The cluster modular group of the dimer model
Abstract
Associated to a convex integral polygon $N$ is a cluster integrable system $\mathcal X_N$ constructed from the dimer model. We compute the group $G_N$ of symmetries of $\mathcal X_N$, called the (22) cluster modular group, showing that it is a certain abelian group conjectured by Fock and Marshakov. Combinatorially, nontorsion elements of $G_N$ are ways of shuffling the underlying bipartite graph, generalizing dominoshuffling. Algebrogeometrically, $G_N$ is a subgroup of the Picard group of a certain algebraic surface associated to $N$.
 Publication:

arXiv eprints
 Pub Date:
 September 2019
 DOI:
 10.48550/arXiv.1909.12896
 arXiv:
 arXiv:1909.12896
 Bibcode:
 2019arXiv190912896G
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Algebraic Geometry
 EPrint:
 Added a background section 2.4, and a new section 6.1. Appendix is now section 5 and contains more details