Q-Systems, Factorization Dynamics, and the Twist Automorphism

TL;DR

This paper connects Q-systems with cluster algebras via double Bruhat cells, proving their integrability and relating twist automorphisms to cluster dynamics in algebraic groups.

Contribution

It provides a cluster-algebraic framework for Q-systems, including twisted types, and demonstrates their integrability through factorization mappings and twist automorphisms.

Findings

Cluster structures on double Bruhat cells realize Q-systems.

Discrete integrability of Q-systems is established via factorization dynamics.

Explicit formulas relate twisted and untwisted cluster variables.

Abstract

We provide a concrete realization of the cluster algebras associated with Q-systems as amalgamations of cluster structures on double Bruhat cells in simple algebraic groups. For nonsimply-laced groups, this provides a cluster-algebraic formulation of Q-systems of twisted type. It also yields a uniform proof of the discrete integrability of these Q-systems by identifying them with the dynamics of factorization mappings on quotients of double Bruhat cells. On the double Bruhat cell itself, we find these dynamics are closely related to those of the Fomin-Zelevinsky twist map. This leads to an explicit formula expressing twisted cluster variables as Laurent monomials in the untwisted cluster variables obtained from the corresponding mutation sequence. This holds for Coxeter double Bruhat cells in any symmetrizable Kac-Moody group, and we show that in affine type the analogous factorization mapping is also integrable.