-system, which is the map in dimension N given by…”

“…We also present a tropical (ultradiscrete) version of the multilinear equation, as well as a corresponding tropical version of (2), and show how this can be used to obtain an explicit formula for degree growth (quadratic in n). For any fixed N, it is then possible to derive the bilinear equation (5), as well as (6) or (7), either numerically (with specific initial data) or symbolically (working with rational functions of initial data). The complete proof of theorem 1.1, for arbitrary N, is reserved until section 3, where we begin by deriving (5) via a modified version of (2) (see equation (22) below), before treating the rest of the result and its detailed consequences for even/odd N separately.…”

Laminations from the symplectic double

“…6, we will see how to associate, to any element of C or C • , a polynomial F c . For curves such as the ones below, which come from an intersecting curve, these F c are in fact the F-polynomials of Fomin and Zelevinsky [10]. We will prove the following theorem, which expresses the function I D (l) in terms of these polynomials.…”

“…5, we review some results on cluster algebras and F-polynomials that play a role in our discussion of the canonical pairing. The references for this section are [7,10,12]. In addition, we introduce a particular cluster algebra that we call the cluster D-algebra. Finally, in Sect.…”