Translations in affine Weyl groups and their applications in discrete integrable systems

TL;DR

This paper explores the properties of affine Weyl groups, including non-simply-laced types, and demonstrates their applications in understanding complex discrete integrable systems like those of types E8 and F4.

Contribution

It extends previous work on Weyl groups to include non-simply-laced types and develops formulas for translational elements relevant to integrable systems.

Findings

Clarified the role of affine Weyl group translations in integrable systems

Provided new formulas for non-simply-laced affine Weyl groups

Applied these methods to systems of type E8 and F4

Abstract

In this paper, we review the properties and representations of the Weyl groups relevant in the study of discrete integrable systems. Previously in \cite{jns4, Shi:19}, properties of Weyl groups of type $ADE$ (known as simply-laced) were shown to be useful in characterizing and establishing relations between different integrable systems. Here we extend the formulations and discussions to include non-simply-laced types, giving special attention to developing formulas related to the translational elements of the affine Weyl groups. As applications, we show how these are used to clarify the natures of some integrable systems of type $E_8$ \cite{NN_elliptic} and $F_4$ \cite{ahjn:16} appeared recently in the literature.