We present a combinatorial model for cluster algebras of type $D_n$ in terms of centrally symmetric pseudotriangulations of a regular $2n$ gon with a small disk in the centre. This model provides convenient and uniform interpretations for clusters, cluster variables and their exchange relations, as well as for quivers and their mutations. We also present a new combinatorial interpretation of cluster variables in terms of perfect matchings of a graph after deleting two of its vertices. This interpretation differs from known interpretations in the literature. Its main feature, in contrast with other interpretations, is that for a fixed initial cluster seed, one or two graphs serve for the computation of all cluster variables. Finally, we discuss applications of our model to polytopal realizations of type $D$ associahedra and connections to subword complexes and $c$-cluster complexes.
@article{10_37236_5282,
author = {Cesar Ceballos and Vincent Pilaud},
title = {Cluster algebras of type {\(D\):} pseudotriangulations approach},
journal = {The electronic journal of combinatorics},
year = {2015},
volume = {22},
number = {4},
doi = {10.37236/5282},
zbl = {1350.13021},
url = {http://geodesic.mathdoc.fr/articles/10.37236/5282/}
}
TY - JOUR
AU - Cesar Ceballos
AU - Vincent Pilaud
TI - Cluster algebras of type \(D\): pseudotriangulations approach
JO - The electronic journal of combinatorics
PY - 2015
VL - 22
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/5282/
DO - 10.37236/5282
ID - 10_37236_5282
ER -
%0 Journal Article
%A Cesar Ceballos
%A Vincent Pilaud
%T Cluster algebras of type \(D\): pseudotriangulations approach
%J The electronic journal of combinatorics
%D 2015
%V 22
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/5282/
%R 10.37236/5282
%F 10_37236_5282
Cesar Ceballos; Vincent Pilaud. Cluster algebras of type \(D\): pseudotriangulations approach. The electronic journal of combinatorics, Tome 22 (2015) no. 4. doi: 10.37236/5282
