Generic bases for cluster algebras and the Chamber Ansatz

TL;DR

This paper establishes a new connection between generating functions of module varieties and cluster characters in the context of cluster algebras associated with unipotent cells, providing a new basis description and proving a conjecture.

Contribution

It shows that generating functions of module varieties coincide with cluster characters, leading to a new description of the generic basis in cluster algebras and proving Dupont's conjecture.

Findings

Generated functions match cluster characters after variable changes.

New basis description for cluster algebras from unipotent cells.

Proof of Dupont's conjecture for coefficient-free acyclic cluster algebras.

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.