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

TL;DR

This paper explores the extended symmetries of cluster integrable systems, including local and non-local transformations, and their impact on spectral data, advancing understanding of discrete dynamics in these models.

Contribution

It characterizes the generalized cluster modular group incorporating geometric R-matrix transformations using extended affine symmetric groups.

Findings

Extended affine symmetric groups describe the generalized cluster modular group.

The action of this group on spectral data is explicitly characterized.

The work unifies local and non-local transformations within a common framework.

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.