Cluster Ensembles and Kac-Moody Groups

Abstract

We study the relationship between two sets of coordinates on a double Bruhat cell, the cluster variables introduced by Berenstein, Fomin, and Zelevinsky and the $\CX$-coordinates defined by the coweight parametrization of Fock and Goncharov. In these coordinates, we show that the generalized Chamber Ansatz of Fomin and Zelevinsky is a nondegenerate version of the canonical monomial transformation between the cluster variables and $\CX$-coordinates defined by a common exchange matrix. We prove this in the setting of an arbitrary symmetrizable Kac-Moody group, generalizing along the way many previous results on the double Bruhat cells of a semisimple algebraic group. In particular, we construct an upper cluster algebra structure on the coordinate ring of any double Bruhat cell in a symmetrizable Kac-Moody group, proving a conjecture of Berenstein, Fomin, and Zelevinsky.