Dyck Paths Categories And Its Relationships With Cluster Algebras

TL;DR

This paper introduces Dyck paths categories as a combinatorial model for Dynkin type An quiver representations, establishing connections with cluster algebras and perfect matchings, and providing formulas for cluster variables.

Contribution

It establishes a novel combinatorial framework linking Dyck paths, snake graph matchings, and cluster algebra variables for Dynkin type An.

Findings

Bijection between Dyck paths and snake graph matchings

Formulas for cluster variables in terms of Dyck paths

New combinatorial model for quiver representations

Abstract

Dyck paths categories are introduced as a combinatorial model of the category of representations of quivers of Dynkin type An. 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 An in terms of Dyck paths.