Geometric Reductions of ABS equations on an n-cube to discrete Painlev\'e systems

In this paper, we show how to relate $n$-dimensional cubes on which ABS equations hold to the symmetry groups of discrete Painlev\'e equations. We here focus on the reduction from the 4-dimensional cube to the $q$-discrete third Painlev\'e equation, which is a dynamical system on a rational surface of type $A_5^{(1)}$ with the extended affine Weyl group $\widetilde{\mathcal W}\bigl((A_2+A_1)^{(1)}\bigr)$. We provide general theorems to show that this reduction also extends to other discrete Painlev\'e equations at least of type A.