Universal geometric cluster algebras from surfaces
نویسندگان
چکیده
منابع مشابه
Positivity for cluster algebras from surfaces
We give combinatorial formulas for the Laurent expansion of any cluster variable in any cluster algebra coming from a triangulated surface (with or without punctures), with respect to an arbitrary seed. Moreover, we work in the generality of principal coefficients. An immediate corollary of our formulas is a proof of the positivity conjecture of Fomin and Zelevinsky for cluster algebras from su...
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We construct two bases for each cluster algebra coming from a triangulated surface without punctures. We work in the context of a coefficient system coming from a full-rank exchange matrix, for example, principal coefficients.
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These are notes from introductory survey lectures given at the Institute for Studies in Theoretical Physics and Mathematics (IPM), Teheran, in 2008 and 2010. We present the definition and the fundamental properties of Fomin-Zelevinsky’s cluster algebras. Then, we introduce quiver representations and show how they can be used to construct cluster variables, which are the canonical generator...
متن کاملOn Cluster Algebras Arising from Unpunctured Surfaces Ii
We study cluster algebras with principal and arbitrary coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of certain paths on a triangulation of the surface. As an immediate consequence, we prove the positivity conjecture of Fomin and Zelevinsky for these cluster a...
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This paper concerns cluster algebras with principal coefficients A•(S,M) associated to bordered surfaces (S,M), and is a companion to a concurrent work of the authors with Schiffler [MSW2]. Given any (generalized) arc or loop in the surface – with or without self-intersections – we associate an element of (the fraction field of) A•(S,M), using products of elements of PSL2(R). We give a direct p...
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ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 2014
ISSN: 0002-9947,1088-6850
DOI: 10.1090/s0002-9947-2014-06156-4