A Lattice Isomorphism Theorem for Cluster Groups of Mutation-Dynkin Type $A_{n}$

Each quiver appearing in a seed of a cluster algebra determines a corresponding group, which we call a cluster group, which is defined via a presentation. Grant and Marsh showed that, for quivers appearing in seeds of cluster algebras of finite type, the associated cluster groups are isomorphic to finite reflection groups. As for finite Coxeter groups, we can consider parabolic subgroups of cluster groups. We prove that, in the type $A_{n}$ case, there exists an isomorphism between the lattice of subsets of the defining generators of the cluster group and the lattice of its parabolic subgroups. Moreover, each parabolic subgroup has a presentation given by restricting the presentation of the whole group.