Tropical friezes and cluster-additive functions via Fock-Goncharov duality and a conjecture of Ringel

TL;DR

This paper explores the relationship between tropical friezes, cluster-additive functions, and Fock-Goncharov duality in cluster algebras, proving a conjecture of Ringel for finite type Cartan matrices and providing explicit pairings.

Contribution

It generalizes and proves Ringel's conjecture on cluster-additive functions for arbitrary finite type Cartan matrices within the Fock-Goncharov duality framework.

Findings

Proved Ringel's conjecture for finite type Cartan matrices.

Explicitly expressed Fock-Goncharov pairings between tropical points.

Established bijections between global monomials and tropical points.

Abstract

We study tropical friezes and cluster-additive functions associated to symmetrizable generalized Cartan matrices in the framework of Fock-Goncharov duality in cluster algebras. In particular, we generalize and prove a conjecture of C. M. Ringel on cluster-additive functions associated to arbitrary Cartan matrices of finite type. For a Fock-Goncharov dual pair of positive spaces of finite type, we use tropical friezes and cluster-additive functions to explicitly express the Fock-Goncharov pairing between their tropical points and the bijections between global monomials on one and tropical points of the other.