Translation in affine Weyl groups and its application in discrete integrable systems

Properties of ADE-type Weyl groups (known as simplylaced) were explored and shown to be useful in characterizing and establishing relations between different integrable systemspreviously in Joshi et al. 2015 Reflection groups and discrete integrable systems. J. Integrable Syst. 1, xyw006. (doi:10.1093/integr/xyw006) and Shi Y. 2019 Two Variations on (A3 A1 A1)(1) Type Discrete Painlevé Equations. Proc. A 475, 20190299. (doi:10.1098/rspa.2019.0299). Here, we extend the formulations to include non-simplylaced types, paying special attention to the translational elements of the group. As applications, we show how these formulas can be used in clarifying the nature of some integrable systems of type E8(1)that appeared recently in the literature (Joshi and Nakazono 2017 Elliptic Painlevé equations from next-nearest-neighbor translations on the E8(1) lattice. J. Phys. A Math. Gen. 50, 305205. (doi:10.1088/1751-8121/aa7915) and F4(1)Atkinson J, Howes P, Joshi N, Nakazono N. 2016 Geometry of an elliptic difference equation related to Q4. J. Lond. Math. Soc. 93, 763-784. (doi:10.1112/jlms/jdw020)).