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

Terrence George (1) , Sanjay Ramassamy (2, 3, 4, 5)

1 Department of Mathematics - University of Michigan
2 Université Paris-Saclay
3 CNRS - Centre National de la Recherche Scientifique
4 CEA - CEA- Saclay
5 IPHT - Institut de Physique Théorique - UMR CNRS 3681

Cluster integrable systems are a broad class of integrable systems modelled on bipartite dimer models on the torus. Many discrete integrable dynamics arise by applying sequences of local transformations, which form the cluster modular group of the cluster integrable system. This cluster modular group was recently characterized by the first author and Inchiostro. There exist some discrete integrable dynamics that make use of non-local transformations associated with geometric R-matrices. In this article we characterize the generalized cluster modular group -- which includes both local and non-local transformations -- in terms of extended affine symmetric groups. We also describe the action of the generalized cluster modular group on the spectral data associated with cluster integrable systems.

Domains: Exactly Solvable and Integrable Systems [nlin.SI]; Mathematical Physics [math-ph]; Combinatorics [math.CO]

Terrence George, Sanjay Ramassamy. Discrete dynamics in cluster integrable systems from geometric R-matrix transformations. 2022. ⟨hal-03815673⟩