We define a combinatorial model for $F$-polynomials and $g$-vectors for type $D_n$ cluster algebras where the associated quiver is acyclic. Our model utilizes a combination of dimer configurations and double dimer configurations which we refer to as mixed dimer configurations. In particular, we give a graph theoretic recipe that describes which monomials appear in such $F$-polynomials, as well as a graph theoretic way to determine the coefficients of each of these monomials. In addition, we prove that a weighting on our mixed dimer configuration model yields the associated $g$-vector. To prove this formula, we use a combinatorial formula due to Thao Tran (arXiv:0911.4462, 2009) and provide explicit bijections between her combinatorial model and our own.
@article{10_37236_10437,
author = {Gregg Musiker and Kayla Wright},
title = {Mixed dimer configuration model in type {D} cluster algebras},
journal = {The electronic journal of combinatorics},
year = {2023},
volume = {30},
number = {2},
doi = {10.37236/10437},
zbl = {1523.13037},
url = {http://geodesic.mathdoc.fr/articles/10.37236/10437/}
}
TY - JOUR
AU - Gregg Musiker
AU - Kayla Wright
TI - Mixed dimer configuration model in type D cluster algebras
JO - The electronic journal of combinatorics
PY - 2023
VL - 30
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/10437/
DO - 10.37236/10437
ID - 10_37236_10437
ER -
%0 Journal Article
%A Gregg Musiker
%A Kayla Wright
%T Mixed dimer configuration model in type D cluster algebras
%J The electronic journal of combinatorics
%D 2023
%V 30
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/10437/
%R 10.37236/10437
%F 10_37236_10437
Gregg Musiker; Kayla Wright. Mixed dimer configuration model in type D cluster algebras. The electronic journal of combinatorics, Tome 30 (2023) no. 2. doi: 10.37236/10437
