New Combinatorial Formulas for Cluster Monomials of Type A Quivers
Combinatorics
2017-06-07 v3
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.
Cite
@article{arxiv.1604.06728,
title = {New Combinatorial Formulas for Cluster Monomials of Type A Quivers},
author = {Kyungyong Lee and Li Li and Ba Nguyen},
journal= {arXiv preprint arXiv:1604.06728},
year = {2017}
}