Positivity for cluster algebras from surfaces
نویسندگان
چکیده
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 surfaces, in geometric type.
منابع مشابه
On 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...
متن کاملTotal positivity and cluster algebras
This is a brief and informal introduction to cluster algebras. It roughly follows the historical path of their discovery, made jointly with A. Zelevinsky. Total positivity serves as the main motivation. Mathematics Subject Classification (2000). Primary 13F60, Secondary 05E10, 05E15, 14M15, 15A23, 15B48, 20F55, 22E46.
متن کاملCLUSTER ALGEBRAS AND CLUSTER CATEGORIES
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 کاملUnfolding of Acyclic Sign-skew-symmetric Cluster Algebras and Applications to Positivity and F -polynomials
In this paper, we build the unfolding approach from acyclic sign-skew-symmetric matrices of finite rank to skew-symmetric matrices of infinite rank, which can be regard as an improvement of that in the skew-symmetrizable case. Using this approach, we give a positive answer to the problem by Berenstein, Fomin and Zelevinsky in [6] which asks whether an acyclic signskew-symmetric matrix is always...
متن کاملFrom Littlewood-richardson Coefficients to Cluster Algebras in Three Lectures
This is an expanded version of the notes of my three lectures at a NATO Advanced Study Institute “Symmetric functions 2001: surveys of developments and perspectives” (Isaac Newton Institute for Mathematical Sciences, Cambridge, UK; June 25 – July 6, 2001). Lecture I presents a unified expression from [4] for generalized Littlewood-Richardson coefficients (= tensor product multiplicities) for an...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2009