Dipartimento di Scienze di Base e Applicate per l'Ingegneria

Acyclic cluster algebras via Coxeter double Bruhat cells and generalized minors

 

Abstract:

Cluster algebras come with a canonical partial basis: the cluster monomials. Extending this partial basis to a full basis has been one of the central problems in the theory giving raise to a varied zoo of constructions.  In this talk we will explain how Lie theory can be used to relate them. Specifically, after recalling the basic definitions, we will explain how any acyclic cluster algebra can be seen as the ring of coordinates of a suitable double Bruhat cell in the associated Kac-Moody group. Under this identification  we will interpret cluster monomials as generalized minors - certain functions on a Kac-Moody group defined in terms of its representations - and explain how one can use generalized minors to extend cluster monomials to a continuous family of bases of the cluster algebra in the affine cases.

This talk is based on joint works with D. Rupel and H. Williams