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 formula 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.
@article{10_37236_6464,
author = {Kyungyong Lee and Li Li and Ba Nguyen},
title = {New combinatorial formulas for cluster monomials of type {\(A\)} quivers},
journal = {The electronic journal of combinatorics},
year = {2017},
volume = {24},
number = {2},
doi = {10.37236/6464},
zbl = {1401.13068},
url = {http://geodesic.mathdoc.fr/articles/10.37236/6464/}
}
TY - JOUR
AU - Kyungyong Lee
AU - Li Li
AU - Ba Nguyen
TI - New combinatorial formulas for cluster monomials of type \(A\) quivers
JO - The electronic journal of combinatorics
PY - 2017
VL - 24
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/6464/
DO - 10.37236/6464
ID - 10_37236_6464
ER -
%0 Journal Article
%A Kyungyong Lee
%A Li Li
%A Ba Nguyen
%T New combinatorial formulas for cluster monomials of type \(A\) quivers
%J The electronic journal of combinatorics
%D 2017
%V 24
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/6464/
%R 10.37236/6464
%F 10_37236_6464
Kyungyong Lee; Li Li; Ba Nguyen. New combinatorial formulas for cluster monomials of type \(A\) quivers. The electronic journal of combinatorics, Tome 24 (2017) no. 2. doi: 10.37236/6464
