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Generic bases for cluster algebras and the Chamber Ansatz

Authors: Christof Geiß, Bernard Leclerc and Jan Schröer
Journal: J. Amer. Math. Soc. 25 (2012), 21-76
MSC (2010): Primary 13F60, 14M15, 14M99, 16G20, 20G44
Published electronically: August 10, 2011
MathSciNet review: 2833478
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Abstract: Let $ Q$ be a finite quiver without oriented cycles, and let $ \Lambda$ be the corresponding preprojective algebra. Let $ \mathfrak{g}$ be the Kac-Moody Lie algebra with Cartan datum given by $ Q$, and let $ W$ be its Weyl group. With $ w \in W$, there is associated a unipotent cell $ N^w$ of the Kac-Moody group with Lie algebra $ \mathfrak{g}$. In previous work we proved that the coordinate ring $ \mathbb{C}[N^w]$ of $ N^w$ is a cluster algebra in a natural way. A central role is played by generating functions $ \varphi_X$ of Euler characteristics of certain varieties of partial composition series of $ X$, where $ X$ runs through all modules in a Frobenius subcategory $ \mathcal{C}_w$ of the category of nilpotent $ \Lambda$-modules. The first aim of this article is to compare the function $ \varphi_X$ with the so-called cluster character of $ X$, which is defined in terms of the Euler characteristics of quiver Grassmannians. We show that for every $ X$ in $ \mathcal{C}_w$, $ \varphi_X$ coincides, after an appropriate change of variables, with the cluster character of Fu and Keller associated with $ X$ using any cluster-tilting object $ T$ of $ \mathcal{C}_w$. A crucial ingredient of the proof is the construction of an isomorphism between varieties of partial composition series of $ X$ and certain quiver Grassmannians. This isomorphism is obtained in a very general setup and should be of interest in itself. Another important tool of the proof is a representation-theoretic version of the Chamber Ansatz of Berenstein, Fomin and Zelevinsky, adapted to Kac-Moody groups. As an application, we get a new description of a generic basis of the cluster algebra $ \mathcal{A}(\underline{\Gamma}_T)$ obtained from $ \mathcal{C}[N^w]$ via specialization of coefficients to 1. Here generic refers to the representation varieties of a quiver potential arising from the cluster-tilting module $ T$. For the special case of coefficient-free acyclic cluster algebras this proves a conjecture by Dupont.

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Christof Geiß
Affiliation: Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, 04510 México D.F., México

Bernard Leclerc
Affiliation: LMNO, Université de Caen, CNRS, UMR 6139, F-14032 Caen Cedex, France

Jan Schröer
Affiliation: Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany

Received by editor(s): May 13, 2010
Received by editor(s) in revised form: May 13, 2011
Published electronically: August 10, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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