Mutation commutativity between geometric realizations and related quivers of classical cluster algebras and applications

Cluster algebras, introduced by Fomin and Zelevinsky, are connected to dual canonical bases. Geometric realization is an effective way to solve algebraic problems. In 2002, Fomin and Zelevinsky proposed geometric realizations for root systems and finite-type cluster algebras. This study examines the regulation of mutation in geometric realization. We establish a fascinating link between centrally symmetric triangulations and quivers in classical cluster algebras, showing that these triangulations and quivers are interchangeable under mutation. This connection leads to a significant finding: any cluster variable of a given cluster algebra can induce a connected subquiver within the corresponding initial quiver. Furthermore, we demonstrate that any two quivers generated by cluster variables intersect, emphasizing their connectedness. To illustrate the practical implications of this geometric realization, we present a convenient method to compute seed mutations of type [Formula: see text] and define a category based on centrally symmetric triangulations. This category’s objects are [Formula: see text]-linear combinations of positive roots of the root system of type [Formula: see text], and we propose an equivalent condition for morphisms. This condition is a generalization of the case of type [Formula: see text] by [P. Caldero, F. Chapoton and R. Schiffler, Quivers with relations arising from clusters [Formula: see text] case), Trans. Amer. Math. Soc. 358(3) (2006) 1347–1364]. Finally, we propose a geometric realization for cluster algebras: weighted triangulations. This approach does not require cluster algebras to be of finite type and can verify whether a matrix from a cluster algebra is equivalent to a subdiagonal matrix.