Cluster modular groups of dimer models and networks

Terrence George

Published 2019-09-27Version 1

Associated to a convex integral polygon $P$ is a dimer cluster integrable system $\mathcal X_P$. We compute the group of symmetries of $\mathcal X_P$, called the cluster modular group, showing that it matches a conjecture of Fock and Marshakov. Probabilistically, non-torsion elements of $G_P$ are ways of shuffling the underlying bipartite graph, generalizing domino-shuffling. Algebro-geometrically, $G_P$ is the degree zero subgroup of the Picard group of a certain algebraic surface associated to $P$. We also prove analogous results for resistor networks.