Reduction of quad-equations consistent around a cuboctahedron I: additive case

TL;DR

This paper demonstrates how a new system of partial difference equations, consistent around a cuboctahedron, reduces to $A_2^{(1) ext{*}}$-type discrete Painlevé equations through a periodic reduction of a 3D lattice.

Contribution

It introduces a reduction method for a new system of equations consistent around a cuboctahedron, linking it to discrete Painlevé equations.

Findings

Reduction to $A_2^{(1) ext{*}}$-type discrete Painlevé equations

Establishment of a periodic reduction technique

Connection between cuboctahedron consistency and Painlevé equations

Abstract

In this paper, we consider a reduction of a new system of partial difference equations, which was obtained in our previous paper (Joshi and Nakazono, arXiv:1906.06650) and shown to be consistent around a cuboctahedron. We show that this system reduces to $A_2^{(1)\ast}$-type discrete Painlev\'e equations by considering a periodic reduction of a three-dimensional lattice constructed from overlapping cuboctahedra.