# Two variations on $(A_3\times A_1\times A_1)^{(1)}$ type discrete Painlev\'e equations

## TL;DR

This paper investigates two symmetry variations of discrete Painlevé equations derived from specific Coxeter group normalizers, clarifying their structure and role in integrability using Brink-Howlett theory.

## Contribution

It introduces two new symmetry structures for discrete Painlevé equations from Coxeter group normalizers, expanding understanding of their algebraic properties.

## Key findings

- Constructed two normalizer-based symmetry groups from a $(A_3 imes A_1 imes A_1)^{(1)}$ subroot system.
- Connected non-translational elements to discrete Painlevé equations.
- Applied Brink-Howlett theory to clarify the nature of these symmetry elements.

## Abstract

By considering the normalizers of reflection subgroups of types $A_1^{(1)}$ and $A_3^{(1)}$ in $\widetilde{W}\left(D_5^{(1)}\right)$, two normalizers: $\widetilde{W}\left(A_3\times A_1\right)^{(1)}\ltimes {W}(A_1^{(1)})$ and $\widetilde{W}\left(A_1\times A_1\right)^{(1)}\ltimes {W}(A_3^{(1)})$ 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.