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Cluster integrable systems are a broad class of integrable systems modelledon bipartite dimer models on the torus. Many discrete integrable dynamics ariseby applying sequences of local transformations, which form the cluster modulargroup of the cluster integrable system. This cluster modular group was recentlycharacterized by the first author and Inchiostro. There exist some discreteintegrable dynamics that make use of non-local transformations associated withgeometric $R$-matrices. In this article we characterize the generalized clustermodular group -- which includes both local and non-local transformations -- interms of extended affine symmetric groups. We also describe the action of thegeneralized cluster modular group on the spectral data associated with clusterintegrable systems.

Discrete dynamics in cluster integrable systems from geometric $R$-matrix transformations