Dyck path categories and its relationships with cluster algebras

Abstract

Dyck path categories are introduced as a combinatorial model of the category of representations of quivers of Dynkin-type [Formula: see text]. In particular, it is proved that there is a bijection between some Dyck paths and perfect matchings of some snake graphs. The approach allows us to give formulas for cluster variables in cluster algebras of Dynkin-type [Formula: see text] in terms of Dyck paths.