Two variations on $(A_3\times A_1\times A_1{)}^{(1)}$ type discrete Painlevé equations

Abstract

By considering the normalizers of reflection subgroups of types $A_1^{(1)}$ and $A_3^{(1)}$ in $\tilde{W}\left(D_5^{(1)}\right)$, two normalizers: $\tilde{W}{\left(A_3\times A_1\right)}^{(1)}⋉W(A_1^{(1)})$ and $\tilde{W}{\left(A_1\times A_1\right)}^{(1)}⋉W(A_3^{(1)})$ can be constructed from a $(A_3\times A_1\times A_1{)}^{(1)}$ type subroot system. These two symmetries arose in the studies of discrete \Pa equations \cite{KNY:2002, Takenawa:03, OS:18}, where certain non-translational elements of infinite order were shown to give rise to discrete \Pa equations. We clarify the nature of these elements in terms of Brink-Howlett theory of normalizers of Coxeter groups \cite{BH}. This is the first of a series of studies which investigates the properties of discrete integrable equations via the theory of normalizers.