Exchange Maps of Cluster Algebras

TL;DR

This paper studies exchange automorphisms in cluster algebras, providing conditions under which they are cluster isomorphisms, and explores the structure of automorphism groups for low-rank seeds.

Contribution

It establishes equivalent conditions for exchange automorphisms to be cluster isomorphisms in positive cluster algebras and analyzes the symmetric group actions involved.

Findings

Conditions for exchange automorphisms to be cluster isomorphisms

Group presentations for automorphisms of rank 1 and 2 seeds

Analysis of symmetric group actions on seed sets

Abstract

Every two seeds in a field of fractions $\mathcal{F}$ together with a symmetric group element gives rise to an automorphism of $\mathcal{F}$ called an exchange automorphism. For positive cluster algebras, we provide equivalent conditions for exchange automorphisms to be cluster isomorphisms of the corresponding cluster algebras. The equivalent conditions are given in terms of a symmetric group action on the set of seeds, which is also studied. Presentations of groups of cluster automorphisms of some seeds of ranks 1 and 2 are given.