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

Abstract

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.

arXiv comment26 pages, 11 figures, final version to appear in Combinatorial Theory

References in corpus (10)

• Total positivity, Grassmannians, and networks
• Random Domino Tilings and the Arctic Circle Theorem
• Random tilings and Markov chains for interlacing particles
• Y-meshes and generalized pentagram maps
• Inverse spectral problem for GK integrable system
• Minimal bipartite dimers and higher genus Harnack curves
• Toric networks, geometric $R$-matrices and generalized discrete Toda lattices
• Dungeons and Dragons: Combinatorics for the $d{P}_3$ Quiver
• The Schwarzian octahedron recurrence (dSKP equation) II: geometric systems
• The cluster modular group of the dimer model