Cluster Ensembles and Kac-Moody Groups

TL;DR

This paper establishes a deep connection between cluster variables and $ ext{X}$-coordinates on double Bruhat cells within symmetrizable Kac-Moody groups, generalizing previous results and confirming a conjecture about their algebraic structure.

Contribution

It introduces an upper cluster algebra structure on double Bruhat cells in Kac-Moody groups and links the Chamber Ansatz to monomial transformations, extending prior semisimple cases.

Findings

Proves the Chamber Ansatz as a nondegenerate monomial transformation.

Constructs an upper cluster algebra structure on double Bruhat cells.

Generalizes results from semisimple to symmetrizable 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.