A ug 2 00 6 CLUSTER ALGEBRAS IV : COEFFICIENTS
نویسنده
چکیده
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. 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 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.
منابع مشابه
A ug 2 00 6 CLUSTER ALGEBRAS AND TRIANGULATED SURFACES PART I : CLUSTER COMPLEXES
We establish basic properties of cluster algebras associated with oriented bordered surfaces with marked points. In particular, we show that the underlying cluster complex of such a cluster algebra does not depend on the choice of coefficients, describe this complex explicitly in terms of “tagged triangulations” of the surface, and determine its homotopy type and its growth rate.
متن کامل1 5 A ug 2 00 6 CLUSTER ALGEBRAS AND TRIANGULATED SURFACES PART I : CLUSTER COMPLEXES
We establish basic properties of cluster algebras associated with oriented bordered surfaces with marked points. In particular, we show that the underlying cluster complex of such a cluster algebra does not depend on the choice of coefficients, describe this complex explicitly in terms of “tagged triangulations” of the surface, and determine its homotopy type and its growth rate.
متن کامل0 A ug 2 00 7 CLUSTER ALGEBRAS AND TRIANGULATED SURFACES PART I : CLUSTER COMPLEXES
We establish basic properties of cluster algebras associated with oriented bordered surfaces with marked points. In particular, we show that the underlying cluster complex of such a cluster algebra does not depend on the choice of coefficients, describe this complex explicitly in terms of “tagged triangulations” of the surface, and determine its homotopy type and its growth rate.
متن کاملar X iv : m at h / 06 08 37 7 v 1 [ m at h . R T ] 1 5 A ug 2 00 6 ALGEBRAS OF DERIVED DIMENSION ZERO
متن کامل
ar X iv : m at h / 02 08 03 3 v 1 [ m at h . Q A ] 5 A ug 2 00 2 CLUSTER ALGEBRAS AND POISSON GEOMETRY
We introduce a Poisson variety compatible with a cluster algebra structure and a compatible toric action on this variety. We study Poisson and topological properties of the union of generic orbits of this toric action. In particular, we compute the number of connected components of the union of generic toric orbits for cluster algebras over real numbers. As a corollary we compute the number of ...
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تاریخ انتشار 2008