Symmetric mutation algebras in the context of subcluster algebras

Abstract

For a rooted cluster algebra [Formula: see text] over a valued quiver [Formula: see text], a symmetric cluster variable is any cluster variable belonging to a cluster associated with a quiver [Formula: see text], for some permutation [Formula: see text]. The subalgebra of [Formula: see text] generated by all symmetric cluster variables, is called the symmetric mutation subalgebra and is denoted by [Formula: see text]. In this paper, we identify the class of cluster algebras that satisfy [Formula: see text], which contains almost every quiver of finite mutation type. In the process of proving the main result, we provide a classification of quivers mutations classes that relates their maximum weights to the shapes of the initial quivers. Furthermore, some properties of symmetric mutation subalgebras are given.