Cluster automorphism groups of cluster algebras with coefficients

TL;DR

This paper investigates the automorphism groups of skew-symmetric cluster algebras with coefficients, introducing the concept of gluing free cluster algebras, and establishes conditions under which these groups relate to those of principal parts or initial quivers.

Contribution

It introduces the notion of gluing free cluster algebras and demonstrates their automorphism groups relate to principal parts, expanding understanding of symmetries in cluster algebras with coefficients.

Findings

Gluing free cluster algebras have automorphism groups as subgroups of principal parts.

Several classes of cluster algebras with coefficients are identified as gluing free.

Automorphism groups of surface cluster algebras are mostly isomorphic to those of their principal parts.

Abstract

We study the cluster automorphism group of a skew-symmetric cluster algebra with geometric coefficients. For this, we introduce the notion of gluing free cluster algebra, and show that under a weak condition the cluster automorphism group of a gluing free cluster algebra is a subgroup of the cluster automorphism group of its principal part cluster algebra (i.e. the corresponding cluster algebra without coefficients). We show that several classes of cluster algebras with coefficients are gluing free, for example, cluster algebras with principal coefficients, cluster algebras with universal geometric coefficients, and cluster algebras from surfaces (except a 4-gon) with coefficients from boundaries. Moreover, except four kinds of surfaces, the cluster automorphism group of a cluster algebra from a surface with coefficients from boundaries is isomorphic to the cluster automorphism group of its principal part cluster algebra; for a cluster algebra with principal coefficients, its cluster automorphism group is isomorphic to the automorphism group of its initial quiver.