Cluster Algebras , Paths and Total Positivity
نویسنده
چکیده
We review the solution of the Ar Q-systems in terms of the partition function of paths on a weighted graph, and show that it is possible to modify the graphs and transfer matrices so as to provide an explicit connection to the theory of planar networks introduced in the context of totally positive matrices by Fomin and Zelevinsky. We also apply our method to give a simple solution for the rank 2 affine cluster algebras studied by Caldero and Zelevinsky.
منابع مشابه
Q - system Cluster Algebras , Paths and Total Positivity ?
In the first part of this paper, we provide a concise review of our method of solution of the Ar Q-systems in terms of the partition function of paths on a weighted graph. In the second part, we show that it is possible to modify the graphs and transfer matrices so as to provide an explicit connection to the theory of planar networks introduced in the context of totally positive matrices by Fom...
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 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...
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 کاملPositivity and Canonical Bases in Rank 2 Cluster Algebras of Finite and Affine Types
The main motivation for the study of cluster algebras initiated in [4, 6, 1] was to design an algebraic framework for understanding total positivity and canonical bases in semisimple algebraic groups. In this paper, we introduce and explicitly construct the canonical basis for a special family of cluster algebras of rank 2. ju-bi-lee 1 : a year of emancipation and restoration provided by ancien...
متن کامل