New Combinatorial Formulas for Cluster Monomials of Type A Quivers

TL;DR

This paper introduces new combinatorial formulas for cluster monomials in type A cluster algebras, connecting them with existing models and theta functions, thereby advancing the understanding of their combinatorial structure.

Contribution

It presents novel combinatorial formulas for cluster monomials in type A cluster algebras and establishes bijections with known models and theta functions.

Findings

New formulas for cluster monomials using globally compatible collections

Bijections with T-paths and perfect matchings models

Connection between formulas and theta functions of Gross, Hacking, Keel, and Kontsevich

Abstract

Lots of research focuses on the combinatorics behind various bases of cluster algebras. This paper studies the natural basis of a type A cluster algebra, which consists of all cluster monomials. We introduce a new kind of combinatorial formulas for the cluster monomials in terms of the so-called globally compatible collections. We give bijective proofs of these formulas by comparing with the well-known combinatorial models of the T-paths and of the perfect matchings in a snake diagram. For cluster variables of a type A cluster algebra, we give a bijection that relates our new formula with the theta functions constructed by Gross, Hacking, Keel and Kontsevich.