Geometric Reductions of ABS equations on an $n$-cube to discrete Painlevé systems

Abstract

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

References in corpus (2)

• q-Difference equations of KdV type and "Chazy-type" second-degree difference equations
• Lattice equations arising from discrete Painlevé systems (I): $(A_2+A_1{)}^{(1)}$ and $(A_1+A_1^{\prime }{)}^{(1)}$ cases

Cited by in corpus (8)

• Commutation Relations and Discrete Garnier Systems
• Lattice equations arising from discrete Painlevé systems (I): $(A_2+A_1{)}^{(1)}$ and $(A_1+A_1^{\prime }{)}^{(1)}$ cases
• Lax pairs of discrete Painlevé equations: $(A_2+A_1{)}^{(1)}$ case
• Lattice equations arising from discrete Painlevé systems. II. $A_4^{(1)}$ case
• Classification of quad-equations on a cuboctahedron
• Constructing Initial Value Spaces of Lattice Equations
• Reflection groups and discrete integrable systems
• Reduction of lattice equations to the Painlevé equations: P${}_{\mathrm{IV}}$ and P${}_{\mathrm{V}}$