A multiplication formula for cluster characters in gentle algebras

TL;DR

This paper establishes a multiplication formula for cluster characters in gentle algebras, linking representation theory and cluster algebra exchange relations, with applications to surface triangulations and variable relations.

Contribution

It generalizes existing multiplication formulas for cluster characters in gentle algebras and provides a representation-theoretic interpretation of exchange relations in cluster algebras.

Findings

Derived a multiplication formula for cluster characters in gentle algebras.

Connected cluster algebra exchange relations with gentle algebra representations.

Related cluster variables of different types via surface triangulations.

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, 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.