Generic bases for cluster algebras and the Chamber Ansatz

Abstract

Let Q Q be a finite quiver without oriented cycles, and let Λ \Lambda be the corresponding preprojective algebra. Let g \mathfrak {g} be the Kac-Moody Lie algebra with Cartan datum given by Q Q , and let W W be its Weyl group. With w ∈ W w \in W , there is associated a unipotent cell N w N^w of the Kac-Moody group with Lie algebra g \mathfrak {g} . In previous work we proved that the coordinate ring C [ N w ] \mathbb {C}[N^w] of N w N^w is a cluster algebra in a natural way. A central role is played by generating functions φ X \varphi _X of Euler characteristics of certain varieties of partial composition series of X X , where X X runs through all modules in a Frobenius subcategory C w \mathcal {C}_w of the category of nilpotent Λ \Lambda -modules. The first aim of this article is to compare the function φ X \varphi _X with the so-called cluster character of X X , which is defined in terms of the Euler characteristics of quiver Grassmannians. We show that for every X X in C w \mathcal {C}_w , φ X \varphi _X coincides, after an appropriate change of variables, with the cluster character of Fu and Keller associated with X X using any cluster-tilting object T T of C w \mathcal {C}_w . A crucial ingredient of the proof is the construction of an isomorphism between varieties of partial composition series of X 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 A ( Γ _ T ) \mathcal {A}(\underline {\Gamma }_T) obtained from C [ N w ] \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 T . For the special case of coefficient-free acyclic cluster algebras this proves a conjecture by Dupont.