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

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.