Geometric description of a discrete power function associated with the sixth Painlevé equation

1. Introduction

The cross-ratio equation is a key discrete integrable equation, which connects discrete integrable systems and discrete differential geometry. One of its powerful roles is to provide a discrete analogue of the Cauchy–Riemann relation, thereby leading to a discrete theory of complex analytic functions. In this paper, we focus on the following system of partial difference equations:

and where $(n,m)\in {\mathbb{Z}}^2$ are independent variables, z is a dependent variable and x, a_i, i=0,1,2,4, are parameters. Equation (1.1) is a special case of the cross-ratio equation and equation (1.2) is its similarity constraint [1]. The purpose of this paper is to provide a full geometric description of the system of equations (1.1) and (1.2) and its expression in terms of τ-variables.

Nijhoff et al. [1] showed that equations (1.1) and (1.2) can be regarded as a part of the Bäcklund transformations of the Painlevé VI equation (P_VI). In this context, x is identified as the independent variable of P_VI, and a_i, i=0,…,4 (a₃ does not appear in the equations) are parameters corresponding to the simple roots of the affine root system of type $D_4^{(1)}$ that relates to the symmetry group of P_VI [2–5]. However, the complete characterization of equations (1.1) and (1.2) in relation to the Bäcklund transformations of P_VI remains unclear.

The solution of equations (1.1) and (1.2) for $(n,m)\in {\mathbb{Z}}_{+}^2$ with the initial condition

where $\mathrm{i}=\sqrt{-1}$ and r=a₀+a₂+a₄, gives rise to a discrete power function [6–8] when the parameters take special values: (There is more than one definition of what constitutes a discrete power function. See Boole [9] for classical definitions.) Note that each of the ratios on the right-hand side of (1.2) is proportional to a harmonic mean of the forward and backward differences of z_n,m, so that the continuum limit of equation (1.2) is which is an equation satisfied by the power function F(Z)=Z^2r, where $Z=X+\mathrm{i}\hspace{0.17em}Y\in \mathbb{C}$. Equations (1.1) and (1.2) also appear as conditions for consistency of quadrilaterals on surfaces and enable a definition of discrete conformality [8,10]. It was shown in [11] that the special cases z_n,0, z_n,1 are given, respectively, by the Gamma function and Gauss hypergeometric functions. By applying the Bäcklund transformations, Ando et al. [12] and Hay et al. [13] found explicit formulae for the discrete power function in terms of the hypergeometric τ functions of P_VI as stated in theorem 1.1.

Theorem 1.1 [12,13].

Define the function${\tau }_{\nu }(a,b,c;t)(c\notin \mathbb{Z},\nu \in {\mathbb{Z}}_{+})$by

whereHere, F(a,b,c;x) is the Gauss hypergeometric function, Γ(x) is the Gamma function, and C₀and C₁are arbitrary constants. Then, for$(n,m)\in {\mathbb{Z}}_{+}^2$, z_n,msatisfying equations (1.1) and (1.2) withand the initial conditionwhere r=a₀+a₂+a₄is given as follows.

• (1) Case where n≤m.

  $$z_{n,m}=\{\begin{array}{ll}{C}_1{x}^{r-n}N\frac{{(r+1)}_{N-1}}{{(-r+1)}_N}\frac{{\tau }_n(-N,-r-N+1,-r;x)}{{\tau }_n(-N+1,-r-N+2,-r+2;x)},& n+m\text{ is even},\\ C_1{x}^{r-n}\frac{{(r+1)}_{M-1}}{{(-r+1)}_{M-1}}\frac{{\tau }_n(-M+1,-r-M+1,-r;x)}{{\tau }_n(-M+2,-r-M+2,-r+2;x)},& n+m\text{ is odd}.\end{array}$$1.10

• (2) Case where n≥m.

  $$z_{n,m}=\{\begin{array}{ll}{C}_{0}N\frac{{(r+1)}_{N-1}}{{(-r+1)}_N}\frac{{\tau }_m(-N+2,-r-N+1,-r+2;x)}{{\tau }_m(-N+1,-r-N+2,-r+2;x)},& n+m\text{ is even},\\ C_0\frac{{(r+1)}_{M-1}}{{(-r+1)}_{M-1}}\frac{{\tau }_m(-M+2,-r-M+1,-r+1;x)}{{\tau }_m(-M+1,-r-M+2,-r+1;x)},& n+m\text{ is odd}.\end{array}$$1.11

Here, N=(n+m)/2, M=(n+m+1)/2 and (u)_j=u(u+1)⋯(u+j−1) is the Pochhammer symbol.

We remark that, in the case of generic values of parameters a_i, the explicit expressions of z_n,m in terms of τ-variables are given in [13]. This result was obtained by a heuristic approach and verified by the bilinear formulation of τ-variables. However, the appearance of the odd–even structure in the explicit formulae has not yet been fully understood. In §§4 and 5, we will see that this odd–even structure can be explained by the notion of projective reduction and the Weyl group symmetry on τ-variables.

In this paper, we characterize equations (1.1) and (1.2) in relation to the Bäcklund transformations of P_VI completely, and clarify the odd–even structure appearing in equations (1.10) and (1.11). The starting point is the appearance of the cross-ratio equation in a cubic lattice, as described in the ABS theory [14–18]. We show how the cubic lattice is embedded in the lattice on which τ-variables of P_VI are assigned. Four copies of the weight lattice of D₄ will be needed to describe the lattice of τ-variables (see §3).

In this paper, we refer to several lattices associated with affine reflection groups. A detailed description of (and notations for) these lattices is provided in the following section.

(a) Background and notation

For completeness, we explain the notation used to describe reflection groups, Coxeter groups and associated lattices in this paper. The notation is illustrated here for reflection groups arising from symmetries of cubes, but it applies to reflection groups of any dimension and type.

A cube is left unchanged by certain reflections across hyperplanes. All such reflections can be expressed in terms of basic operations on vectors, which form a group, denoted by B₃ [19]. In higher dimensions, we would have the reflection group B_n, n≥1, where the subscript refers to an integer number of dimensions. The group is generated by simple reflections, which we denote s_j, j=1,…,n. The simple roots associated with these reflections are denoted α_j, j=1,…,n, while corresponding co-roots are denoted ${\alpha }_j^{\vee }$.

Repeated translation of a fundamental cube leads to a space-filling cubic lattice, on which equations (1.1) and (1.2) are iterated. Reflections across hyperplanes in the cubic lattice form an affine reflection group, which is denoted by $B_n^{(1)}$. We will refer also to the root lattice denoted by $Q(B_n^{(1)})={\oplus }_{j=0}^n\mathbb{Z}{\alpha }_j$. When interpreted as a Coxeter group, the reflection group is denoted by $W(B_n^{(1)})$.

In a Dynkin diagram, the simple reflections s_j are represented by nodes. Obviously, such a diagram remains unchanged when the nodes are permuted. Such an operation is called a diagram automorphism. Let Ω be the group of such automorphisms. Incorporating these automorphisms into the space of allowable operations, we get a lift of the Coxeter group $W(B_n^{(1)})$ to an extended Coxeter group denoted $\tilde{W}(B_n^{(1)})=W(B_n^{(1)})⋊\mathit{\Omega }$ [20].

The classification of finite reflection groups leads to several fundamental types. We will encounter A_n, B_n and D_n in this paper, along with their corresponding lattices, affine Coxeter groups and extended Coxeter groups. Where direct sums of the same type of groups occur, we use an integer coefficient as an abbreviation, e.g. A₁⊕A₁⊕A₁⊕A₁=:4A₁. For conciseness, we replace ⊕ by +, e.g. D₄⊕A₁⊕A₁=D₄+2A₁.

(b) Main result and plan of the paper

The main contribution of this paper, given in theorem 4.1, is to clarify the relation of the discrete power function to Bäcklund transformations of the P_VI by including the system of defining difference equations into a lattice with $\tilde{W}(3A_1^{(1)})$ symmetry. In general, a cubic lattice has $W(B_3^{(1)})$ as a symmetry group, but we chose to consider the symmetry group as $\tilde{W}(3A_1^{(1)})$ by taking different selections of reflection hyperplanes. This idea allows us to realize $\tilde{W}(3A_1^{(1)})$ as a subgroup of $\tilde{W}(D_4^{(1)})$, which is the symmetry group of P_VI. To achieve this realization, we use the combination of the cross-ratio equation together with its similarity relation, which is associated with the symmetry of the cubic lattice.

The plan of this paper is as follows. In §2, we introduce a multi-dimensionally consistent cubic lattice with the symmetry $\tilde{W}(3A_1^{(1)})$ on which the cross-ratio equation is naturally defined. In §3, we give actions of $\tilde{W}(D_4^{(1)})$ on the parameters and the τ-variables. Based on this material, we construct a realization of $\tilde{W}(3A_1^{(1)})$ in $\tilde{W}(D_4^{(1)})$ and derive equations (1.1) and (1.2) in §4. We then explain in §5 the odd–even structures appearing in the explicit formula in [12,13]. Concluding remarks will be given in §6.

2. Cubic lattice

In this section, we describe a multi-dimensionally consistent cubic lattice in terms of a symmetry structure, which will turn out to be related to the symmetry group of P_VI.

While each cube inside the lattice is composed of six component faces, we will regard these as composed of two triple faces, i.e. a pair of three faces around a common vertex. We will see in §4 how this perspective is embedded in the symmetry group lattice of P_VI. The symmetry group $\tilde{W}(3A_1^{(1)})$ is suggested by the similarity equation

Here, Indeed, the similarity equation has three parameters, β₁, γ₁ and ζ₀, and two directions, ρ₁ and ρ₂, defined by $z_{l_1,l_2}={{\rho }_1}^{l_1}{{\rho }_2}^{l_2}(z_{0,0})$, which shift the parameters as follows: It is, therefore, natural to expect that there exists a Bäcklund transformation of the similarity equation, which shifts the parameters as follows:

This implies that the three directions ρ₀,ρ₁,ρ₂ are translations in $\tilde{W}(3A_1^{(1)})$ and {β₁,γ₁,ζ₀} are the parameters associated with the simple roots (or co-roots) of type $3A_1^{(1)}$. Consequently, we take the symmetry group of the cubic lattice to be $\tilde{W}(3A_1^{(1)})$.

Consider the cubic lattice constructed by the directions ρ_i, i=0,1,2. We place the following equations on each respective face of a triple of faces associated with each cube (referred to as the face equations):

where ${\{\dots ,{\kappa }_{-1}^{(i)},{\kappa }_0^{(i)},{\kappa }_1^{(i)},\dots \}}_{i=1,2,3}$ are parameters and It is well known that this system of equations is multi-dimensionally consistent in the cubic lattice. Indeed, equation (2.5a) and equations (2.5b) and (2.5c) are called Q1 and H1, respectively, in the Adler–Bobenko–Suris (ABS) list (figure 1) [14–18].

• Download figure
• Open in new tab
• Download PowerPoint

Using the geometric theory of P_VI, we can realize the action of $\tilde{W}(3A_1^{(1)})$ associated with these equations (see §4).

3. Actions of affine Weyl group $\tilde{W}(D_4^{(1)})$

In this section, we provide the basic ingredients needed to describe actions of the symmetry group of P_VI, i.e. $\tilde{W}(D_4^{(1)})$. The explicit formulation of the τ function of P_VI in terms of the weight lattice is provided here; to our knowledge, such an explicit formulation has not appeared in the literature. More precisely, four copies of the weight lattice of D₄ will be needed to define the lattice of τ-variables. We will consider a root system of type D₄+2A₁ instead of D₄ to give a full description.

(a) Linear action

In this subsection, we define the transformation group $\tilde{W}(D_4^{(1)})$ and describe its linear actions on the weight lattice.

We consider the following $\mathbb{Z}$-modules:

and with the bilinear form $⟨\cdot ,\cdot ⟩:Q^{\vee }\times P\to \mathbb{Z}$ defined by The weight lattice of type $D_4^{(1)}$ is spanned by h_i, i=1,3,4 and h₂−h₀. The generators $\{{\alpha }_0^{\vee },\dots ,{\alpha }_4^{\vee }\}$ and $\{{\alpha }_5^{\vee },{\alpha }_6^{\vee }\}$ are identified with the co-root of type $D_4^{(1)}$ and type 2A₁, respectively. Let us define the roots of type $D_4^{(1)}+2A_1$: {α₀,…,α₆}, which satisfy by the following: Here, ${(A_{ij})}_{i,j=0}^4$ is the generalized Cartan matrix of type $D_4^{(1)}$, We note that

We define the transformations s_i, i=0,…,4, which are reflections for the roots {α₀,…,α₄}, by

which give

Note that the h_i’s which are not explicitly shown in equation (3.9) remain unchanged. Also, define the transformations σ_i, i=1,2,3, which are the automorphisms of the Dynkin diagram of type $D_4^{(1)}$ (figure 2) and the reflections for the simple roots α₅ and α₆, by

From definitions (3.3), (3.5), (3.8) and (3.10), we can compute actions on the simple roots α_i, i=0,…,6, Under the linear actions on the weight lattice (3.8) and (3.10), $\tilde{W}(D_4^{(1)})=⟨s_0,\dots ,s_4,{\sigma }_1,{\sigma }_2⟩$ forms an extended affine Weyl group of type $D_4^{(1)}$. Indeed, the following fundamental relations hold: and where

• Download figure
• Open in new tab
• Download PowerPoint

Remark 3.1

We can also define the action of $\tilde{W}(D_4^{(1)})$ on the co-root lattice Q^∨ by replacing α_i with ${\alpha }_i^{\vee }$ in the actions (3.11). Then, the transformations in $\tilde{W}(D_4^{(1)})$ preserve the form 〈⋅,⋅〉, that is, the following holds:

for arbitrary $w\in \tilde{W}(D_4^{(1)})$, γ∈Q^∨ and λ∈P.

(b) Action on τ-variables

It is well known that we can extend the linear action of $\tilde{W}(D_4^{(1)})$ to the action of the τ function of the Painlevé VI equation [2,3]. In this section, we define the τ function on the weight lattice and give the action of $\tilde{W}(D_4^{(1)})$ on it.

Let M_k, k=0,…,4, be the orbits of h_i, i=0,…,4, defined by

and M be their disjoint union defined by Each of M_k is decomposed to the orbit of $W(D_4^{(1)})$ by where and For fixed k∈{0,1,2,3,4}, $M_k^{(ij)}$ are transformed to each other by the action of Dynkin automorphisms σ_i, i=1,2,3. For example, the weight h₀ lies in $M_0^{(00)}$ and the orbit of h₀ under σ₁ and σ₂ can be calculated by equations (3.3) and (3.10), which give

Remark 3.2

• (i) Given any $w\in \tilde{W}(D_4^{(1)})$, the element w(h_k), k=0,1,3,4, can be expressed as $\sum _{i=0}^4{n}_i{h}_i+l{h}_5+m{h}_6$, where $n_i\in \mathbb{Z}$, l,m∈{0,1}. By computing bilinear forms with ${\alpha }_5^{\vee }$ and ${\alpha }_6^{\vee }$, we see that w(h_k) lies in $M_k^{(lm)}$.

• (ii) From the relations (3.12b), we can express any element of $\tilde{W}(D_4^{(1)})$ as one of

  $$w_1,{\sigma }_1{w}_2,{\sigma }_2{w}_3,{\sigma }_1{\sigma }_2{w}_4,$$
  where $w_i\in W(D_4^{(1)})$, i=1,…,4. Note that the action of w_i, i=1,…,4, on $\sum _{i=0}^4{n}_i{h}_i+l{h}_5+m{h}_6$ does not change l and m but σ_i, i=1,2, does change them.

• (iii) Based on these observations in (i) and (ii), it follows that if $w(h_k)\in M_k^{(lm)}$, where k=0,1,3,4 and $w\in \tilde{W}(D_4^{(1)})$, the superscript ‘(lm)’ of $M_k^{(lm)}$ is determined by the number of iterations of σ₁ and σ₂ occurring in w as

  $$l=\{\begin{array}{l}0(\text{if number of }{\sigma }_1\text{ is even}),\\ 1(\text{if number of }{\sigma }_1\text{ is odd})\end{array}\mathrm{and}m=\{\begin{array}{l}0(\text{if number of }{\sigma }_2\text{ is even}),\\ 1(\text{if number of }{\sigma }_2\text{ is odd}).\end{array}$$3.21
  This is easily verified from the actions of σ_i, i=1,2, on h_k given in (3.10).

• (iv) It follows that for any element $w(h_k)=\sum _{i=0}^4{n}_i{h}_i+l{h}_5+m{h}_6$, where k=0,1,3,4 and $w\in \tilde{W}(D_4^{(1)})$, l and m are uniquely determined from n_i, i=0,…,4. For example, suppose that $w\in \tilde{W}(D_4^{(1)})$ is decomposed as σ₁w₁, where $w_1\in W(D_4^{(1)})$. Then we have from (3.10) that

  $$\begin{array}{r}{w}_1(h_0)=n_1{h}_0+n_0{h}_1+n_2{h}_2+n_4{h}_3+n_3{h}_4+(n_0+n_1+2n_2+n_3+n_4-l)h_5+m{h}_6.\end{array}$$3.22
  Since w₁(h₀) should not include h₅ and h₆ as seen in equation (3.9), we have l=n₀+n₁+2n₂+n₃+n₄ and m=0.

Lemma 3.3.

The following properties hold.

• (i) Lattices M_kare mutually disjoint.

• (ii) Each of M_k, k=0,1,3,4, is isomorphic to the weight lattice of type$D_4^{(1)}$.

Proof.

First we prove the property (i). Define

which is invariant under the action of $\tilde{W}(D_4^{(1)})$. From remark 3.1, it is obvious that M₂ is disjoint from the others, since 〈δ^∨,λ〉=2 for any λ∈M₂ and 〈δ^∨,λ〉=1 for any λ∈M_k, k=0,1,3,4. Let us consider the 16 lattices $M_k^{(ij)}$, k=0,1,3,4. It is easy to see that these are mutually disjoint if the superscripts (ij) are different from each other. Then it is sufficient to show that $M_k^{(00)}$, k=0,1,3,4, are mutually disjoint. We prove it by contradiction. Suppose that there exists an element of $M_0^{(00)}\cap M_1^{(00)}$. Since we are now restricting our attention to $M_k^{(00)}$, we use only the actions of $W(D_4^{(1)})$. Then, there exists an element $w\in W(D_4^{(1)})$ such that h₁=w(h₀), which is the reflection with respect to the root vector h₁−h₀. Therefore, it contradicts the assumption.

Next, we show the property (ii). We prove it for M₀, by constructing a map defined by

where every element of M₀ is given by $\sum _{i=0}^4{n}_i{h}_i+l{h}_5+m{h}_6$, with the proviso that l and m are determined by {n_i}, as mentioned in remark 3.2. We define the image of such an element under p as $\sum _{i=0}^4{n}_i{h}_i$. It follows from this construction that p is injective, because two elements in the domain of p with the same image must therefore be identical. Since p is injective and 〈δ^∨,p(λ)〉=1 for λ∈M₀, lattice M₀ is isomorphic to the weight lattice of type D₄. Note that we have In a similar manner, we can prove the property (ii) for M_k, k=1,3,4. Therefore, we have completed the proof. ▪

We now consider the τ-variables assigned on the lattice M, and the action of $\tilde{W}(D_4^{(1)})$ on them. As we will show in §4, the dependent variable of the system of partial difference equations (1.1) and (1.2) is defined by a ratio of τ-variables, and the system is derived from the action of the Weyl group on the τ-variables.

First, let ${\{{\tau }_i\}}_{i=0}^4$, ${\tau }_2^{({\sigma }_1)}$, ${\tau }_2^{({\sigma }_2)}$, ${\tau }_2^{({\sigma }_3)}$, ${\tau }_2^{(s_2)}$ be the variables and a_i, i=0,…,4, x be the complex parameters satisfying the following conditions:

and Therefore, the number of essential variables and parameters are 7 and 5, respectively. We now proceed to give the actions of $\tilde{W}(D_4^{(1)})$ on these τ-variables, which were deduced in [2,3], where Here, we define the action on the parameters a_i, i=0,…,4, by and where A_ij is given in (3.6). Note that each parameter a_i is associated with the root α_i, i=0,…,4, and the condition (3.27) corresponds to (3.7).

The fundamental relations (3.12) also hold under the actions s_i, i=0,…,4, on the τ-variables and the parameters. Note that the extended parts σ_i, i=1,2, are modified to become

Remark 3.4

The correspondence between the notations in this paper and those in [2,3] is given by

We then define a mapping τ on M by

where i=0,…,4, j=1,2,3 and $w\in \tilde{W}(D_4^{(1)})$.

4. Geometric description of discrete power function

In this section, we provide our main result, which is based on the action of $\tilde{W}(D_4^{(1)})$ constructed in the previous sections.

Theorem 4.1.

• I. There exist elements${\rho }_i\in \tilde{W}(D_4^{(1)})$, i=0,1,2, which satisfy the following properties.

  • (i) (Projective reduction) For each i=0,1,2, the element ρ_i²is a translation of$\tilde{W}(D_4^{(1)})$, while ρ_iitself is not a translation.

  • (ii) (Commutativity) ρ_iρ_j=ρ_jρ_i, for i,j=0,1,2.

  • (iii) (Bilinear forms) There exists a subspace$\mathbb{Z}{\beta }_1\oplus \mathbb{Z}{\gamma }_1\oplus \mathbb{Z}{\zeta }_1$, of the parameter space$\mathbb{Z}{a}_1\oplus \mathbb{Z}{a}_2\oplus \mathbb{Z}{a}_3\oplus \mathbb{Z}{a}_4$, on which the actions become translations. That is, the projections of ρ_i, i=0,1,2, on the subspace satisfy the following properties:

    $$({\beta }_1|{\gamma }_1)=({\gamma }_1|{\zeta }_1)=({\zeta }_1|{\beta }_1)=0\mathrm{and}({\beta }_1|{\beta }_1)=({\gamma }_1|{\gamma }_1)=({\zeta }_1|{\zeta }_1)=2,$$4.1
    where the bilinear form ( | ) on the parameter space is defined by
    $$(a_i|a_j)=⟨{\alpha }_i^{\vee },{\alpha }_j⟩=A_{ij}.$$4.2

  • (iv) (Shift actions) Each ρ_i, i=0,1,2, shifts one of {β₁,γ₁,ζ₁} but not the others, i.e.

    $$\begin{array}{rl}& {\rho }_1({\beta }_1)-{\beta }_1\in {\mathbb{Z}}_{\ne 0},{\rho }_1:({\gamma }_1,{\zeta }_1)\mapsto ({\gamma }_1,{\zeta }_1),\end{array}$$4.3a
    $$\begin{array}{rl}& {\rho }_2({\gamma }_1)-{\gamma }_1\in {\mathbb{Z}}_{\ne 0},{\rho }_2:({\beta }_1,{\zeta }_1)\mapsto ({\beta }_1,{\zeta }_1)\end{array}$$4.3b
    $$\begin{array}{rl}\text{and}& {\rho }_0({\zeta }_1)-{\zeta }_1\in {\mathbb{Z}}_{\ne 0},{\rho }_0:({\beta }_1,{\gamma }_1)\mapsto ({\beta }_1,{\gamma }_1).\end{array}$$4.3c

• II. Define z_l1,l2by

  $$z_{l_1,l_2}={(-1)}^{l_1+l_2}{\rho }_1{\hspace{0.17em}}^{l_1}{\rho }_2{\hspace{0.17em}}^{l_2}(z_0),$$4.4
  where
  $$z_0=\frac{{{\rho }_0}^2({\tau }_0)}{{\tau }_0}.$$4.5
  Then z_l1,l2satisfies the cross-ratio equation
  $$\frac{(z_{l_1,l_2}-z_{l_1+1,l_2})(z_{l_1+1,l_2+1}-z_{l_1,l_2+1})}{(z_{l_1+1,l_2}-z_{l_1+1,l_2+1})(z_{l_1,l_2+1}-z_{l_1,l_2})}=\frac{1}{x}$$4.6
  and similarity equation (2.1).

Remark 4.2

• (a) We note that the $\mathbb{Z}$-module spanned by the parameters a_i, i=0,…,4, gives a co-root- or root-lattice of type $D_4^{(1)}$ denoted by $Q(D_4^{(1)})$, with the pairing ( | ).

• (b) The properties (iii) and (iv) are natural requirements since (iii) is the property that β₁, γ₁, ζ₁ form simple roots of type $3A_1^{(1)}$ and (iv) comes from the idea that ρ_i are the fundamental translations in $\tilde{W}(3A_1^{(1)})$.

• (c) The choice of z₀ was motivated by the observation that the solution of the cross-ratio equation is given by a ratio of two τ functions, where one of the τ functions lies in a direction orthogonal to the plane in which the equation is iterated [13]. This observation also shows that the variable must be defined in terms of a square of ρ₀. The remaining directions ρ₁ and ρ₂ act as the two shift directions.

• (d) By choosing τ-variables appropriately as in [12], we can reproduce the initial values (1.3) of the discrete power function solution of (1.1) and (1.2). In particular, note that the choice τ₀=1 (up to gauge) and ρ₀²(τ₀)=0 leads to the value z₀=0.

• (e) The property (i) is called a projective reduction, in the sense that we took ρ_i, which were not translations in $\tilde{W}(D_4^{(1)})$, and considered them in a subgroup in which they became translations. Many discrete integrable systems can be derived by using such a procedure [21–24].

Proof.

We begin with the construction of the subspace of parameters needed in item (iii). Choose

The parameters γ₁ and ζ₁ can be chosen from to satisfy the conditions in item (iii). For example, take γ₁=a₁+a₂+a₄. Then (β₁|γ₁)=0 and the other required conditions follow from equations (3.6) and (4.2). There is one remaining choice, which we label as μ₁. To be specific, from now on, we define and also define an additional set of four parameters which are needed as simple roots of $4A_1^{(1)}$.

The $\mathbb{Z}$-module spanned by the parameters β_i, γ_i, ζ_i, μ_i, i=0,1, denoted by $Q(4A_1^{(1)})$, gives a co-root- or root-lattice of type $4A_1^{(1)}$, with the pairing ( | ). The transformation group for $Q(4A_1^{(1)})$ forms an extended affine Weyl group of type $4A_1^{(1)}$. (See appendix A for more detail.) Notice that by relinquishing one pair of parameters, say μ_i, i=0,1, we obtain $Q(3A_1^{(1)})$ from $Q(4A_1^{(1)})$. We here focus on the sublattice spanned by the parameters β_i, γ_i, ζ_i, denoted by $Q(3A_1^{(1)})$. Nevertheless, we continue to use the notations including μ such that

for consistency with appendix A.

We next construct ρ_i in $\tilde{W}(3A_1^{(1)})$ by using the elements of $\tilde{W}(D_4^{(1)})$. Recall that $\tilde{W}(D_4^{(1)})=⟨s_0,\dots ,s_4,{\sigma }_1,{\sigma }_2,{\sigma }_3⟩$. The extended affine Weyl group $\tilde{W}(3A_1^{(1)})$ corresponding to the parameters chosen above is given by

where The action of $\tilde{W}(3A_1^{(1)})$ on the parameters β_i, γ_i, ζ_i, μ_i, i=0,1, is given by and (4.11). We are now in a position to define ρ_i by using the translations in $\tilde{W}(3A_1^{(1)})$, whose actions on the parameters β_i, γ_i, ζ_i, μ_i, i=0,1, are given by We can easily verify that the transformations ρ_i, i=0,1,2, satisfy the properties (i), (ii) and (iv).

For the second part of the theorem, note that, with z₀ as defined in (4.5), the remaining directions ρ₁ and ρ₂ act as the two shift directions. Now we introduce

We can verify that these variables z₀, z₁, z₂ and z₁₂ are related by by using the framework constructed, namely (3.28), (4.13), (4.15), (4.5) and (4.17). For example, to express z₁ in terms of τ-variables, we need the composition of ρ₁ with ρ₀. The starting point is (4.15), which in turn are given by (4.13), and the actions on τ functions (3.28). In this way, we deduce and where $q=-\mathrm{i}{x}^{1/2}{\tau }_2^{({\sigma }_2)}/{\tau }_2$. Then we immediately obtain equation (4.18) from equations (4.19).

Furthermore, from the actions of the inverses of ρ₁ and ρ₂, we find

Using equations (4.18) and (4.20), respectively, we then obtain the cross-ratio equation (4.6) and similarity equation (2.1) by defining the dependent variable by (4.4). ▪

Remark 4.3

In the proof of part II of the theorem, we used the action of $\tilde{W}(D_4^{(1)})$ on τ functions. However, an alternative approach is available by starting with the definitions of the variables z₀, z₁, z₂, z₁₂, i.e. (4.5) and (4.17). For this purpose, action (3.28) is interpreted in terms of the variables z₀, z₁, z₂, z₁₂. The resulting expressions are very large and to simplify these we can instead introduce z₁₂ by using (4.18), namely

Complete details are provided in appendix B.

Remark 4.4

Let

From equation (4.18) and the following actions of ρ_i, 1,2,3: and we obtain These equations are equivalent to the lattice equations of ABS type discussed in §2. Indeed, equations (4.24) can be obtained from equations (2.5) by the following specialization of parameters: where $l\in \mathbb{Z}$.

5. Description by weight lattice

In this section, we relate our results to those obtained from a different perspective, namely the determinantal structure of hypergeometric solutions. We show that our z-variable (4.4) is identical to that found in [12,13].

To make this correspondence, we consider the actions of ρ_i, i=0,1,2, on the sublattice of the weight lattice. We here refer to a vector (w−id).x for $w\in \tilde{W}(D_4^{(1)})$, x∈M₀, as a displacement vector corresponding to w. Since the dependent variable of the discrete power function in terms of the τ function is given by

and ρ_i, i=0,1,2, satisfy the properties (i) and (ii) in §4, we here consider the displacement vectors of ρ₁ and ρ₂ on the following sublattice of M₀ (3.15): where Here, v_i, i=0,1,2, are the displacement vectors on M₀ given by which correspond to ρ_i², i=0,1,2, respectively. As illustrated in figure 3, the transformations ρ₁ and ρ₂ correspond to the vectors on L as the following: and

• Download figure
• Open in new tab
• Download PowerPoint

Let us define the following translations in $\tilde{W}(D_4^{(1)})$ by

and Using these translations, the actions of ρ₀², ρ₁ and ρ₂ on L are expressed by Therefore, the dependent variable of the discrete power function can be expressed by where This expression is equivalent to that in [12,13]. Note that the difference of the coefficients between (5.1) and (5.9) arises from equations (3.31).

6. Concluding remarks

In this paper, we have investigated the geometric structure of the system of partial difference equations (1.1) and (1.2). The cross-ratio equation (1.1) is naturally defined on a multi-dimensionally consistent cubic lattice as a face equation from the ABS theory. On the other hand, we have shown in this paper that its similarity constraint (1.2) is derived from the symmetries of the lattice.

However, it was known that the system (1.1) and (1.2) could be derived from the Bäcklund transformations of P_VI, which form $\tilde{W}(D_4^{(1)})$. The question of how to relate the symmetry group $\tilde{W}(D_4^{(1)})$ to the symmetry group of the lattice remained open in the literature. We have answered this question in this paper by constructing the action of $\tilde{W}(3A_1^{(1)})$ as a subgroup of $\tilde{W}(D_4^{(1)})$. Moreover, considering the realization on the level of the τ function simultaneously, we gave the geometric characterizations of the discrete power function. In particular, we explained the odd–even structure appearing in the explicit formula in [12,13] in terms of the projective reduction.

It is interesting to relate the geometric structures reported in this paper to other properties of (1.1) and (1.2) reported in the literature. For example, the discrete part of the Lax pair given in [25] can be obtained from translations on the $\tilde{W}(3A_1^{(1)})$. However, the remaining part of the Lax pair involves a monodromy variable, which is not visible from the point of view of the Painlevé VI equation itself. We believe that this is related to the choice of gauge we took in defining the τ functions. This is an open subject for further investigation.

Not many examples of discrete complex analytic functions are known. In [26], one was constructed on the hexagonal lattice. Further interesting directions concern the remaining Painlevé and discrete Painlevé equations. We expect our analysis to be useful for such equations, not only of second order, to give geometric characterization of known examples or to identify new examples. Our results in these directions will be reported in forthcoming publications.

Data accessibility

This paper does not contain any additional data.

Authors' contributions

All authors contributed equally to writing the paper.

Competing interests

We have no competing interests.

Funding

This research was supported by an Australian Laureate Fellowship no. FL120100094 and grant no. DP160101728 from the Australian Research Council and JSPS KAKENHI grant nos. JP16H03941, JP16K13763 and JP17J00092.

Footnotes