Unistructurality of cluster algebras of type ${\widetilde{\mathbb{A}}}$

TL;DR

This paper proves the unistructurality conjecture for cluster algebras of type A, showing that their cluster variables uniquely determine their structure, and confirms the automorphism conjecture for these types.

Contribution

It establishes unistructurality and automorphism conjectures for cluster algebras of type A, extending previous results beyond Dynkin and rank 2 cases.

Findings

Unistructurality holds for A cluster algebras.

Automorphism conjecture is validated for A types.

Uses triangulations of annuli and algebraic independence in proofs.

Abstract

It is conjectured by Ibrahim Assem, Ralf Schiffler and Vasilisa Shramchenko in "Cluster Automorphisms and Compatibility of Cluster Variables" that every cluster algebra is unistructural, that is to say, that the set of cluster variables determines uniquely the cluster algebra structure. In other words, there exists a unique decomposition of the set of cluster variables into clusters. This conjecture has been proven to hold true for algebras of type Dynkin or rank 2 by Assem, Schiffler and Shramchenko. The aim of this paper is to prove it for algebras of type ${\widetilde{\mathbb{A}}}$. We use triangulations of annuli and algebraic independence of clusters to prove unistructurality for algebras arising from annuli, which are of type ${\widetilde{\mathbb{A}}}$. We also prove the automorphism conjecture from Assem, Schiffler and Shramchenko for algebras of type ${\widetilde{\mathbb{A}}}$ as a direct consequence.