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 $\mathfrak {g}$ 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.