Unistructurality of cluster algebras

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 $\mathcal{A}(S)$.

8 pages. In this second version, the authors revised a symbol error and an omission of coefficients in a formula in the proof of Proposition 3.2. arXiv admin note: text overlap with arXiv:1803.05281