Unistructurality of cluster algebras

TL;DR

This paper proves that all skew-symmetrizable cluster algebras are unistructural, confirming a conjecture and showing that cluster automorphisms correspond to permutations of cluster variables.

Contribution

It establishes the unistructurality of skew-symmetrizable cluster algebras, a significant conjecture in the field, and characterizes cluster automorphisms as permutations of variables.

Findings

Proves skew-symmetrizable cluster algebras are unistructural.

Shows cluster automorphisms are permutations of cluster variables.

Confirms a conjecture by Assem, Schiffler, and Shramchenko.

Abstract

We prove that any skew-symmetrizable cluster algebra is unistructural, which is a conjecture by Assem, Schiffler, and Shramchenko. As a corollary, we obtain that a cluster automorphism of a cluster algebra $\mathcal A(\mathcal S)$ is just an automorphism of the ambient field $\mathcal F$ which restricts to a permutation of cluster variables of $\mathcal A(S)$.