Geometric description of discrete power function associated with the sixth Painlev\'e equation
Nalini Joshi, Kenji Kajiwara, Tetsu Masuda, Nobutaka Nakazono, Yang, Shi
TL;DR
This paper explores the geometric structure of a discrete power function linked to the sixth Painlevé equation, revealing its embedding in a symmetric lattice and clarifying the odd-even pattern in related formulas.
Contribution
It demonstrates the embedding of the discrete power function in a cubic lattice with specific symmetry and relates the symmetry groups of the Painlevé equation to the lattice structure.
Findings
The discrete power function is embedded in a cubic lattice with tenilde;W(3A_1^{(1)}) symmetry.
The symmetry group tenilde;W(D_4^{(1)}) of P_VI is related to the lattice symmetry group.
The odd-even structure in explicit formulas is explained via translations in the symmetry group.
Abstract
In this paper, we consider the discrete power function associated with the sixth Painlev\'e 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 symmetry. By constructing the action of as a subgroup of , i.e., the symmetry group of P, we show how to relate to the symmetry group of the lattice. Moreover, by using translations in , we explain the odd-even structure appearing in previously known explicit formulas in terms of the function.
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Geometric description of discrete power function associated with the sixth Painlevé equation
Nalini Joshi
School of Mathematics and Statistics, The University of Sydney, New South Wales 2006, Australia.
,
Kenji Kajiwara
Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Fukuoka 819-0395, Japan.
,
Tetsu Masuda
Department of Physics and Mathematics, Aoyama Gakuin University, Sagamihara, Kanagawa 252-5258, Japan.
,
Nobutaka Nakazono
Department of Physics and Mathematics, Aoyama Gakuin University, Sagamihara, Kanagawa 252-5258, Japan.
and
Yang Shi
School of Mathematics and Statistics, The University of Sydney, New South Wales 2006, Australia.
Abstract.
In this paper, we consider the discrete power function associated with the sixth Painlevé 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 symmetry. By constructing the action of as a subgroup of , i.e., the symmetry group of PVI, we show how to relate to the symmetry group of the lattice. Moreover, by using translations in , we explain the odd-even structure appearing in previously known explicit formulas in terms of the function.
Key words and phrases:
discrete power function; Painlevé equation; affine Weyl group; projective reduction; ABS equation; function
2010 Mathematics Subject Classification:
14H70, 33E17, 34M55, 39A14
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:
[TABLE]
where are independent variables, is a dependent variable and , , , are parameters. Equation (1.1) is a special case of the cross-ratio equation and Equation (1.2) is its similarity constraint [23]. The purpose of this paper is to provide a full geometric description of the system of equations (1.1), (1.2) and its expression in terms of -variables.
Nijhoff et al. [23] showed that Equations (1.1) and (1.2) can be regarded as a part of the Bäcklund transformations of the Painlevé VI equation (PVI). In this context, is identified as the independent variable of PVI, and , ( does not appear in the equations) are parameters corresponding to the simple roots of the affine root system of type that relates to the symmetry group of PVI [21, 20, 19, 25]. However, the complete characterization of Equations (1.1) and (1.2) in relation to the Bäcklund transformations of PVI remains unclear.
The solution of Equations (1.1) and (1.2) for with the initial condition,
[TABLE]
where and , gives rise to a discrete power function [4, 7, 8] when the parameters take special values:
[TABLE]
(There is more than one definition of what constitutes a discrete power function. See [13] 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 , so that the continuum limit of Equation (1.2) is
[TABLE]
which is an equation satisfied by the power function , where . Equations (1.1) and (1.2) also appear as conditions for consistency of quadrilaterals on surfaces and enable a definition of discrete conformality [8, 9]. It was shown in [3] that the special cases , are given respectively by the Gamma function and Gauss hypergeometric functions. By applying the Bäcklund transformations, [6, 14] found explicit formulas for the discrete power function in terms of the hypergeometric functions of PVI as stated in Theorem 1.1 below.
Theorem 1.1** ([6, 14]).**
Define the function by
[TABLE]
where
[TABLE]
Here, is the Gauss hypergeometric function, is the Gamma function, and and are arbitrary constants. Then, for , satisfying Equations (1.1) and (1.2) with
[TABLE]
and the initial condition
[TABLE]
where , is given as follows.
Case where .
[TABLE] 2.
Case where .
[TABLE]
Here, , and is the Pochhammer symbol.
We remark that in the case of generic values of parameters , the explicit expressions of in terms of -variables are given in [14]. This result was obtained by a heuristic approach and verified by the bilinear formulation of -variables. However, the appearance of the odd-even structure in the explicit formulas 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 -variables.
In this paper, we characterize Equations (1.1) and (1.2) in relation to the Bäcklund transformations of PVI 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[1, 2, 10, 11, 12]. We show how the cubic lattice is embedded in the lattice on which -variables of PVI are assigned. Four copies of the weight lattice of will be needed to describe the lattice of -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 §1.1 below.
1.1. 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 they apply 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 . (See [15].) In higher dimensions, we would have the reflection group , , where the subscript refers to an integer number of dimensions. The group is generated by simple reflections, which we denote , . The simple roots associated with these reflections are denoted , , while corresponding co-roots are denoted .
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 . We will refer also to the root lattice denoted by . When interpreted as a Coxeter group, the reflection group is denoted by .
In a Dynkin diagram, the simple reflections are represented by nodes. Obviously, such a diagram remains unchanged when the nodes are permuted. Such an operation is called a diagram automorphism. Let be the group of such automorphisms. Incorporating these automorphisms into the space of allowable operations, we get a lift of the Coxeter group to an extended Coxeter group denoted . (See [24].)
The classification of finite reflection groups leads to several fundamental types. We will encounter , , and 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., . For conciseness, we replace by , e.g., .
1.2. 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 sixth Painlevé equation PVI by including the system of defining difference equations into a lattice with symmetry. In general, a cubic lattice has as a symmetry group, but we chose to consider the symmetry group as by taking different selections of reflection hyperplanes. This idea allows us to realize as a subgroup of , which is the symmetry group of PVI. 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 multidimensionally consistent cubic lattice with the symmetry on which the cross-ratio equation is naturally defined. In §3 we give actions of on the parameters and the -variables. Based on this material, we construct a realization of in 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 [6, 14]. Concluding remarks will be given in §6.
2. Cubic Lattice
In this section, we describe a multidimensionally consistent cubic lattice in terms of a symmetry structure, which will turn out to be related to the symmetry group of PVI.
While each cube inside the lattice is composed of 6 component faces, we will regard these as composed of two triple faces, i.e., a pair of 3 faces around a common vertex. We will see in §4 how this perspective is embedded in the symmetry-group-lattice of PVI. The symmetry group is suggested by the similarity equation:
[TABLE]
Here,
[TABLE]
Indeed, the similarity equation has three parameters , and and two directions and , defined by , which shift the parameters as follows:
[TABLE]
It is therefore natural to expect that there exists a Bäcklund transformation of the similarity equation, which shifts the parameters as follows:
[TABLE]
This implies that the three directions are translations in and are the parameters associated with the simple roots (or, coroots) of type . Consequently, we take the symmetry group of the cubic lattice to be .
Consider the cubic lattice constructed by the directions , . We place the following equations on each respective face of a triple of faces associated with each cube (referred to as the face equations):
[TABLE]
where are parameters and
[TABLE]
It is well known that this system of equations is multidimensionally consistent in the cubic lattice. Indeed, Equation (2.5a) and Equations (2.5b) and (2.5c) are called Q1 and H1 respectively in the ABS list (see Figure 1)[1, 2, 10, 11, 12].
Using the geometric theory of PVI, we can realize the action of associated with these equations (see §4).
3. Actions of affine Weyl group
In this section, we provide the basic ingredients needed to describe actions of the symmetry group of PVI, i.e., . The explicit formulation of the function of PVI 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 will be needed to define the lattice of -variables. We will consider a root system of type instead of to give a full description.
3.1. Linear action
In this subsection, we define the transformation group and describe its linear actions on the weight lattice.
We consider the following -modules:
[TABLE]
with the bilinear form defined by
[TABLE]
The weight lattice of type is spanned by , , and . The generators and are identified with the coroot of type and type , respectively. Let us define the roots of type : which satisfy
[TABLE]
by the following:
[TABLE]
Here, is the Generalized Cartan matrix of type :
[TABLE]
We note that
[TABLE]
We define the transformations , , which are reflections for the roots , by
[TABLE]
which give
[TABLE]
Note that the “”s which are not explicitly shown in Equation (3.9) remain unchanged. Also, define the transformations , , which are the automorphisms of the Dynkin diagram of type (see Figure 2) and the reflections for the simple roots and , by
[TABLE]
From definitions (3.3), (3.5), (3.8) and (3.10), we can compute actions on the simple roots , :
[TABLE]
Under the linear actions on the weight lattice (3.8) and (3.10), forms an extended affine Weyl group of type . Indeed, the following fundamental relations hold:
[TABLE]
where
[TABLE]
Remark 3.1**.**
We can also define the action of on the coroot lattice by replacing with in the actions (3.11). Then, the transformations in preserve the form , that is, the following holds:
[TABLE]
for arbitrary , and .
3.2. Action on -variables
It is well known that we can extend the linear action of to the action of the function of the Painlevé VI equation[21, 20]. In this section, we define the function on the weight lattice and give the action of on it.
Let , , be the orbits of , , defined by
[TABLE]
and be their disjoint union defined by
[TABLE]
Each of is decomposed to the orbit of by
[TABLE]
where
[TABLE]
For fixed , are transformed to each other by the action of Dynkin automorphisms , . For example, the weight lies in and the orbit of under and can be calculated by Equations (3.3) and (3.10), which give
[TABLE]
Remark 3.2**.**
- (i)
Given any , the element , , can be expressed as , where , . By computing bilinear forms with and , we see that lies in . 2. (ii)
From the relations (3.12b), we can express any element of as one of
[TABLE]
where , . Note that the action of , , on , does not change and but , , does change them. 3. (iii)
Based on these observations in (i) and (ii), it follows that if , where and , the superscript “” of is determined by the number of iterations of and occurring in as:
[TABLE]
This is easily verified from the actions of , , on given in (3.10). 4. (iv)
It follows that for any element , where and , and are uniquely determined from , . For example, suppose that is decomposed as , where . Then we have from (3.10) that
[TABLE]
Since should not include and as seen in Equation (3.9), we have and .
Lemma 3.3**.**
The following properties hold.
**(i): **
Lattices are mutually disjoint.
**(ii): **
Each of , , is isomorphic to the weight lattice of type .
Proof.
First we prove the property (i). Define
[TABLE]
which is invariant under the action of . From Remark 3.1, it is obvious that is disjoint from the others, since for any and for any , . Let us consider the 16 lattices , . It is easy to see that these are mutually disjoint if the superscripts are different from each other. Then it is sufficient to show that , , are mutually disjoint. We prove it by contradiction. Suppose that there exists an element of . Since we are now restricting our attention to , we use only the actions of . Then, there exists an element such that which is the reflection with respect to the root vector . Therefore, it contradicts the assumption.
Next, we show the property (ii). We prove it for , by constructing a map defined by
[TABLE]
where every element of is given by , with the proviso that and are determined by , as mentioned in Remark 3.2. We define the image of such an element under as . It follows from this construction that is injective, because two elements in the domain of with the same image must therefore be identical. Since is injective and for , lattice is isomorphic to the weight lattice of type . Note that we have
[TABLE]
In a similar manner, we can prove the property (ii) for , . Therefore, we have completed the proof. ∎
We now consider the -variables assigned on the lattice , and action of 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 -variables, and the system is derived from the action of the Weyl group on the -variables.
First, let , , , , be the variables and , , be the complex parameters satisfying the following conditions:
[TABLE]
Therefore, the number of essential variables and parameters are 7 and 5, respectively. We now proceed to give the actions of on these -variables, which were deduced in [21, 20]:
[TABLE]
where
[TABLE]
Here, we define the action on the parameters , , by
[TABLE]
where is given in (3.6). Note that each parameter is associated with the root , , and the condition (3.27) corresponds to (3.7).
The fundamental relations (3.12) also hold under the actions , , on the -variables and the parameters. Note that the extended part , , are modified to become
[TABLE]
Remark 3.4**.**
The correspondence between the notations in this paper and those in [21, 20] is given by
[TABLE]
We then define a mapping on by
[TABLE]
where , and .
4. Geometric description of discrete power function
In this section, we provide our main result, which is based on the action of constructed in the previous sections.
Theorem 4.1**.**
- I.
There exist elements , , which satisfy the following properties.
**(i): **
(Projective reduction)* For each , the element is a translation of , while itself is not a translation.*
**(ii): **
(Commutativity)* , for .*
**(iii): **
(Bilinear forms)* There exists a subspace , of the parameter space , on which the actions become translations. That is, the projections of , on the sub-space satisfy the following properties:*
[TABLE]
where the bilinear form on the parameter space is defined by
[TABLE]
**(iv): **
(Shift actions)* Each , , shifts one of but not the others, i.e.,*
[TABLE] 2. II.
Define by
[TABLE]
where
[TABLE]
Then satisfies the cross-ratio equation
[TABLE]
and similarity equation (2.1).
Remark 4.2**.**
**a: **
We note that -module spanned by the parameters , , gives coroot- or root-lattice of type denoted by , with the pairing .
**b: **
The properties (iii) and (iv) are natural requirements since (iii) is the property that , , form simple roots of type and (iv) comes from the idea such that are the fundamental translations in .
**c: **
The choice of was motivated by the observation that the solution of the cross-ratio equation is given by a ratio of two functions, where one of the functions lies in a direction orthogonal to the plane in which the equation is iterated **[14]**. This observation also shows that the variable must be defined in terms of a square of . The remaining directions and act as the two shift directions.
**d: **
By choosing -variables appropriately as in **[6]**, 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 (up to gauge) and leads to the value .
**e: **
The property (i) is called a projective reduction, in the sense that we took , which were not translations in , and considered them in a subgroup in which they became translations. Many discrete integrable systems can be derived by using such a procedure (see **[16, 17, 18, 26]**).
Proof.
We begin with the construction of the subspace of parameters needed in item (iii). Choose
[TABLE]
The parameters and can be chosen from
[TABLE]
to satisfy the conditions in item (iii). For example, take . Then and the other required conditions follow from Equations (3.6) and (4.2). There is one remaining choice, which we label as . To be specific, from now on, we define
[TABLE]
and also define an additional set of 4 parameters
[TABLE]
which are needed as simple roots of .
The -module spanned by the parameters , , , , , denoted by gives a coroot- or root-lattice of type , with the pairing . The transformation group for forms an extended affine Weyl group of type . (See Appendix A for more detail.) Notice that by relinquishing one pair of parameters, say , , we obtain from . We here focus on the sublattice spanned by the parameters , , , denoted by . Nevertheless, we continue to use the notations including such that
[TABLE]
for consistency with Appendix A.
We next construct in by using the elements of . Recall that . The extended affine Weyl group corresponding to the parameters chosen above is given by
[TABLE]
where
[TABLE]
The action of on the parameters , , , , , is given by
[TABLE]
and (4.11). We are now in a position to define by using the translations in :
[TABLE]
whose actions on the parameters , , , , , are given by
[TABLE]
We can easily verify that the transformations , , satisfy the properties (i), (ii) and (iv).
For the second part of the theorem, note that with as defined in (4.5), the remaining directions and acts as the two shift directions. Now we introduce
[TABLE]
We can verify that these variables , , and are related by
[TABLE]
by using the framework constructed, namely, (3.28), (4.13), (4.15), (4.5) and (4.17). For example, to express in terms of -variables, we need the composition of with . The starting point is (4.15) which in turn are given by (4.13) and the actions on functions (3.28). In this way, we deduce
[TABLE]
where . Then we immediately obtain Equation (4.18) from Equations (4.19).
Furthermore, from the actions of the inverses of and , we find
[TABLE]
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 on functions. However, an alternative approach is available by starting with the definitions of the variables , , , , i.e. (4.5) and (4.17). For this purpose, action (3.28) is interpreted in terms of the variables , , , . The resulting expressions are very large and to simplify these we can instead introduce by using (4.18), namely,
[TABLE]
Complete details are provided in Appendix B.
Remark 4.4**.**
Let
[TABLE]
From Equation (4.18) and the following actions of , 1,2,3$$:
[TABLE]
we obtain
[TABLE]
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:
[TABLE]
where .
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 -variable (4.4) is identical to that found in [6, 14].
To make this correspondence, we consider the actions of , , on the sublattice of the weight lattice. We here refer to a vector for , as a displacement vector corresponding to . Since the dependent variable of discrete power function in terms of the function is given by
[TABLE]
and , , satisfy the properties (i) and (ii) in §4, we here consider the displacement vectors of and on the following sublattice of (3.15):
[TABLE]
where
[TABLE]
Here, , , are displacement vectors on given by
[TABLE]
which correspond to , , respectively. The transformations and correspond to the vectors on as the following:
[TABLE]
Let us define the following translations in by
[TABLE]
Using these translations, the actions of , and on are expressed by
[TABLE]
Therefore, the dependent variable of discrete power function can be expressed by
[TABLE]
where
[TABLE]
This expression is equivalent to that in [6, 14]. 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 multidimensionally consistent cubic lattice as a face equation from the ABS theory. On the other hand, we showed 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 PVI, which form . The question of how to relate the symmetry group to the symmetry group of the lattice remained open in the literature. We answered the question in this paper by constructing the action of as a subgroup of . Moreover, considering the realization on the level of 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 [6, 14] 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 [22] can be obtained from translations on the . However, the remaining part of the Lax pair involves a monodromy variable, which is not visible from the point of view of the sixth Painlevé equation itself. We believe that this is related to the choice of gauge we took in defining the functions. This is an open subject for further investigation.
Not many examples of discrete complex analytic functions are known. In [5], 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 data.
Competing interests
We have no competing interests.
Authors’ contributions
All authors contributed equally to writing the paper.
Acknowledgment
N/A
Funding statement
This research was supported by an Australian Laureate Fellowship # FL120100094 and grant # DP160101728 from the Australian Research Council and JSPS KAKENHI Grant Numbers JP16H03941, JP16K13763 and JP17J00092.
Ethics statement
This work does not include research on humans or animals.
Appendix A Extended affine Weyl group
In this section, we consider an affine Weyl group for spanned by the parameters , , , , .
Let
[TABLE]
whose actions on the parameters , , , , , are given by
[TABLE]
where . They collectively form the extended affine Weyl group of type :
[TABLE]
Indeed, satisfies the following fundamental relations:
[TABLE]
Here, the relation for transformations and means that there is no positive integer such that . Note that from the definitions (4.13) and (A.1), holds.
Appendix B Birational actions of on the -variables
The action of on the variables , , , , where is related to other variables as (4.21), is given by the following:
[TABLE]
Under the actions on the parameters (4.11) and (4.13) and -variables (B.1), the following fundamental relations hold:
[TABLE]
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