Generic bases for cluster algebras and the Chamber Ansatz

Abstract

Let Q be a finite quiver without oriented cycles, and let $\Lambda$ be the corresponding preprojective algebra. Let g be the Kac-Moody Lie algebra with Cartan datum given by Q, and let W be its Weyl group. With w in W is associated a unipotent cell N^w of the Kac-Moody group with Lie algebra g. In previous work we proved that the coordinate ring \C[N^w] of N^w is a cluster algebra in a natural way. A central role is played by generating functions \vphi_X of Euler characteristics of certain varieties of partial composition series of X, where X runs through all modules in a Frobenius subcategory C_w of the category of nilpotent $\Lambda$-modules. We show that for every X in C_w, \vphi_X coincides after appropriate changes of variables with the cluster characters of Fu and Keller associated with any cluster-tilting module T of C_w. As an application, we get a new description of a generic basis of the cluster algebra obtained from \C[N^w] via specialization of coefficients to 1. For the special case of coefficient-free acyclic cluster algebras this proves a conjecture by Dupont.

48 pages. Version 2: Minor improvements, and a few typos corrected. v3: New section 2 (reminder on cluster algebras), so subsequent sections renumbered; section 6 (was section 5) reorganized and extended; several small corrections; references updated. Final version, now 55 pages, to appear JAMS

References in corpus (2)

• Kac-Moody groups and cluster algebras
• Cluster algebras via cluster categories with infinite-dimensional morphism spaces

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