Affine cluster monomials are generalized minors

TL;DR

This paper connects cluster monomials in acyclic cluster algebras to generalized minors in Kac-Moody groups, providing a new algebraic group representation perspective for finite and affine types.

Contribution

It proves that cluster monomials with g-vectors in the doubled Cambrian fan are restrictions of generalized minors, offering a non-recursive algebraic description of these cluster algebras.

Findings

Cluster monomials correspond to restrictions of generalized minors.

Finite and affine type cluster algebras admit algebraic group representation descriptions.

In type A_1^{(1)}, bases of cluster monomials are composed of minors.

Abstract

We study the realization of acyclic cluster algebras as coordinate rings of Coxeter double Bruhat cells in Kac-Moody groups. We prove that all cluster monomials with g-vector lying in the doubled Cambrian fan are restrictions of principal generalized minors. As a corollary, cluster algebras of finite and affine type admit a complete and non-recursive description via (ind-)algebraic group representations, in a way similar in spirit to the Caldero-Chapoton description via quiver representations. In type A_1^{(1)}, we further show that elements of several canonical bases (generic, triangular, and theta) which complete the partial basis of cluster monomials are composed entirely of restrictions of minors. The discrepancy among these bases is accounted for by continuous parameters appearing in the classification of irreducible level-zero representations of affine Lie groups. We discuss how our results illuminate certain parallels between the classification of representations of finite-dimensional algebras and of integrable weight representations of Kac-Moody algebras.