A Multiplication Formula for Cluster Characters in Gentle Algebras

Abstract We prove a multiplication formula for cluster characters induced by generating extensions in a gentle algebra $A$, generalizing a result of Cerulli Irelli, Esposito, Franzen, and Reineke. In the case where $A$ is the gentle algebra of a triangulation $T$ of an unpunctured marked surface, this provides a representation-theoretic interpretation of the exchange relations in the cluster algebra with principal coefficients in $T$. As an application, we interpret a formula that relates cluster variables of type $B$ to cluster variables of type $A$ in the symmetric module category of the algebras arising from special triangulations of a regular polygon.