A multiplication formula for cluster characters in gentle algebras

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, 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.