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

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 Painleve equations by considering a periodic reduction of a three-dimensional lattice constructed from overlapping cuboctahedra.

Discrete Power Functions on a Hexagonal Lattice I: Derivation of defining equations from the symmetry of the Garnier System in two variables

The discrete power function on the hexagonal lattice proposed by Bobenko et al is considered, whose defining equations consist of three cross-ratio equations and a similarity constraint. We show that the defining equations are derived from the discrete symmetry of the Garnier system in two variables.

A review of elliptic difference Painleve equations

Discrete Painleve equations are nonlinear, nonautonomous difference equations of second-order. They have coefficients that are explicit functions of the independent variable $n$ and there are three different types of equations according to whether the coefficient functions are linear, exponential or elliptic functions of $n$. In this paper, we focus on the elliptic type and give a review of the construction of such equations on the $E_8$ lattice. The first such construction was given by Sakai cite{SakaiH2001:MR1882403}. We focus on recent developments giving rise to more examples of elliptic discrete Painleve equations.

Geometric description of discrete power function associated with the sixth Painleve equation

In this paper, we consider the discrete power function associated with the sixth Painleve equation. This function is a special solution of the so-called cross-ratio equation with a similarity constraint. We show in this paper that this system is embedded in a cubic lattice with $widetilde{W}(3A_1^{(1)})$ symmetry. By constructing the action of $widetilde{W}(3A_1^{(1)})$ as a subgroup of $widetilde{W}(D_4^{(1)})$, i.e., the symmetry group of P$_{rm VI}$, we show how to relate $widetilde{W}(D_4^{(1)})$ to the symmetry group of the lattice. Moreover, by using translations in $widetilde{W}(3A_1^{(1)})$, we explain the odd-even structure appearing in previously known explicit formulas in terms of the $tau$ function.

Lattice equations arising from discrete Painleve systems (I): $(A_2+A_1)^{(1)}$ and $(A_1+A_1)^{(1)}$ cases

We introduce the concept of $omega$-lattice, constructed from $tau$ functions of Painleve systems, on which quad-equations of ABS type appear. In particular, we consider the $A_5^{(1)}$- and $A_6^{(1)}$-surface $q$-Painleve systems corresponding affine Weyl group symmetries are of $(A_2+A_1)^{(1)}$- and $(A_1+A_1)^{(1)}$-types, respectively.