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

By Nalini Joshi and Nobutaka Nakazono

Abstract

In this paper, we consider a reduction of a new system of partial difference equations, which was obtained in our previous paper [Classification of quad-equations on a cuboctahedron, arXiv:1906.06650, 2019] and shown to be consistent around a cuboctahedron. We show that this system reduces to $A_2^{(1)\ast }$-type discrete Painlevé equations by considering a periodic reduction of a three-dimensional lattice constructed from overlapping cuboctahedra.

1. Introduction

In this paper, we consider a system of partial difference equations (P$\Delta$Es) governing a function $u=u({\boldsymbol{l}})$ taking values on the vertices of a face-centered cubic lattice $\Omega$, given by

$$\begin{equation} \Omega =\left\{\left. {\boldsymbol{l}}=\sum _{i=1}^3l_i\boldsymbol{\epsilon }_i\nobreakspace \right|\nobreakspace l_i\in \mathbb{Z},\nobreakspace l_1+l_2+l_3\in 2\mathbb{Z} \right\}, \cssId{texmlid31}{\tag{1.1}} \end{equation}$$

where $\{\boldsymbol{\epsilon }_1,\boldsymbol{\epsilon }_2,\boldsymbol{\epsilon }_3\}$ is a standard basis of $\mathbb{R}^3$. The system consists of 6 equations:

$$\begin{equation} {\dfrac{u_{\overline{ik}}}{u_{\underline{i}\overline{k}}} \!\!=\!\!\dfrac{(\alpha _{ij}+\gamma _i)u_{\overline{jk}}-(\alpha _{ij}+\gamma _j-\gamma _k)u_{\underline{j}\overline{k}}}{(\alpha _{ij}-\gamma _j+\gamma _k)u_{\overline{jk}}-(\alpha _{ij}-\gamma _i)u_{\underline{j}\overline{k}}},\; \dfrac{u_{\underline{jk}}}{u_{\overline{jk}}} \!\!=\!\!\dfrac{(\alpha _{ij}+\gamma _i)u_{\underline{ik}}-(\alpha _{ij}-\gamma _j+\gamma _k)u_{\overline{ik}}}{(\alpha _{ij}+\gamma _j-\gamma _k)u_{\underline{ik}}-(\alpha _{ij}-\gamma _i)u_{\overline{ik}}}, } \cssId{texmlid1}{\tag{1.2}} \end{equation}$$

where $(i,j,k)=(1,2,3),\,(2,3,1),\,(3,1,2)$, and the bars $\bar{i}$ and $\underline{j}$ denote ${\boldsymbol{l}}\to {\boldsymbol{l}}+\boldsymbol{\epsilon }_i$ and ${\boldsymbol{l}}\to {\boldsymbol{l}}-\boldsymbol{\epsilon }_j$ respectively and the coefficients are given by

$$\begin{align} &\alpha _{ij}=\alpha _i(l_i)-\alpha _j(l_j), &&\alpha _i(k)=\alpha _i(0)+k, &&i,j\in \{1,2,3\},\nobreakspace k\in \mathbb{Z},\tag{1.3a}\\ &\gamma _1=-c+(-1)^{l_1+l_2}\delta _1, &&\gamma _2=-c+(-1)^{l_2+l_3}\delta _2, &&\gamma _3=-c+(-1)^{l_1+l_3}\delta _3, \tag{1.3b} \end{align}$$

with $\alpha _i(0)$, $i=1,2,3$, $c$, and $\delta _j$, $j=1,2,3$, being complex parameters. Figure 1.1 shows a unit cell in $\Omega$.

Graphic without alt text — Figure 1.1.
A unit cell of the $\Omega$ lattice

Our study is motivated by two considerations. Firstly, the system Equation 1.2 satisfies the consistency around a cuboctahedron (CACO) property Reference 10, which is a generalization of the famous consistency around a cube (CAC) property Reference 17. (See Appendix A for a summary of the details of the CACO property and §1.2 for those of the CAC property.) Secondly, we are motivated by finding relations between partial difference equations and ordinary difference equations known as the discrete Painlevé equations.

In this paper, we show that the system Equation 1.2 reduces to discrete Painlevé equations with initial value space characterised as $A_2^{(1)\ast }$ in the sense of Sakai Reference 22. The latter equations have two forms in the literature given respectively by Tsuda Reference 23 and Ramani et al. Reference 21 and are explicitly given by:

$$\begin{align} &\begin{cases} \Big (\underline{Y}+X\Big )\Big (X+Y\Big ) =\dfrac{\left((X+c_3)^2-c_1\right)\left((X-c_3)^2-c_2\right)}{(X+t)^2-c_4},\\ \Big (\overline{X}+Y)\Big (X+Y\Big ) =\dfrac{\left((Y-c_3)^2-c_1\right)\left((Y+c_3)^2-c_2\right)}{\left(Y+t+\frac{1}{2}\right)^2-c_5}, \end{cases}\cssId{texmlid7}{\tag{1.4a}}\\ &\Big (\overline{X}+X\Big )\Big (\underline{X}+X\Big )=\dfrac{\left(X^2-c_1\right)\left(X^2-c_2\right)}{(X+t)^2-c_3}. \cssId{texmlid29}{\tag{1.4b}} \end{align}$$

Here, $t\in \mathbb{C}$ is an independent variable, $c_i$, $i=1$, …, $5$, are complex parameters and $X$, $Y$ are dependent variables:

$$\begin{equation} X=X(t),\quad Y=Y(t),\quad \overline{X}=X(t+1),\quad \underline{X}=X(t-1),\quad \underline{Y}=Y(t-1). \tag{1.5} \end{equation}$$

We note that discrete Painlevé equations admit special solutions when parameters take special values. For example, Equation Equation 1.4a has the special solution given by the generalized hypergeometric series ${}_3F_2$ when $4c_3+2\sqrt {c_4}+2\sqrt {c_5}=1$ Reference 11.

Our main result is Theorem 1.1. To state the theorem, we first explain how to take the reduction on the lattice $\Omega$. To be explicit, consider a vertex ${\boldsymbol{l}}\in \Omega$, given by $l_1\epsilon _1+l_2\epsilon _2+l_3\epsilon _3$. Define the plane $H_k\subset \Omega$ given by $l_3=k$. We project the vertices of $H_1$ to the adjacent horizontal plane $H_0$ by taking $(l_1,l_2,1) \mapsto (l_1-1,l_2-1,0)$. The union of the projection with the lattice points on $H_0$ forms $\mathbb{Z}^2$. We can define such a projection from every plane $H_k$ to $H_0$ by the following:

$$\begin{equation*} (l_1,l_2,k) \mapsto (l_1-k,l_2-k,0). \end{equation*}$$

We call the result of this operation a $(1,1,1)$-periodic reduction.

Theorem 1.1.

The $A_2^{(1)\ast }$-type discrete Painlevé equations Equation 1.4 can be obtained from the system of P$\Delta$Es Equation 1.2 via the $(1,1,1)$-periodic reduction.

1.1. Notation and definitions

Throughout the paper, we use terminology to describe polynomials and quad-equations that is common in the literature. Readers who are unfamiliar with this notation may wish to consult Reference 1Reference 8Reference 10. We use $Q=Q(x,y,z,w)$ to denote a multivariable polynomial over $\mathbb{C}$. Under certain conditions, i.e., $Q$ be affine linear and irreducible, we will refer to the equation $Q=0$ as a quad-equation or sometimes, for succinctness, refer to the polynomial $Q$ as a quad-equation. We remind the reader that the condition of irreducibility implies that $Q(x,y,z,w)=0$ can be solved for each argument, and that the solution is a rational function of the other three arguments.

1.2. Background

Integrable systems are widely applicable models of science, occurring in fluid dynamics, particle physics and optics. The prototypical example is the famous Korteweg-de Vries (KdV) equation whose solitary wave-like solutions interact with elastically like particles, leading to the invention of the term soliton. It is then natural to ask what discrete versions of such equations are also integrable. This question turns out to be related to consistency conditions for polynomials associated to faces of cubes as we explain below.

Integrable discrete systems were discovered Reference 15Reference 16Reference 18Reference 20 from mappings that turn out to be consistent on multi-dimensional cubes. (We note that there are additional systems that do not fall into this class; see e.g., Reference 8, Chapter 3.) These are quad-equations in the sense in §1.1. In Reference 1Reference 2Reference 3Reference 4, Adler-Bobenko-Suris et al. classified quad-equations satisfying the consistency around a cube (CAC) property, which lead to integrable P$\Delta$Es. We refer to such P$\Delta$Es as ABS equations. It turns out that ABS equations contain many well known integrable P$\Delta$Es Reference 9Reference 14Reference 15Reference 16.

Reductions of integrable PDEs lead to Painlevé equations, which first arose in the search for new transcendental functions in the early 1900’s Reference 5Reference 6Reference 19. Again a natural question is to ask whether discrete versions exist with analogous properties. This question led to the discovery of second-order difference equations called the discrete Painlevé equations Reference 7Reference 13Reference 20.

It is now well-known that discrete Painlevé equations have initial value spaces with geometric structures that can be identified with root systems and affine Weyl groups Reference 22. Sakai showed that there are 22 types of initial value spaces as shown in Table 1.1.

Table 1.1.

Types of spaces of initial values

Discrete type	Type of space of initial values
Elliptic	$A_0^{(1)}$
Multiplicative	$A_0^{(1)\ast }$, $A_1^{(1)}$, $A_2^{(1)}$, $A_3^{(1)}$, …, $A_8^{(1)}$, $A_7^{(1)'}$
Additive	$A_0^{(1)\ast \ast }$, $A_1^{(1)\ast }$, $A_2^{(1)\ast }$, $D_4^{(1)}$, …, $D_8^{(1)}$, $E_6^{(1)}$, $E_7^{(1)}$, $E_8^{(1)}$

1.3. Outline of the paper

This paper is organized as follows. In §2, we show the extended affine Weyl group of type $E_6^{(1)}$ and its subgroup which forms that of type $A_2^{(1)}$. Moreover, from those birational actions we obtain the discrete Painlevé equations Equation 1.4 and the P$\Delta$Es Equation 2.16, which are periodically reduced equations of the system Equation 1.2. In §3, using the results in §2 we give the proof of Theorem 1.1. Finally, we give some concluding remarks in §4.

2. Derivation of the discrete integrable systems from an extended affine Weyl group of type $E_6^{(1)}$

In this section, we derive the partial/ordinary discrete integrable systems from the birational actions of an extended affine Weyl group of type $E_6^{(1)}$, denoted by $\widetilde{W}(E_6^{(1)})$. Note that details of $\widetilde{W}(E_6^{(1)})$ are given in Appendix B.

2.1. Extended affine Weyl group of type $A_2^{(1)}$

Let $a_i$, $i=0$, …, $6$, be parameters satisfying the condition

$$\begin{equation} a_1+2a_2+3a_3+2a_4+a_5+2a_6+a_0=1, \cssId{texmlid32}{\tag{2.1}} \end{equation}$$

and $\tau ^{(i)}_j$, $i=1$, $2$, $3$, $j=0$, $1$, $2$, $3$, be variables. Moreover, we define the transformations $s_i$, $i=0$, …, $6$, $\iota _j$, $j=1$, $2$, $3$, by isomorphisms from the field of rational functions $K(\{\tau ^{(i)}_j\})$, where $K=\mathbb{C}(\{a_i\})$, to itself. These transformations collectively form the extended affine Weyl group of type $E_6^{(1)}$, denoted by $\widetilde{W}(E_6^{(1)})$:

$$\begin{equation} \widetilde{W}(E_6^{(1)})=\langle s_0,\dots ,s_6\rangle \rtimes \langle \iota _1,\iota _2,\iota _3\rangle . \cssId{texmlid33}{\tag{2.2}} \end{equation}$$

See Appendix B for more details.

Let us define the transformations $w_i$, $i=0,1,2$, and $\pi$ by

$$\begin{equation} w_0=s_2s_1s_3s_2,\quad w_1=s_4s_5s_3s_4,\quad w_2=s_6s_0s_3s_6,\quad \pi =\iota _3\iota _1. \tag{2.3} \end{equation}$$

They collectively form the extended affine Weyl group of type $A_2^{(1)}$:

$$\begin{equation} \widetilde{W}(A_2^{(1)})=\langle w_0,w_1,w_2\rangle \rtimes \langle \pi \rangle . \tag{2.4} \end{equation}$$

Indeed, the following fundamental relations hold:

$$\begin{equation} (w_iw_j)^{a_{ij}}=1,\quad i,j\in \{0,1,2\},\quad \pi ^3=1,\quad \pi w_{\{0,1,2\}}=w_{\{1,2,0\}}\pi , \tag{2.5} \end{equation}$$

where

$$\begin{equation} (a_{ij})_{i,j=0}^2 =\begin{pmatrix} 2&3&3\\[-8.00003pt] 3&2&3\\[-8.00003pt] 3&3&2 \end{pmatrix}. \tag{2.6} \end{equation}$$

Introduce the parameters and variables that go well with $\widetilde{W}(A_2^{(1)})$ as follows. Let

$$\begin{align} &b_0=a_1+2a_2+a_3,\quad b_1=a_3+2a_4+a_5,\quad b_2=a_3+2a_6+a_0,\tag{2.7a}\\ &c=\dfrac{a_0+a_1+a_3+a_5}{2},\quad d_{12}=\dfrac{a_0+a_1-a_3-a_5}{2},\quad d_{23}=\dfrac{a_0-a_1+a_3-a_5}{2},\tag{2.7b}\\ &d_{13}=\dfrac{a_0-a_1-a_3+a_5}{2}, \tag{2.7c} \end{align}$$

where $b_0+b_1+b_2=1$, and

$$\begin{equation} y_1=\frac{\tau ^{(1)}_1}{\tau ^{(1)}_0},\quad y_2=\frac{\tau ^{(3)}_3}{\tau ^{(3)}_2},\quad y_3=\frac{\tau ^{(2)}_1}{\tau ^{(2)}_0},\quad y_4=\frac{\tau ^{(2)}_3}{\tau ^{(2)}_2},\quad y_5=\frac{\tau ^{(1)}_3}{\tau ^{(1)}_2},\quad y_6=\frac{\tau ^{(3)}_1}{\tau ^{(3)}_0}. \tag{2.8} \end{equation}$$

Then, the actions of $\widetilde{W}(A_2^{(1)})$ on the parameters $b_0$, $b_1$, $b_2$, $c$, $d_{12}$, $d_{23}$, $d_{13}$ are given by

$$\begin{align} &w_i(b_j) =\begin{cases} -b_i&\text{if }\nobreakspace i=j,\\ b_j+b_i&\text{if }\nobreakspace i\neq j, \end{cases}\qquad w_0:(d_{12},d_{23})\mapsto (d_{23},d_{12}),\tag{2.9a}\\ &w_1:(d_{23},d_{13})\mapsto (d_{13},d_{23}),\qquad w_2:(d_{12},d_{13})\mapsto (-d_{13},-d_{12}),\tag{2.9b}\\ &\pi :(b_0,b_1,b_2,d_{12},d_{23},d_{13})\mapsto (b_1,b_2,b_0,-d_{23},-d_{13},d_{12}), \tag{2.9c} \end{align}$$

where $i,j\in \mathbb{Z}/(3\mathbb{Z})$, while those on the $y$-variables $y_i$, $i=1$, …, $6$, are given by

$$\begin{align} &w_0: \begin{pmatrix} y_1,\nobreakspace y_3\\ y_5,\nobreakspace y_6 \end{pmatrix} \mapsto \left(\begin{matrix} y_5,\nobreakspace \dfrac{(b_0-c+d_{13})y_3-(b_0-d_{12}+d_{23})y_1}{(b_0+d_{12}-d_{23})y_3-(b_0+c-d_{13})y_1}y_5\\[15.0pt] y_1,\nobreakspace \dfrac{(b_0-c+d_{13})y_6-(b_0-d_{12}+d_{23})y_1}{(b_0+d_{12}-d_{23})y_6-(b_0+c-d_{13})y_1}y_5 \end{matrix}\right),\tag{2.10a}\\ &w_1: \begin{pmatrix} y_1,\nobreakspace y_3\\ y_4,\nobreakspace y_6 \end{pmatrix} \mapsto \left(\begin{matrix} \dfrac{(b_1-c+d_{12})y_1-(b_1-d_{13}+d_{23})y_3}{(b_1+d_{13}-d_{23})y_1-(b_1+c-d_{12})y_3}y_4,\nobreakspace y_4\\[15.0pt] y_3,\nobreakspace \dfrac{(b_1-c+d_{12})y_6-(b_1-d_{13}+d_{23})y_3}{(b_1+d_{13}-d_{23})y_6-(b_1+c-d_{12})y_3}y_4 \end{matrix}\right),\tag{2.10b}\\ &w_2: \begin{pmatrix} y_1,\nobreakspace y_2\\ y_3,\nobreakspace y_6 \end{pmatrix} \mapsto \left(\begin{matrix} \dfrac{(b_2-c-d_{23})y_1-(b_2-d_{12}-d_{13})y_6}{(b_2+d_{12}+d_{13})y_1-(b_2+c+d_{23})y_6}y_2,\nobreakspace y_6\\[15.0pt] \dfrac{(b_2-d_{12}-d_{13})y_6-(b_2-c-d_{23})y_3}{(b_2+c+d_{23})y_6-(b_2+d_{12}+d_{13})y_3}y_2,\nobreakspace y_2 \end{matrix}\right),\tag{2.10c}\\ &\pi :(y_1,y_2,y_3,y_4,y_5,y_6)\mapsto (y_3,y_5,y_6,y_2,y_4,y_1). \tag{2.10d} \end{align}$$

Remark 2.1.

We follow the convention that the parameters and $y$-variables not explicitly included in the actions listed in Equations Equation 2.9 and Equation 2.10 are the ones that remain unchanged under the action of the corresponding transformation. That is, the transformation acts as an identity on those parameters or variables.

For later convenience, we here define the translations in $\widetilde{W}(A_2^{(1)})$ by

$$\begin{equation} T_1=w_1w_2\pi ^2,\quad T_2=w_2w_0\pi ^2,\quad T_3=w_0w_1\pi ^2, \tag{2.11} \end{equation}$$

whose actions on the parameters $b_0$, $b_1$, $b_2$, $c$, $d_{12}$, $d_{23}$, $d_{13}$ are given by

$$\begin{align} T_1:&(b_0,b_1,d_{12},d_{13})\mapsto (b_0-1,b_1+1,-d_{12},-d_{13}),\tag{2.12a}\\ T_2:&(b_1,b_2,d_{12},d_{23})\mapsto (b_1-1,b_2+1,-d_{12},-d_{23}),\tag{2.12b}\\ T_3:&(b_2,b_0,d_{23},d_{13})\mapsto (b_2-1,b_0+1,-d_{23},-d_{13}). \tag{2.12c} \end{align}$$

Note that $T_1T_2T_3=1$ and $T_iT_j=T_jT_i$, where $i,j=1,2,3$.

2.2. Derivation of the partial difference equations from $\widetilde{W}(A_2^{(1)})$

In this subsection, we derive the P$\Delta$Es Equation 2.16 from the birational action of $\widetilde{W}(A_2^{(1)})$.

Let

$$\begin{equation} u_{l_1,l_2,l_3}={T_1}^{l_1}{T_2}^{l_2}{T_3}^{l_3}(y_2). \tag{2.13} \end{equation}$$

Note that

$$\begin{equation} u_{0,1,1}=y_1,\quad u_{0,0,0}=y_2,\quad u_{1,0,0}=y_3,\quad u_{0,1,0}=y_4,\quad u_{1,1,0}=y_5,\quad u_{1,2,0}=y_6. \tag{2.14} \end{equation}$$

We assign the variable $u_{l_1,l_2,l_3}$ on the vertices $(l_1,l_2,l_3)$ of the triangle lattice

$$\begin{equation} \mathbb{Z}^3/(1,1,1)\coloneq \left\{\left. (l_1,l_2,l_3)\in \mathbb{Z}^3\nobreakspace \right|\nobreakspace l_1+l_2+l_3=0 \right\}. \tag{2.15} \end{equation}$$

Then, we obtain Lemma 2.2.

Lemma 2.2.

On the triangle lattice there are three fundamental relations (essentially two):

$$\begin{equation} \dfrac{u_{\overline{i}}}{u_{\underline{i}}} =\dfrac{\Big (b^{(i)}_{l_i,l_j}-c+(-1)^{l_i+l_j}d_{ij}\Big )u_{\overline{j}}-\Big (b^{(i)}_{l_i,l_j}-(-1)^{l_j+l_k}d_{jk}+(-1)^{l_i+l_k}d_{ik}\Big )u_{\underline{j}}}{\Big (b^{(i)}_{l_i,l_j}+(-1)^{l_j+l_k}d_{jk}-(-1)^{l_i+l_k}d_{ik}\Big )u_{\overline{j}}-\Big (b^{(i)}_{l_i,l_j}+c-(-1)^{l_i+l_j}d_{ij}\Big )u_{\underline{j}}}, \cssId{texmlid24}{\tag{2.16}} \end{equation}$$

where $(i,j,k)=(1,2,3),\,(2,3,1),\,(3,1,2)$ and

$$\begin{equation} b^{(1)}_{l_1,l_2}=b_1+l_1-l_2,\quad b^{(2)}_{l_2,l_3}=b_2+l_2-l_3-1,\quad b^{(0)}_{l_1,l_3}=b_0+l_3-l_1. \tag{2.17} \end{equation}$$

Here, $u=u_{l_1,l_2,l_3}$ and the subscript ${\overline{i}}$ (or, $\underline{i}$ ) for a function $u=u_{l_1,l_2,l_3}$ means $+1$ shift (or, $-1$ shift) in the $l_i$-direction.

Proof.

Equation Equation 2.16 with $(i,j,k)=(1,2,3),\,(2,3,1),\,(3,1,2)$ are respectively obtained from the following actions:

$$\begin{align} &\dfrac{T_1(y_5)}{y_4} =\dfrac{(b_1-c+d_{12}) y_6-(b_1+d_{23}-d_{13}) y_3}{(b_1-d_{23}+d_{13}) y_6-(b_1+c-d_{12}) y_3}, \cssId{texmlid25}{\tag{2.18a}}\\ &\dfrac{T_2(y_4)}{y_2} =\dfrac{(b_2-c-d_{23}) y_1-(b_2-d_{13}-d_{12}) y_6}{(b_2+d_{13}+d_{12}) y_1-(b_2+c+d_{23}) y_6}, \cssId{texmlid26}{\tag{2.18b}}\\ &\dfrac{T_3(y_2)}{y_5} =\dfrac{(b_0-c+d_{13}) y_3-(b_0-d_{12}+d_{23}) y_1}{(b_0+d_{12}-d_{23}) y_3-(b_0+c-d_{13}) y_1}. \cssId{texmlid27}{\tag{2.18c}} \end{align}$$

Moreover, we can easily verify that using Equations Equation 2.16 we can express any $u_{l_1,l_2,l_3}$ on the lattice by the six initial variables $y_i$, $i=1$, …, $6$, and one of the equations Equation 2.16 can be obtained from the other two equations. Therefore, we have completed the proof.

■

Remark 2.3.

Because of the following relations:

$$\begin{align} &w_0(u_{l_1,l_2,l_3})=u_{l_3,l_2,l_1},\quad w_1(u_{l_1,l_2,l_3})=u_{l_2,l_1,l_3},\quad w_2(u_{l_1,l_2,l_3})=u_{l_1+1,l_3+2,l_2},\tag{2.19a}\\ &\pi (u_{l_1,l_2,l_3})=u_{l_3+1,l_1+1,l_2}, \tag{2.19b} \end{align}$$

which follow from

$$\begin{align*} &w_0T_{\{1,2,3\}}=T_{\{3,2,1\}}w_0,\quad w_1T_{\{1,2,3\}}=T_{\{2,1,3\}}w_1,\quad w_2T_{\{1,2,3\}}=T_{\{1,3,2\}}w_2,\\ &\pi T_{\{1,2,3\}}=T_{\{2,3,1\}}\pi ,\quad w_0(u_{0,0,0})=u_{0,0,0},\quad w_1(u_{0,0,0})=u_{0,0,0},\\ &w_2(u_{0,0,0})=u_{1,2,0},\quad \pi (u_{0,0,0})=u_{1,1,0}, \end{align*}$$

the transformation group $\widetilde{W}(A_2^{(1)})$ can be also regarded as a symmetry of the triangle lattice (see Figure 2.1).

2.3. Derivation of the $A_2^{(1)\ast }$-type discrete Painlevé equations from $\widetilde{W}(A_2^{(1)})$

In this subsection, we derive the $A_2^{(1)\ast }$-type discrete Painlevé equations Equation 1.4 from the birational action of $\widetilde{W}(A_2^{(1)})$.

Let

$$\begin{align} &f=\dfrac{(c-d_{12}+d_{23}-d_{13})y_1}{2(y_6-y_1)}+\frac{b_2+c-d_{12}+d_{23}-d_{13}}{4},\tag{2.20a}\\ &g=\dfrac{(c-d_{12}+d_{23}-d_{13})y_3}{2(y_3-y_6)}-\frac{b_2+c-d_{12}+d_{23}-d_{13}}{4}. \tag{2.20b} \end{align}$$

Then, the action of $\widetilde{W}(A_2^{(1)})$ on the variables $f$ and $g$ is given by

$$\begin{align*} &w_0(f)=f-\frac{b_0}{4},\quad w_1(g)=g+\frac{b_1}{4},\quad \pi (g)=f-\frac{b_2+b_0}{4},\\ &\frac{4(c-d_{12}+d_{23}-d_{13})}{4 f-2 b_0-b_2-c-d_{12}+d_{23}+d_{13}} \left(w_0(g)+\frac{b_2+b_0+c+d_{12}-d_{23}-d_{13}}{4}\right)\\ &=\frac{(b_0-d_{12}+d_{23})(4g+b_2-c+d_{12}-d_{23}+d_{13})}{4f-b_2+c-d_{12}+d_{23}-d_{13}}\\ &-\frac{(b_0-c+d_{13})(4g+b_2+c-d_{12}+d_{23}-d_{13})}{4f-b_2-c+d_{12}-d_{23}+d_{13}},\\ &\frac{4(c-d_{12}+d_{23}-d_{13})}{4g+2b_1+b_2+c-d_{12}-d_{23}+d_{13}} \left(w_1(f)-\frac{b_1+b_2+c-d_{12}-d_{23}+d_{13}}{4}\right)\\ &=\frac{(b_1+d_{23}-d_{13})(4f-b_2+c-d_{12}+d_{23}-d_{13})}{4g+b_2-c+d_{12}-d_{23}+d_{13}}\\ &-\frac{(b_1-c+d_{12})(4f-b_2-c+d_{12}-d_{23}+d_{13})}{4g+b_2+c-d_{12}+d_{23}-d_{13}},\\ &\pi (f)=-\frac{(4f-b_2+c-d_{12}+d_{23}-d_{13})(4g+b_2+c-d_{12}+d_{23}-d_{13})}{16(f+g)}\\ &+\frac{b_0+c-d_{12}+d_{23}-d_{13}}{4}. \end{align*}$$

Using the transformation ${T_1}^2$ whose action on the parameter space $\{b_0,b_1,b_2,c,d_{12}, d_{23},d_{13}\}$ is translational as ${T_1}^2:(b_0,b_1)\mapsto (b_0-2,b_1+2)$ shows, we obtain the discrete Painlevé equation Equation 1.4a with the following correspondence:

$$\begin{align} &X=f,\quad Y=g,\quad \overline{X}={T_1}^2(f),\quad \underline{Y}={T_1}^{-2}(g),\quad t=\dfrac{2b_1+b_2-2}{4},\tag{2.21a}\\ &c_1=\dfrac{(b_2+c+d_{23})^2}{16},\quad c_2=\dfrac{(b_2-c-d_{23})^2}{16},\quad c_3=\dfrac{d_{12}+d_{13}}{4},\tag{2.21b}\\ &c_4=\dfrac{(c+d_{12}-d_{23}-d_{13})^2}{16},\quad c_5=\dfrac{(c-d_{12}-d_{23}+d_{13})^2}{16}. \tag{2.21c} \end{align}$$

We can also obtain the discrete Painlevé equations from another operation on the parameter space as follows Reference 12. The action of $T_1$ on the parameter space:

$$\begin{equation*} T_1:(b_0,b_1,d_{12},d_{13})\mapsto (b_0-1,b_1+1,-d_{12},-d_{13}), \end{equation*}$$

is not translational, but when the parameters take the special values $d_{12}=d_{13}=0$, it becomes translational motion on the parameter sub-space $\{b_0,b_1,b_2,c,d_{23}\}$: $T_1:(b_0,b_1)\mapsto (b_0-1,b_1+1)$. Under the specialization of the parameters, the action of $T_1$ gives the discrete Painlevé equation Equation 1.4b with the following correspondence:

$$\begin{align} &X=2f,\quad \overline{X}=T_1(2f),\quad \underline{X}={T_1}^{-1}(2f),\quad t=\dfrac{2b_1+b_2-2}{2},\tag{2.22a}\\ &c_1=\dfrac{(b_2+c+d_{23})^2}{4},\quad c_2=\dfrac{(b_2-c-d_{23})^2}{4},\quad c_3=\dfrac{(c-d_{23})^2}{4}. \tag{2.22b} \end{align}$$

3. Proof of Theorem 1.1

In this section, we give the proof of Theorem 1.1 via the reduction from the system of P$\Delta$Es Equation 1.2 to the system of P$\Delta$Es Equation 2.16.

Lemma 3.1 holds.

Lemma 3.1.

By imposing the $(1,1,1)$-periodic condition: $u({\boldsymbol{l}}+\boldsymbol{\epsilon }_1+\boldsymbol{\epsilon }_2+\boldsymbol{\epsilon }_3)=u({\boldsymbol{l}})$ for ${\boldsymbol{l}}\in \Omega$, the system Equation 1.2 can be reduced to the following system of P$\Delta$Es:

$$\begin{equation} \dfrac{u_{\overline{i}}}{u_{\underline{i}}} =\dfrac{\Big (\alpha _{ij}-c+(-1)^{l_i+l_j}\delta _i\Big )u_{\overline{j}}-\Big (\alpha _{ij}-(-1)^{l_j+l_k}\delta _j+(-1)^{l_i+l_k}\delta _k\Big )u_{\underline{j}}}{\Big (\alpha _{ij}+(-1)^{l_j+l_k}\delta _j-(-1)^{l_i+l_k}\delta _k\Big )u_{\overline{j}}-\Big (\alpha _{ij}+c-(-1)^{l_i+l_j}\delta _i\Big )u_{\underline{j}}}, \cssId{texmlid30}{\tag{3.1}} \end{equation}$$

where $(i,j,k)=(1,2,3),\,(2,3,1),\,(3,1,2)$, $u=u({\boldsymbol{l}})$ and ${\boldsymbol{l}}=\sum _{i=1}^3l_i\boldsymbol{\epsilon }_i\in \mathbb{Z}^3/(\boldsymbol{\epsilon }_1+\boldsymbol{\epsilon }_2+\boldsymbol{\epsilon }_3)$.

Proof.

Applying the $(1,1,1)$-periodic condition to the system Equation 1.2, we obtain Equation Equation 3.1 with $(i,j,k)=(1,2,3)$, $(2,3,1)$ and $(3,1,2)$ from Equation Equation 1.2 with $(i,j,k)=(1,2,3)$, $(2,3,1)$ and $(3,1,2)$, respectively. Therefore, we have completed the proof.

■

Remark 3.2.

(i): The number of essential equations in the system Equation 3.1 is two.
(ii): By the $(1,1,1)$-reduction, each cuboctahedron is reduced to a hexagram (see Figure 3.1), which causes the reduction from the face-centred cubic lattice $\Omega$ to the triangle lattice $\mathbb{Z}^3/(\boldsymbol{\epsilon }_1+\boldsymbol{\epsilon }_2+\boldsymbol{\epsilon }_3)$.

Lemma 3.3.

The reduced system Equation 3.1 is equivalent to equations in the system Equation 2.16.

Proof.

The statement follows from the following correspondences:

$$\begin{align*} &b^{(1)}_{l_1,l_2}=\alpha _{12},\quad b^{(2)}_{l_2,l_3}=\alpha _{23},\quad b^{(0)}_{l_1,l_3}=\alpha _{31},\quad d_{12}=\delta _1,\quad d_{23}=\delta _2,\quad d_{13}=\delta _3,\\ &u_{l_1,l_2,l_3}=u(l_1\boldsymbol{\epsilon }_1+l_2\boldsymbol{\epsilon }_2+l_3\boldsymbol{\epsilon }_3). \end{align*}$$ ■

Remark 3.4.

Lemma 3.3 means that the reduced system Equation 3.1 can be obtained from the theory of the $\tau$-function associated with $A_2^{(1)\ast }$-type discrete Painlevé equations.

We are now ready to prove Theorem 1.1. The $(1,1,1)$-periodic reduction from the system Equation 1.2 to the system Equation 3.1 given in Lemma 3.1, the relation between the system Equation 3.1 and the system Equation 2.16 given in Lemma 3.3, and that between the system Equation 2.16 and the $A_2^{(1)\ast }$-type discrete Painlevé equations Equation 1.4 given in §2.2 and §2.3 collectively give the proof of Theorem 1.1.

4. Concluding remarks

In this paper, we considered a reduction of a system of P$\Delta$Es, which is unusual in the sense that it has the CACO property but not the widely studied CAC property. We showed how the system Equation 1.2 can be reduced to the $A_2^{(1)\ast }$-type discrete Painlevé equations Equation 1.4 using the affine Weyl group associated with the discrete Painlevé equations.

In a forthcoming paper (N. Joshi and N. Nakazono), we will show how another system of P$\Delta$Es, which also has the CACO property, can be reduced to the $A_2^{(1)}$-type discrete Painlevé equations (see Table 1.1 for the distinction between $A_2^{(1)}$ and $A_2^{(1)\ast }$).

Appendix A. Consistency around a cuboctahedron property

In this appendix, we recall the definition of consistency around a cuboctahedron. To define it, we also introduce an additional important property called consistency around an octahedron. We refer the reader to Reference 10 for detailed information about these properties.

A.1. Consistency around an octahedron property

In this subsection, we give a definition of a consistency around an octahedron.

Let $u_i$, $i=1$, …, $6$, be variables and consider the octahedron shown in Figure A.1. The planes that pass through the vertices $\{u_4,u_2,u_1,u_5\}$, $\{u_2,u_6,u_5,u_3\}$ and $\{u_6,u_4,u_3,u_1\}$ give 3 quadrilaterals that lie in the interior of the octahedron and we assign the quad-equations $Q_i$, $i=1,2,3$, to the quadrilaterals as the following:

$$\begin{equation} Q_1\left(u_4,u_2,u_1,u_5\right)=0,\quad Q_2\left(u_2,u_6,u_5,u_3\right)=0,\quad Q_3\left(u_6,u_4,u_3,u_1\right)=0. \cssId{texmlid34}{\tag{A.1}} \end{equation}$$

The consistency around an octahedron property is defined by the following.

Definition A.1 (CAO property Reference 10).

The octahedron with quad-equations $\{Q_1, Q_2, Q_3\}$ is said to have a consistency around an octahedron (CAO) property if each quad-equation can be obtained from the other two equations. An octahedron is said to be a CAO octahedron if it has the CAO property.

A.2. Consistency around a cuboctahedron property

In this subsection, we give a definition of consistency around a cuboctahedron.

We consider the cuboctahedron centered around the origin whose twelve vertices are given by $V=\left\{\left. \pm \boldsymbol{\epsilon }_i\pm \boldsymbol{\epsilon }_j\nobreakspace \right|\nobreakspace i,j\in \mathbb{Z},\nobreakspace 1\leq i<j\leq 3 \right\}$, where $\{\boldsymbol{\epsilon }_1,\boldsymbol{\epsilon }_2,\boldsymbol{\epsilon }_3\}$ form the standard basis of $\mathbb{R}^3$. We assign the variables $u({\boldsymbol{l}})$ to the vertices ${\boldsymbol{l}}\in V$ and impose the following relations:

$$\begin{align} &Q_1\left(u_5,u_1,v_5,v_4\right)=0,\quad Q_2\left(v_2,v_1,u_2,u_4\right)=0,\quad Q_3\left(u_3,u_5,v_3,v_2\right)=0, \cssId{texmlid35}{\tag{A.2a}}\\ &Q_4\left(v_6,v_5,u_6,u_2\right)=0,\quad Q_5\left(u_1,u_3,v_1,v_6\right)=0,\quad Q_6\left(v_4,v_3,u_4,u_6\right)=0, \cssId{texmlid36}{\tag{A.2b}}\\ &Q_7\left(u_4,u_2,u_1,u_5\right)=0,\quad Q_8\left(u_2,u_6,u_5,u_3\right)=0,\quad Q_9\left(u_6,u_4,u_3,u_1\right)=0, \cssId{texmlid37}{\tag{A.2c}} \end{align}$$

where $Q_i$, $i=1$, …, $9$, are quad-equations and

$$\begin{align} &u_1=u(\boldsymbol{\epsilon }_2+\boldsymbol{\epsilon }_3), &&u_2=u(-\boldsymbol{\epsilon }_1-\boldsymbol{\epsilon }_3), &&u_3=u(\boldsymbol{\epsilon }_1+\boldsymbol{\epsilon }_2), &&u_4=u(-\boldsymbol{\epsilon }_2-\boldsymbol{\epsilon }_3),\tag{A.3a}\\ &u_5=u(\boldsymbol{\epsilon }_1+\boldsymbol{\epsilon }_3), &&u_6=u(-\boldsymbol{\epsilon }_1-\boldsymbol{\epsilon }_2), &&v_1=u(\boldsymbol{\epsilon }_2-\boldsymbol{\epsilon }_3), &&v_2=u(\boldsymbol{\epsilon }_1-\boldsymbol{\epsilon }_3),\tag{A.3b}\\ &v_3=u(\boldsymbol{\epsilon }_1-\boldsymbol{\epsilon }_2), &&v_4=u(-\boldsymbol{\epsilon }_2+\boldsymbol{\epsilon }_3), &&v_5=u(-\boldsymbol{\epsilon }_1+\boldsymbol{\epsilon }_3), &&v_6=u(-\boldsymbol{\epsilon }_1+\boldsymbol{\epsilon }_2). \tag{A.3c} \end{align}$$

Note that quad-equations $Q_i$, $i=1$, …, $6$, are assigned to the faces of the cuboctahedron (see Figure A.2a). Moreover, $u_i$, $i=1$, …, $6$, collectively form the vertices of an octahedron and quad-equations $Q_i$, $i=7$, $8$, $9$, are assigned to the quadrilaterals that appear as sections passing through four vertices of the octahedron (see Figure A.2b).

We are now in a position to give the following definitions.

Definition A.2 (CACO property Reference 10).

The cuboctahedron with quad-equations $\{Q_1,\dots ,Q_9\}$ is said to have a consistency around a cuboctahedron (CACO) property if the following properties hold.

(i): The octahedron with quad-equations $\{Q_7,Q_8,Q_9\}$ has the CAO property.
(ii): Assume that $u_1$, …, $u_6$ are given so as to satisfy $Q_i=0$, $i=7$, $8$, $9$, and, in addition, $v_k$ is given, for some $k\in \{1,\dots ,6\}$. Then, quad-equations $Q_i$, $i=1$, …, $6$, determine the variables $v_j$, $j\in \{1,\dots ,6\}\backslash \{k\}$, uniquely.

A cuboctahedron is said to be a CACO cuboctahedron if it has the CACO property.

Definition A.3 (Square property Reference 10).

The CACO cuboctahedron with quad-equations $\{Q_1,\dots ,Q_9\}$ is said to have a square property if there exist polynomials $K_i=K_i(x,y,z,w)$, $i=1,2,3$, where $\deg _x{K_i}=\deg _w{K_i}=1$ and $1\leq \deg _y{K_i},\,\deg _z{K_i}$, satisfying

$$\begin{equation} K_1(v_1,u_1,u_4,v_4)=0,\quad K_2(v_2,u_2,u_5,v_5)=0,\quad K_3(v_3,u_3,u_6,v_6)=0. \tag{A.4} \end{equation}$$

Then, each equation $K_i=0$ is called a square equation.

A.3. CACO property of P$\Delta$Es

We now explain how to associate quad-equations with P$\Delta$Es in three-dimensional space by using the system of P$\Delta$Es Equation 1.2 as an example. This requires us to consider overlapping cuboctahedra that lead to two-dimensional tessellations consisting of quadrilaterals. For each given cuboctahedron, there are twelve overlapping cuboctahedra.

The twelve overlapping cuboctahedra around a given one provide six directions of tiling by quadrilaterals. For later convenience, we label directions by $\boldsymbol{\epsilon }_i\pm \boldsymbol{\epsilon }_j$, $1\leq i<j\leq 3$. Vertices labelled in this way form the set $\Omega$ given by Equation 1.1. Such vertices are interpreted as being iterated on each successive cuboctahedron. We here consider the system of P$\Delta$Es Equation 1.2. For simplicity, we abbreviate each respective equation in Equations Equation 1.2 as

$$\begin{align*} &P_1\left(u_{\overline{13}},u_{\overline{23}},u_{\underline{1}\overline{3}},u_{\underline{2}\overline{3}}\right)=0,\nobreakspace P_2\left(u_{\overline{12}},u_{\overline{13}},u_{\overline{1}\underline{2}},u_{\overline{1}\underline{3}}\right)=0,\nobreakspace P_3\left(u_{\overline{23}},u_{\overline{12}},u_{\overline{2}\underline{3}},u_{\underline{1}\overline{2}}\right)=0,\\ &P_4\left(u_{\underline{23}},u_{\underline{13}},u_{\overline{23}},u_{\overline{13}}\right)=0,\nobreakspace P_5\left(u_{\underline{13}},u_{\underline{12}},u_{\overline{13}},u_{\overline{12}}\right)=0,\nobreakspace P_6\left(u_{\underline{12}},u_{\underline{23}},u_{\overline{12}},u_{\overline{23}}\right)=0. \end{align*}$$

Conversely, given ${\boldsymbol{l}}\in \Omega$, we obtain the cuboctahedron centered around ${\boldsymbol{l}}$. We refer to its quad-equations as before by $\{Q_1({\boldsymbol{l}}),\dots ,Q_9({\boldsymbol{l}})\}$. Moreover, the overlapped region gives an octahedron centred around ${\boldsymbol{l}}+\boldsymbol{\epsilon }_3$, and we label its quad-equations by $\{\hat{Q}_1({\boldsymbol{l}}),\hat{Q}_2({\boldsymbol{l}}),\hat{Q}_3({\boldsymbol{l}})\}$.

Each such quad-equation is identified with the 6 partial difference equations given in Equations Equation 1.2 in the following way. For $Q_1$, …, $Q_9$, we use

$$\begin{align*} &Q_1({\boldsymbol{l}})=P_1\left(u_{\overline{13}},u_{\overline{23}},u_{\underline{1}\overline{3}},u_{\underline{2}\overline{3}}\right)=0,\quad Q_2({\boldsymbol{l}})=P_1\left(u_{\overline{1}\underline{3}},u_{\overline{2}\underline{3}},u_{\underline{13}},u_{\underline{23}}\right)=0,\\ &Q_3({\boldsymbol{l}})=P_2\left(u_{\overline{12}},u_{\overline{13}},u_{\overline{1}\underline{2}},u_{\overline{1}\underline{3}}\right)=0,\quad Q_4({\boldsymbol{l}})=P_2\left(u_{\underline{1}\overline{2}},u_{\underline{1}\overline{3}},u_{\underline{12}},u_{\underline{13}}\right)=0,\\ &Q_5({\boldsymbol{l}})=P_3\left(u_{\overline{23}},u_{\overline{12}},u_{\overline{2}\underline{3}},u_{\underline{1}\overline{2}}\right)=0,\quad Q_6({\boldsymbol{l}})=P_3\left(u_{\underline{2}\overline{3}},u_{\overline{1}\underline{2}},u_{\underline{23}},u_{\underline{12}}\right)=0,\\ &Q_7({\boldsymbol{l}})=P_4\left(u_{\underline{23}},u_{\underline{13}},u_{\overline{23}},u_{\overline{13}}\right)=0,\quad Q_8({\boldsymbol{l}})=P_5\left(u_{\underline{13}},u_{\underline{12}},u_{\overline{13}},u_{\overline{12}}\right)=0,\\ &Q_9({\boldsymbol{l}})=P_6\left(u_{\underline{12}},u_{\underline{23}},u_{\overline{12}},u_{\overline{23}}\right)=0, \end{align*}$$

and for $\hat{Q}_1,\hat{Q}_2,\hat{Q}_3$, we use

$$\begin{align*} &\hat{Q}_1({\boldsymbol{l}})=P_1\left(u_{\overline{13}},u_{\overline{23}},u_{\underline{1}\overline{3}},u_{\underline{2}\overline{3}}\right)=0,\quad \hat{Q}_2({\boldsymbol{l}})=P_2\left(u_{\overline{23}},u_{\overline{33}},u_{\underline{2}\overline{3}},u\right)=0,\\ &\hat{Q}_3({\boldsymbol{l}})=P_3\left(u_{\overline{33}},u_{\overline{13}},u,u_{\underline{1}\overline{3}}\right)=0. \end{align*}$$

Then, Proposition A.4 holds.

Proposition A.4 (Reference 10).

The system of P$\Delta$Es Equation 1.2 has the CACO and square properties, that is, the following statements hold.

(i): The cuboctahedra with quad-equations $\{Q_i({\boldsymbol{l}})\}$ have the CACO and square properties.
(ii): The square equations are consistent with the P$\Delta$Es Equation 1.2.
(iii): The octahedra with quad-equations $\{\hat{Q}_i({\boldsymbol{l}})\}$ have the CAO property.

Appendix B. Extended affine Weyl group of type $E_6^{(1)}$ and $\tau$-variables

In this appendix, we review the action of the extended affine Weyl group of type $E_6^{(1)}$ given in Reference 23, which is the symmetry group of $A_2^{(1)\ast }$-type discrete Painlevé equations.

Let $a_i$, $i=0$, …, $6$, be parameters satisfying the condition Equation 2.1 and $\tau ^{(i)}_j$, $i=1$, $2$, $3$, $j=0$, $1$, $2$, $3$, be variables. The actions of transformations $s_i$, $i=0$, …, $6$, and $\iota _j$, $j=1$, $2$, $3$, on the parameters are given by

$$\begin{align*} &s_0:(a_0,a_6)\mapsto (-a_0,a_6+a_0),\qquad s_1:(a_1,a_2)\mapsto (-a_1,a_2+a_1),\\ &s_2:(a_1,a_2,a_3)\mapsto (a_1+a_2,-a_2,a_3+a_2),\\ &s_3:(a_2,a_3,a_4,a_6)\mapsto (a_2+a_3,-a_3,a_4+a_3,a_6+a_3),\\ &s_4:(a_3,a_4,a_5)\mapsto (a_3+a_4,-a_4,a_5+a_4),\qquad s_5:(a_4,a_5)\mapsto (a_4+a_5,-a_5),\\ &s_6:(a_0,a_3,a_6)\mapsto (a_0+a_6,a_3+a_6,-a_6),\qquad \\ &\iota _1 a_{\{0,5,4,6\}}\mapsto a_{\{5,0,4,6\}},\quad \iota _2 a_{\{0,1,2,6\}}\mapsto a_{\{1,0,6,2\}},\quad \iota _3 a_{\{1,5,2,4\}}\mapsto a_{\{5,1,4,2\}}, \end{align*}$$

while those on the $\tau$-variables $\tau ^{(i)}_j$, $i=1$, $2$, $3$, $j=0$, $1$, $2$, $3$, are given by

$$\begin{align*} s_0&:(\tau ^{(3)}_2,\tau ^{(3)}_3)\mapsto (\tau ^{(3)}_3,\tau ^{(3)}_2),\qquad s_1:(\tau ^{(1)}_2,\tau ^{(1)}_3)\mapsto (\tau ^{(1)}_3,\tau ^{(1)}_2),\\ s_2&:(\tau ^{(1)}_1,\tau ^{(1)}_2)\mapsto (\tau ^{(1)}_2,\tau ^{(1)}_1),\\ &:(\tau ^{(2)}_0,\tau ^{(3)}_0)\mapsto \left(\frac{(a_2+a_3) \tau ^{(1)}_1 \tau ^{(2)}_0-a_2 \tau ^{(2)}_1 \tau ^{(1)}_0}{a_3 \tau ^{(1)}_2}, \frac{(a_2+a_3) \tau ^{(1)}_1 \tau ^{(3)}_0-a_2 \tau ^{(3)}_1 \tau ^{(1)}_0}{a_3 \tau ^{(1)}_2}\right),\\ s_3&:(\tau ^{(1)}_1,\tau ^{(2)}_1,\tau ^{(3)}_1,\tau ^{(1)}_0,\tau ^{(2)}_0,\tau ^{(3)}_0)\mapsto (\tau ^{(1)}_0,\tau ^{(2)}_0,\tau ^{(3)}_0,\tau ^{(1)}_1,\tau ^{(2)}_1,\tau ^{(3)}_1),\\ s_4&:(\tau ^{(2)}_1,\tau ^{(2)}_2)\mapsto (\tau ^{(2)}_2,\tau ^{(2)}_1),\\ &:(\tau ^{(1)}_0,\tau ^{(3)}_0)\mapsto \left(\frac{(a_3+a_4) \tau ^{(2)}_1 \tau ^{(1)}_0-a_4 \tau ^{(1)}_1 \tau ^{(2)}_0}{a_3 \tau ^{(2)}_2}, \frac{(a_3+a_4) \tau ^{(2)}_1 \tau ^{(3)}_0-a_4 \tau ^{(3)}_1 \tau ^{(2)}_0}{a_3 \tau ^{(2)}_2}\right),\\ s_5&:(\tau ^{(2)}_2,\tau ^{(2)}_3)\mapsto (\tau ^{(2)}_3,\tau ^{(2)}_2),\\ s_6&:(\tau ^{(3)}_1,\tau ^{(3)}_2)\mapsto (\tau ^{(3)}_2,\tau ^{(3)}_1),\\ &:(\tau ^{(1)}_0,\tau ^{(2)}_0)\mapsto \left(\frac{(a_3+a_6) \tau ^{(3)}_1 \tau ^{(1)}_0-a_6 \tau ^{(1)}_1 \tau ^{(3)}_0}{a_3 \tau ^{(3)}_2}, \frac{(a_3+a_6) \tau ^{(3)}_1 \tau ^{(2)}_0-a_6 \tau ^{(2)}_1 \tau ^{(3)}_0}{a_3 \tau ^{(3)}_2}\right),\\ \iota _1&:(\tau ^{(2)}_j,\tau ^{(3)}_j)\mapsto (\tau ^{(3)}_j,\tau ^{(2)}_j),\qquad \iota _2:(\tau ^{(1)}_j,\tau ^{(3)}_j)\mapsto (\tau ^{(3)}_j,\tau ^{(1)}_j),\\ \iota _3&:(\tau ^{(1)}_j,\tau ^{(2)}_j)\mapsto (\tau ^{(2)}_j,\tau ^{(1)}_j),\qquad j=0,1,2,3. \end{align*}$$

Remark B.1.

(i): Each transformation here defined is an isomorphism from the field of rational functions $K(\{\tau ^{(i)}_j\})$, where $K=\mathbb{C}(\{a_i\})$, to itself.
(ii): We follow the convention of Remark 2.1 for the above equations. That is, each transformation acts as an identity on parameters or variables not appearing in its definition.

The transformations collectively form the extended affine Weyl group of type $E_6^{(1)}$, denoted by Equation 2.2. Indeed, the following fundamental relations hold:

$$\begin{align} &(s_is_j)^{A_{ij}}=1,\quad {\iota _1}^2={\iota _2}^2={\iota _3}^2=1,\quad \iota _1\iota _2=\iota _2\iota _3=\iota _3\iota _1,\quad \iota _2\iota _1=\iota _3\iota _2=\iota _1\iota _3,\tag{B.1a}\\ &\iota _1 s_{\{0,5,4,6\}}= s_{\{5,0,4,6\}}\iota _1,\quad \iota _2 s_{\{0,1,2,6\}}= s_{\{1,0,6,2\}}\iota _2,\quad \iota _3 s_{\{1,5,2,4\}}= s_{\{5,1,4,2\}}\iota _3, \tag{B.1b} \end{align}$$

where $i,j\in \{0,1,\dots ,6\}$ and

$$\begin{equation} (A_{ij})_{i,j=0}^6 =\begin{pmatrix} 2&0&0&0&0&0&3\\[-8.00003pt] 0&2&3&0&0&0&0\\[-8.00003pt] 0&3&2&3&0&0&0\\[-8.00003pt] 0&0&3&2&3&0&3\\[-8.00003pt] 0&0&0&3&2&3&0\\[-8.00003pt] 0&0&0&0&3&2&0\\[-8.00003pt] 3&0&0&3&0&0&2 \end{pmatrix}. \tag{B.2} \end{equation}$$

Remark B.2.

The correspondence between the notations in this paper and those in Reference 23 is given by $\tau ^{(i)}_j\to \tau ^{i}_j$ and $\tau ^{(i)}_0\to s_3(\tau ^i_1)$, where $i,j=1,2,3$.

Acknowledgments

The second author would like to thank Profs M. Noumi, Y. Ohta and Y. Yamada for inspiring and fruitful discussions.

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

Abstract

1. Introduction

1.1. Notation and definitions

1.2. Background

1.3. Outline of the paper

2. Derivation of the discrete integrable systems from an extended affine Weyl group of type $E_6^{(1)}$

2.1. Extended affine Weyl group of type $A_2^{(1)}$

2.2. Derivation of the partial difference equations from $\widetilde{W}(A_2^{(1)})$

2.3. Derivation of the $A_2^{(1)\ast }$-type discrete Painlevé equations from $\widetilde{W}(A_2^{(1)})$

3. Proof of Theorem 1.1

4. Concluding remarks

Appendix A. Consistency around a cuboctahedron property

A.1. Consistency around an octahedron property

A.2. Consistency around a cuboctahedron property

A.3. CACO property of P$\Delta$Es

Appendix B. Extended affine Weyl group of type $E_6^{(1)}$ and $\tau$-variables

Acknowledgments

Table of Contents

Figures

Mathematical Fragments

References

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