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 (2-2) cluster modular group, showing that it is a certain abelian group conjectured by Fock and Marshakov. Combinatorially, non-torsion elements of $G_N$ are ways of shuffling the underlying bipartite graph, generalizing domino-shuffling. Algebro-geometrically, $G_N$ is a subgroup of the Picard group of a certain algebraic surface associated to $N$.
- Publication:
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arXiv e-prints
- Pub Date:
- September 2019
- DOI:
- 10.48550/arXiv.1909.12896
- arXiv:
- arXiv:1909.12896
- Bibcode:
- 2019arXiv190912896G
- Keywords:
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- Mathematics - Combinatorics;
- Mathematics - Algebraic Geometry
- E-Print:
- Added a background section 2.4, and a new section 6.1. Appendix is now section 5 and contains more details