The Lattice Structure of Connection Preserving Deformations for q-Painlevé Equations I

We wish to explore a link between the Lax integrability of the q-Painlevé equations and the symmetries of the q-Painlevé equations. We shall demonstrate that the connection preserving deformations that give rise to the q-Painlevé equations may be thought of as elements of the groups of Schlesinger t...

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Datum:	2011
1. Verfasser:	Ormerod, C.M.
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spelling	irk-123456789-1468622019-02-12T01:24:34Z The Lattice Structure of Connection Preserving Deformations for q-Painlevé Equations I Ormerod, C.M. We wish to explore a link between the Lax integrability of the q-Painlevé equations and the symmetries of the q-Painlevé equations. We shall demonstrate that the connection preserving deformations that give rise to the q-Painlevé equations may be thought of as elements of the groups of Schlesinger transformations of their associated linear problems. These groups admit a very natural lattice structure. Each Schlesinger transformation induces a Bäcklund transformation of the q-Painlevé equation. Each translational Bäcklund transformation may be lifted to the level of the associated linear problem, effectively showing that each translational Bäcklund transformation admits a Lax pair. We will demonstrate this framework for the q-Painlevé equations up to and including q-PVI. 2011 Article The Lattice Structure of Connection Preserving Deformations for q-Painlevé Equations I / C.M. Ormerod // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 32 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 34M55; 39A13 DOI:10.3842/SIGMA.2011.045 http://dspace.nbuv.gov.ua/handle/123456789/146862 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We wish to explore a link between the Lax integrability of the q-Painlevé equations and the symmetries of the q-Painlevé equations. We shall demonstrate that the connection preserving deformations that give rise to the q-Painlevé equations may be thought of as elements of the groups of Schlesinger transformations of their associated linear problems. These groups admit a very natural lattice structure. Each Schlesinger transformation induces a Bäcklund transformation of the q-Painlevé equation. Each translational Bäcklund transformation may be lifted to the level of the associated linear problem, effectively showing that each translational Bäcklund transformation admits a Lax pair. We will demonstrate this framework for the q-Painlevé equations up to and including q-PVI.
format	Article
author	Ormerod, C.M.
spellingShingle	Ormerod, C.M. The Lattice Structure of Connection Preserving Deformations for q-Painlevé Equations I Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Ormerod, C.M.
author_sort	Ormerod, C.M.
title	The Lattice Structure of Connection Preserving Deformations for q-Painlevé Equations I
title_short	The Lattice Structure of Connection Preserving Deformations for q-Painlevé Equations I
title_full	The Lattice Structure of Connection Preserving Deformations for q-Painlevé Equations I
title_fullStr	The Lattice Structure of Connection Preserving Deformations for q-Painlevé Equations I
title_full_unstemmed	The Lattice Structure of Connection Preserving Deformations for q-Painlevé Equations I
title_sort	lattice structure of connection preserving deformations for q-painlevé equations i
publisher	Інститут математики НАН України
publishDate	2011
url	http://dspace.nbuv.gov.ua/handle/123456789/146862
citation_txt	The Lattice Structure of Connection Preserving Deformations for q-Painlevé Equations I / C.M. Ormerod // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 32 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT ormerodcm thelatticestructureofconnectionpreservingdeformationsforqpainleveequationsi AT ormerodcm latticestructureofconnectionpreservingdeformationsforqpainleveequationsi
first_indexed	2025-07-11T00:47:52Z
last_indexed	2025-07-11T00:47:52Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 045, 22 pages The Lattice Structure of Connection Preserving Deformations for q-Painlevé Equations I Christopher M. ORMEROD La Trobe University, Department of Mathematics and Statistics, Bundoora VIC 3086, Australia E-mail: C.Ormerod@latrobe.edu.au Received November 26, 2010, in final form May 03, 2011; Published online May 07, 2011 doi:10.3842/SIGMA.2011.045 Abstract. We wish to explore a link between the Lax integrability of the q-Painlevé equa- tions and the symmetries of the q-Painlevé equations. We shall demonstrate that the con- nection preserving deformations that give rise to the q-Painlevé equations may be thought of as elements of the groups of Schlesinger transformations of their associated linear prob- lems. These groups admit a very natural lattice structure. Each Schlesinger transformation induces a Bäcklund transformation of the q-Painlevé equation. Each translational Bäcklund transformation may be lifted to the level of the associated linear problem, effectively showing that each translational Bäcklund transformation admits a Lax pair. We will demonstrate this framework for the q-Painlevé equations up to and including q-PVI. Key words: q-Painlevé; Lax pairs; q-Schlesinger transformations; connection; isomonodromy 2010 Mathematics Subject Classification: 34M55; 39A13 1 Introduction and outline The discrete Painlevé equations are second order non-linear non-autonomous difference equa- tions admitting the the Painlevé equations as continuum limits [22]. They are considered integrable by the integrability criterion of singularity confinement [22], solvability via asso- ciated linear problems [21] and have zero algebraic entropy [2]. There are three classes of discrete Painlevé equations: the additive difference equations, q-difference equations and ellip- tic difference equations. This classification is in accordance with the way in which the non- autonomous variable evolves with each application of the iterative scheme. The aim of this article is to derive a range of symmetries of the considered q-difference Painlevé equations using the framework of their associated linear problems. The discovery of the solvability of a q-Painlevé equation via an associated linear problem was established by Papageorgiou et al. [21]. It was found that a q-Painlevé equation can be equivalent to the compatibility condition between two systems of linear q-difference equations, given by Y (qx, t) = A(x, t)Y (x, t), (1.1a) Y (x, qt) = B(x, t)Y (x, t), (1.1b) where A(x, t) and B(x, t) are rational in x, but not necessarily t, which is the independent variable in the q-Painlevé equation. A difference analogue of the theory of monodromy preserving deformations [13, 11, 12] was proposed by Jimbo and Sakai [14], who introduced the concept of connection preserving deformations for q-Painlevé equations. Given a system of linear q- difference equations, of the form (1.1a), conditions on the existence of the formal solutions and connection matrix associated with (1.1a) has been the subject of Birkhoff theory [4, 6, 1]. The preservation of this connection matrix leads naturally to the auxiliary linear problem in t, given mailto:C.Ormerod@latrobe.edu.au http://dx.doi.org/10.3842/SIGMA.2011.045 2 C.M. Ormerod by (1.1b). This theory also led to the discovery of the q-analogue the sixth Painlevé equation in an analogous manner to the classical result of Fuchs [9, 8]. There was one unresolved issue in the explanation of the emergence of discrete Painlevé type systems, and this was related to the emergence, and definition, of B(x, t). Unlike the theory of monodromy preserving deformations and the surfaces of initial conditions where there exists just one canonical Hamiltonian flow [27, 13], there are many ways to preserve the connection matrix. According to the existing framework, one may determine B(x, t) from a known change in the discrete analogue of monodromy data, i.e., the characteristic data. However, we would argue that this known change is not canonical, or unique. That is to say that by considering all possible changes in the characteristic data, one formulates a system of Schlesinger transforma- tions inducing a system of Bäcklund transformations of the Painlevé equation. A particular case of this theory was studied by the author in connection with the q-difference equation satisfied by the Big q-Laguerre polynomials [20]. This concept is not entirely new, one only needs to examine Jimbo and Miwa’s second pa- per [11] to see a set of discrete transformations, known as Schlesinger transformations, specified for the Painlevé equations. These Schlesinger transformations can be thought of as a discrete group of monodromy preserving evolutions, in which various elements of the monodromy data are shifted by integer amounts. The compatibility between the discrete evolution and the con- tinuous flow induces a Bäcklund transformation of the Painlevé equation, which appear in the associated linear problem, and in some cases, induce the evolution of some discrete Painlevé equations of additive type [15]. We will show the same type of transformations for systems of q-difference equations describe similar transformations. We consider systems of transformations in which (1.1a) is quadratic. There are only a finite number of cases in which A(x) is quadratic, and these cases cover the q-Painlevé equations up to and including the q-analogue of the sixth Painlevé equation [14]. We will consider the set of transformations of the associated linear problems for the exceptional q-Painlevé equations in a separate article as we have preliminary results, including a Lax pair for one of these equations that seems distinct from those that have appeared in [31] and [25]. These transformations may be derived in an analogous manner to the differential case. If one knows formal expansions of the solutions of (1.1a) at x = 0 and x =∞, then one may formulate the Schlesinger transformations directly in terms of the expansions. Formal expansions for regular q-difference equations are well established in the integrable community [4, 14], however, many associated linear problems for q-Painlevé equations fail to be regular at 0 or ∞. Hence, it is another goal of this article to apply the more general expansions provided by Adams [1] and Guenther [5] to derive the required Schlesinger equations. These expansions seem to be not as well established, however, all known examples of connection preserving deformations seem to fit nicely a framework that encompasses the regular and irregular cases [14, 17, 21]. In previous studies, such as those by by Sakai [24, 14], B(x, t) from (1.1b) and the Painlevé equations seem to have been derived from an overdetermined set of compatibility conditions. In an analogous manner to the continuous case, the formal solutions obtained may or may not converge in any given region of the complex plane [28]. The most important distinction we wish to make here is the absence of a variable t. We remark that this should be seen as a consistent trend in the studies of discrete Painlevé equations, whereby the framework of Sakai [23] and the work of Noumi et al. [18] show us that we should consider the Bäcklund transformations and discrete Painlevé equations on the same footing. The outline of this article is as follows: In Section 2 we introduce some special functions associated with q-linear systems. In Section 3 we shall outline the main results of Birkhoff theory as presented by Adams and Guenther [1, 5] and subsequently, specify the natural lattice structure of the connection preserving deformations. In Section 4 we will explore the quadratic matrix polynomial case in depth. Even at this level, we uncover a series of Lax-pairs. Lattice Structure of Connection Preserving Deformations 3 2 q-special functions A general theory of special q-difference equations, with particular interest in basic hypergeomet- ric series, may be found in the book by Gasper and Rahman [10]. The work of Sauloy and Ramis provide some insight as to how these q-special functions both relate to the special solutions of the linear problems and the connection matrix [26, 7], as does the work of van der Put and Singer [29, 30]. For the following theory, it is convenient to fix a q ∈ C such that \|q\| < 1, in order to have given functions define analytic functions around 0 or ∞. We start by defining the q-Pochhammer symbol [10], given by (a, q)k =  k∏ n=1 ( 1− aqn−1 ) for k ∈ N, ∞∏ n=1 ( 1− aqn−1 ) for k =∞, 1 for k = 0. We note that (a, q)∞ satisfies (qx; q)∞ = 1 1− x (x; q)∞, or equivalently,( x q ; q ) ∞ = ( 1− x q ) (x; q)∞. We define a fundamental building block; the Jacobi theta function [10], given by the bi- infinite expansion θq(x) = ∑ n∈Z q( n 2)xn, which satisfies the q-difference equation xθq(qx) = θq(x). The second function we require to describe the asymptotics of the solutions of (1.1a) is the q-character, given by eq,c(x) = θq(x)θq (c) θq (xc) , which satisfies eq,c(qx) = ceq,c(x), eq,qc(x) = xeq,c(x). Using the above functions, we are able to solve any one dimensional problem. 3 Connection preserving deformations We will take a deeper look into systems of linear q-difference equations of the form y(qnx) + an−1(x)y(qn−1x) + · · ·+ a1(x)y(qx) + a0(x)y(x) = 0, 4 C.M. Ormerod where the ai(x) are rational functions. One may easily express a system of this form as a matrix equation of the form Y (qx) = A(x)Y (x), where A(x) is some rational matrix. We quote the theorem of Adams [1] suitably translated for matrix equations by Birkhoff [5] in the revised language of Sauloy et al. [26, 29]. Theorem 3.1. Under general conditions, the system possesses formal solutions given by Y0(x) = Ŷ0(x)D0(x), Y∞(x) = Ŷ∞(x)D∞(x), (3.1) where Ŷ0(x) and Ŷ∞(x) are series expansions in x around x = 0 and x =∞ respectively and D0(x) = diag ( eq,λi(x) θq (x)νi ) , D∞(x) = diag ( eq,κi(x) θq (x)νi ) . Given the existence and convergence of these solutions, we may form the connection matrix, similar to that of difference equations [3], given by P (x) = Y∞(x)Y0(x)−1. In a similar manner to monodromy preserving deformations, we identify a discrete set of cha- racteristic data, namely M = { κ1 . . . κn λ1 . . . λn a1 . . . am } . Defining this characteristic data may not uniquely define A(x) in general. In the problems we consider, which are also those problems considered previously in the work of Sakai [14], Murata [17] and Yamada [31], the characteristic data defines a three dimensional linear algebraic group as a system of two first order linear q-difference equations, but is two dimensional as one second order q-difference equation. The gauge freedom disappears when one extracts the second order q-difference equation from the system of two first order q-difference equations. Let us suppose the linear algebraic group is of dimension d, then let us introduce variables y1, y2, . . ., yd, which parameterize the linear algebraic group. We may now set a co-ordinate system for this linear algebraic group, hence, we write MA = { κ1 . . . κn λ1 . . . λn a1 . . . am : y1, . . . , yd } , (3.2) as defining a matrix A. By considering the determinant of the left hand side of (1.1a) and the fundamental solutions specified by (3.1), one is able to show n∏ i=1 κi m∏ j=1 (−aj) = n∏ i=1 λi, (3.3) which forms a constraint on the characteristic data. We now explore connection preserving deformations [14]. Let R(x) be a rational matrix in x, then we apply a transformation Y → Ỹ by the matrix equation Ỹ (x) = R(x)Y (x). (3.4) Lattice Structure of Connection Preserving Deformations 5 The matrix Ỹ satisfies a matrix equation of the form (1.1a), given by Ỹ (qx) = [ R(qx)A(x)R(x)−1 ] Ỹ (x) = Ã(x)Ỹ (x). (3.5) We note that if R(x) is rational and invertible, then P̃ (x) = (Ỹ∞(x))−1Ỹ0(x) = (R(x)Y∞(x))−1R(x)Y0(x) = (Y∞(x))−1R(x)−1R(x)Y0(x) = Y∞(x)Y0(x), hence, the system defined by Ỹ (qx) = Ã(x)Ỹ (x), is a system of linear q-difference equations that possesses the same connection matrix. However, the left action of R(x) may have the following effects: The transformation may • change the asymptotic behavior of the fundamental solutions at x = ∞ by letting κi → qnκi; • change the asymptotic behavior of the fundamental solutions at x = 0 by letting λi → qnλi; • change the position of a root of the determinant by letting ai → qnai. We use the same co-ordinate system for Ã(x) as we did for A(x) via (3.2), giving MÃ = { κ̃1 . . . κ̃n λ̃1 . . . λ̃n ã1 . . . ãm : ỹ1, . . . , ỹd } . Naturally, R(x) is a function of the characteristic data and the yi, hence, we write R(x) = Ỹ0(x)Y0(x)−1 = Ỹ∞(x)Y∞(x)−1 = R(x; yi, ỹi, κj , κ̃j , λ̃l, λl, an, ãn). An alternate characterization of (3.5) is the compatibility condition resulting the consistency of (3.4) with (1.1a), which gives Ã(x)R(x) = R(qx)A(x), (3.6) which has appeared many times in the literature [14, 21, 24]. Given a suitable parameterization, we obtain a rational map on the co-ordinate system for A(x): T : MA →MÃ, For each i = 1, . . . , d, we use (3.6) to find a rational mapping φi such that ỹi = φi(MA), where MA includes all of the variables in (3.2). We now draw further correspondence with the framework of Jimbo and Miwa [11]. Let {µ1, . . . , µK} be a collection of elements in the characteristic data, and {m1, . . . ,mK} be a set of integers, then we define the transformation Tµm1 1 ,...,µ mK K : MA →MÃ, to be the transformation that multiplies µi by qmi leaving all other characteristic data fixed. The mi have to be chosen so that MÃ satisfies (3.3). The group of Schlesinger transforma- tions, GA, is the set of these transformation with the operation of composition GA = 〈Tµm1 1 ,...,µ mk K , ◦〉. 6 C.M. Ormerod This group is generated by a set of elementary Schlesinger transformations which only alter two variables, i.e., K = 2. For example, if wish to multiply κ1 by q and a1 by q−1, so to preserve (3.3), this would be the elementary transformation Tκ1,a−1 1 . Since the subscripts define the change in characteristic data, we need only specify the relation between the yi and ỹi, hence, we will write Tµm1 1 ,...,µ mK K : ỹ1 = φ1(MA), . . . , ỹd = φd(MA). We will denote matrix that induces the transformation Tµm1 1 ,...,µ mK K by Rµm1 1 ,...,µ mK K so that the transformation MA →MÃ is specified by the transformation of the linear problem given by Ỹ (x) = Rµm1 1 ,...,µ mK K (x)Y (x). The goal of the the later sections will be to construct a set of elementary Schlesinger transfor- mations that generate GA and describe the corresponding R matrices. 4 q-Painlevé equations We now look at the group of Schlesinger transformations for the q-Painlevé equations. In accor- dance with the classification of Sakai [23], we have ten surfaces considered to be of multiplicative type. Each surface is associated with a root subsystem of a root system of type E (1) 8 . We may label the surface in two different ways; either the type of the root system of E (1) 8 , R, or the type of the root subsystem of the orthogonal complement of R, R⊥. Given a surface of initial conditions associated with a root subsystem of type R, the Painlevé equation itself admits a rep- resentation of an affine Weyl group of type R⊥ as a group of Bäcklund transformations. The degeneration diagram for these surfaces is shown in Fig. 1 where we list both R along with R⊥. Figure 1. This diagram represents the degeneration diagram for the surfaces of initial conditions of multiplicative type [23]. In this article we will discuss the group of connection preserving deformations for q-Painlevé equations for cases up to and including the A (1) 3 . As the A (1) 8 surface does not correspond to any q-Painlevé equation, this gives us six cases to consider here. These are also the set of cases in which A(x) is quadratic in x. We will consider the higher degree cases, which include associated linear problem for the Painlevé equations whose Bäcklund transformations represent the exceptional affine Weyl groups in a separate article. It has been well established that the degree one case of (1.1a), where A(x) = A0 + A1x is completely solvable in general in terms of hypergeometric functions [16]. It is also interesting to note that the basic hypergeometric difference equation fits into the framework of connection preserving deformations. The result of the above transformation group reproduces many known transformations [10]. Lattice Structure of Connection Preserving Deformations 7 Table 1. The correspondence between the data that defines the linear problem and the q-Painlevé equations. This data includes the determinant, and the asymptotic behavior of the solutions at x = 0 and x =∞ in (3.1). detA (µ1, µ2) (ν1, ν2) dimGA P ( A (1) 7 ) κ1κ2x 3 (2, 1) (0, 3) 3 P ( A (1)′ 7 ) κ1κ2x 2 (2, 0) (0, 2) 3 P ( A (1) 6 ) κ1κ2x 2(x− a1) (2, 1) (0, 2) 4 P ( A (1) 5 ) κ1κ2x(x− a1)(x− a2) (2, 1) (0, 1) 5 P ( A (1) 4 ) κ1κ2(x− a1)(x− a2)(x− a3) (2, 1) (0, 0) 6 P ( A (1) 3 ) κ1κ2(x− a1)(x− a2)(x− a3)(x− a4) (2, 2) (0, 0) 7 4.1 q-P ( A (1) 7 ) Let us consider the simplest case in a more precise manner than the others as a test case. The simplest q-difference case is the {b0, b1 : f, g} → { b0 q , b1q : f̃ , g̃ } , where q = b0b1 and the evolution is f̃f = b1(1− g̃) g̃ , g̃g = 1 f , (4.1) which is also known as q-PI. We note that this transformation gives us a copy of Z inside the group of Bäcklund transformations [23]. We note that there exists also exists a Dynkin diagram automorphism in the group of Bäcklund transformations, however, we still do not know how these Dynkin diagram automorphisms manifest themselves in this theory. The associated linear problem, specified by Murata [17], is given by Y (qx) = A(x)Y (x), where A(x) = ((x− y)(x− α) + z1)κ1 κ2w(x− y) (xγ + δ)κ1 w (x− y + z2)κ2  . We fix the determinant detA(x) = κ1κ2x 3, and let A(0) possess one non-zero eigenvalue, λ1. This allows us to specify the values α = λ1 − z1κ1 + (y − z2)κ2 yκ1 , γ = (−2y − α+ z2)κ2, δ = (yα+ z1) (y − z2)κ2 y . We make a choice of parameterization of z1 and z2, given by z1 = y2 z , z2 = yz. We will make a full correspondence between y and z with f and g. To do this, let us first consider the full space of Schlesinger transformations on this space. 8 C.M. Ormerod The problem possesses formal solutions around x = 0 and x =∞ specified by Y0(x) =  −1 −wyκ2 z − 1 w y(z − 1)κ2 − λ +O (x) eq,λ1(x) 0 0 eq,λ2(x) θq (x)3  , Y∞(x) = I + 1 x  q(y + α) q − 1 −wκ2 κ1 qγ w −q(y + α) q − 1 +O ( 1 x2 )  eq,κ1(x) θq (x)2 0 0 eq,κ2(x) θq (x)  . The asymptotics of these solutions and the zeros of the determinant define the characteristic data to be M = { κ1 κ2 λ1 λ2 } , meaning our co-ordinate system for the A matrices should be MA = { κ1 κ2 λ1 λ2 : w, y, z } , where we have the constraint λ1λ2 = κ1κ2. This specifies four parameters with one constraint, however, it is easy to verify on the level of parameters (and much more work to verify on the level of the Painlevé variables) that Tκ1,λ2 ◦ T −1 κ1,λ1 ◦ Tκ2,λ1 = Tκ1,λ2 , hence, we may regard Tκ1,λ2 as an element of the group generated by the other three elements. Upon further examination, the action of Rκ1,κ2,λ1,λ2(x) = xI is represented by an identity on the Painlevé parameters, however, there is a q-shift of the asymptotic behaviors at 0 and ∞. We regard this as a trivial action, Tκ1,κ2,λ1,λ2 , whereby, we have the relation Tκ1,λ2 ◦ Tκ2,λ1 = Tκ1,κ2,λ1,λ2 , hence, if we include Tκ1,κ2,λ1,λ2 , if we include Tκ1,κ2,λ1,λ2 in our generators, we need only consider the two non-trivial translations Tκ1,λ1 and Tκ1,λ2 . We now proceed to calculate Tκ1,λ1 : { κ1 κ2 λ1 λ2 : w, y, z } → { qκ1 κ2 qλ1 λ2 : w̃, ỹ, z̃ } , Tκ1,λ2 : { κ1 κ2 λ1 λ2 : w, y, z } → { qκ1 κ2 λ1 qλ2 : w̃, ỹ, z̃ } , where in these two cases, the action (w, y, z) → (w̃, ỹ, z̃), is obtained from the left action of a Schlesinger matrix, R = Rκ1,λ1(x) and R = Rκ1,λ2(x) respectively. One of the aspects of [20] was that R(x) may be obtained directly from the solutions via expansions of R(x) = Ỹ (x)Y (x)−1, where Y = Y0 or Y = Y∞. Using Y = Y0 an expansion of R(x) around x = 0 gives R(x) = R0 +O(x), Lattice Structure of Connection Preserving Deformations 9 where as using Y = Y∞ an expansion of R around x =∞ gives R(x) = ( x 0 0 0 ) +R0 +O ( 1 x ) . The equality of these gives us that R(x) is linear in x, in fact, expanding R around x = ∞ to the constant term gives R(x) = x+ q(ỹ − y + α̃− α) q − 1 κ2w κ1 qγ̃ w̃ 1  . (4.2) This parameterization is the same for Rκ1,λ1 and Rκ1,λ2 , however, ỹ and w̃ and hence γ̃ and α̃ are different for each case. Computing the compatibility condition for the two non-trivial generators gives the following relations Tκ1,λ1 : w̃ = w ( qλ1(z̃ − 1) yz̃κ1 ) , ỹ = qyλz̃ (z̃ − 1) (qλ+ yκ2z̃) , z̃ = qzλ qy2κ1 − yzκ2 , Tκ1,λ2 : w̃ = w ( κ2 κ1 − qy z ) , ỹ = λ1 qκ1y − κ2z , z̃ = qκ1y κ2z . We now note Z3 ∼= 〈Tκ1,λ1 , Tκ1,λ2 , Tκ1,κ2,λ1,λ2〉 = GA, which is the lattice of connection preserving deformations. We now specify the connection preserving deformation that defines q-PI as q-PI : { κ1 κ2 λ1 λ2 : w, y, z } →  κ1 κ2 qλ1 λ2 q : w̃, ỹ, z̃  . which is decomposed elementary Schlesinger transformations q-PI = Tκ1,λ1 ◦ T −1 κ1,λ2 . A simple calculation reveals Tλ1,λ−1 2 = Tκ1,λ1 ◦ T −1 κ1,λ2 : w̃ = w(1− z̃), ỹy = λ1z̃ κ1(z̃ − 1) , z̃z = qκ1y κ2 . We make the correspondence with (4.1) by letting f = κ2 qκ1y and z = g, b1 = −κ22/(q2λ1κ1) and b0b1 = q. In this way, the one and only translational element of A (1) 1 is identified with Tλ1,λ−1 2 . 4.2 q-P ( A (1)′ 7 ) We note that there are many ways to interpret q-P ( A (1)′ 7 ) . We choose but one element of infinite order that we are able to make a correspondence with, for this, in Sakai’s notation, we choose s1 ◦ σ(1357)(2460), which has the following effect { b0 b1 b2 : f, g } →  b0 q q b1 b2b 2 1 : f̃ , g̃  , where f̃f = q + g̃b0 g̃(g̃ + 1)b0 , g̃g = q b0b1f2 , 10 C.M. Ormerod which is also known as q-P′I. The surface corresponds to the same affine Weyl group as before, however, the technical difference is that it corresponds to a copy of a root subsystem of E (1) 8 where the lengths of the roots are different from those that correspond to q-P ( A (1) 7 ) [23]. The associated linear problem for q-P ( A (1)′ 7 ) , has been given by Murata [17]. In terms of the theory presented above, the same theory applies in that the characteristic data is well defined, and the deformations are all prescribed in the same manner as for q-P ( A (1) 7 ) . In short, we expect a three dimensional lattice of deformations as above. We take A(x) = ( κ1((x− y)(x− α) + z1) κ2w(x− y) κ1 w (γx+ δ) κ2z2 ) , where α = −z1κ1 − z2κ2 + λ1 yκ1 , γ = z2κ2 − κ2, δ = yγ − yz2κ2 − αz2κ2, and the choice of z is specified by z1 = y2 z , z2 = z. We find the solutions are given by Y0(x) = −wy(1− z) zκ2 wy (y(z − 1)κ2 − λ) λ1 − zκ2 1− z y(z − 1)κ2 − λ +O(x)  eq,λ1(x) 0 0 eq,λ2(x) θq (x)2  , Y∞(x) = I + 1 x  q(y + α) q − 1 − w κ1 qγ w − q ( (z − 1)κ1y 2 + z (λ1 − zκ2) ) (q − 1)yzκ1 +O ( 1 x2 ) × eq,κ1(x) θq (x)2 0 0 eq,κ2(x)  , where κ1κ2 = λ1λ2. This allows us to specify the characteristic data to be M = { κ1 κ2 λ1 λ2 } , and the defining co-ordinate system for A(x) to be MA = { κ1 κ2 λ1 λ2 : w, y, z } . For the same reasons as for the previous subsection, it suffices to define Tκ1,λ1 , Tκ1,λ2 and Tκ1,κ2,λ1,λ2 , which is a trivial shift induced by Rκ1,κ2,λ1,λ2(x) = xI. The matrices Rκ1,λ1(x) and Rκ1,λ2(x) are given by (4.2), allowing us compute the following: Tκ1,λ1 : w̃ = qwλ1(z̃ − 1) yκ1z̃ , ỹ = yz̃ (qλ1 − z̃κ2) q(z̃ − 1)λ1 , z̃ = zλ1 y2κ1 , Lattice Structure of Connection Preserving Deformations 11 Tκ1,λ2 : w̃ = −qwy z , ỹy = λ1 qκ1 , z̃z = λ1 qκ2 . In a similar manner to the associated linear problem for P ( A (1) 7 ) , we have Z3 ∼= 〈Tκ1,λ1 , Tκ1,λ2 , Tκ1,κ2,λ1,λ2〉 = GA, forming the lattice of connection preserving deformations. We now identify the evolution of q-P′I as a decomposition into the basis for elementary Schlesinger transformations above. q-P′I = Tκ1,λ1 ◦ T −1 κ1,λ2 . The action is specified by Tκ1,λ1 ◦ T −1 κ1,λ2 : w̃ = w(1− z̃), ỹy = z̃ (qλ1 − κ2z̃) qκ1(z̃ − 1) , z̃z = qκ1 κ2 y2, where we may identify with the above evolutions as z = −1/g, y = √ b0b1f and κ2b0 = q2λ1 − q2κ1. 4.3 q-P ( A (1) 6 ) We now increase the dimension of the underlying lattice by introducing non-zero root of detA. We note that in accordance with the notation of Sakai [23], we choose σ◦σ to define the evolution of q-PII. This system may be equivalently written as{ b0 b1 b2 : f, g } → { b0 b1 qb2 : f̃ , g̃ } , where q = b0b1 and f̃f = − b2g̃ g̃ + b1b2 , g̃g = b1b2(1− f). (4.3) We find a parameterization of the associated linear problem to be given by A(x) = ( κ1((x− α)(x− y) + z1) κ2w(x− y) κ1 w (γx+ δ) κ2(x− y + z2) ) , where by letting detA(x) = κ1κ2x 2(x− a1), we readily find the entries of A(x) are given by α = −z1κ1 + (y − z2)κ2 + λ1 yκ1 , γ = −2y − α+ a1 + z2, δ = (yα+ z1) (y − z2) y . We take a choice of z to be specified by z1 = y (y − a1) z , z2 = yz. The formal symbolic solutions are specified by Y0(x) = −w(1− z) z − 1 −wyκ2 1− z y(z − 1)κ2 − λ1 +O(x)  eq,λ1(x) 0 0 eq,λ2(x) θq (x)2  , 12 C.M. Ormerod Y∞(x) = I + 1 x  q(y + α) q − 1 −wκ2 κ1 qγ w −qy(z − 1) ((y − a1)κ1 − zκ2) + qzλ1 (q − 1)yzκ1 +O ( 1 x2 ) ×  eq,κ1(x) θq (x)2 0 0 eq,κ2(x) θq (x)  and the zeros of detA(x) define the characteristic data to be M = { κ1 κ2 λ1 λ2 a1 } . This allows us to write co-ordinate system for A as MA = { κ1 κ2 λ1 λ2 a1 : w, y, z } , where we have the constraint λ1λ2 = −a1κ1κ2. For the same reasons as previous subsections, it suffices to define Tκ1,λ1 , Tκ1,λ1 and Tκ1,κ2,λ1,λ2 , however, we also need to choose an element that changes a1. We choose Ta1,λ1 , which is induced by some Ra1,λ1 . By solving the scalar determinantal problem, one arrives at the conclusion detRa1,λ1 = x x− qa1 . When we consider the change in κi values, it is clear that by expanding Ra1,λ1 around x = ∞ using Ra1,λ1 = Ỹ∞Y −1 ∞ we have the expansion Ra1,λ1 = I + 1 x  q(y + α) q − 1 −wκ2 κ1 qγ w −q (y + α− a1) q − 1 +O ( 1 x2 ) . This means that we may deduce that Ra1,λ1 is given by Ra1,λ1 = 1 x− qa1 x− q(y − ỹ + α− α̃) q − 1 − qa1 κ2(w − w̃) κ1 q(w̃γ − wγ̃) ww̃ x+ q(y − ỹ + α− α̃) q − 1  . (4.4) We can take Rκ1,λ1 and Rκ1,λ2 to be of the form of (4.2). Computing the compatibility reveals Ta1,λ1 : w̃ = w(1− z̃), ỹy = z̃λ1 (z̃ − 1)κ1 , z̃z = q (y − a1)κ1 κ2 , Tκ1,λ1 : w̃ = w (qλ1 − κ2ỹ(z̃ − 1)) κ1ỹ , ỹy = zλ1 κ1(z̃ − 1) , z̃ = qya1κ1 + qzλ1 qy2κ1 − yzκ2 , Tκ1,λ2 : w̃ = wκ2(1− z̃) κ1 , ỹy = λ1z̃ qκ1(z̃ − 1) , z̃z = q (y − a1)κ1 κ2 . Lattice Structure of Connection Preserving Deformations 13 The full group of Schlesinger transformations is given by Z4 ∼= 〈Ta1,λ1 , Tκ1,λ1 , Tκ1,λ2 , Tκ1,κ2,λ1,λ2〉 = GA, which is the lattice of connection preserving deformations. To obtain a full correspondence of Ta1,λ1 with (4.3) we let y = a1f, z = −g b0 b2 , with the relations b0 = λ1 a1κ2 and b2 = − λ1 qa21κ1 . We know that the action of Ta1,λ1 ◦ Tκ1,λ−1 2 is trivial. I.e., we have Ta1,λ1 ◦ Tκ1,λ−1 2 : w̃ = κ1w κ2 , ỹ = qy, z̃ = z, which leaves f and g invariant. When we restrict our attention to f and g alone we only have a two dimensional lattice of transformations, corresponding to (A1 +A1) (1). 4.4 q-P ( A (1) 5 ) As a natural progression from the above cases, we allow two non-zero roots of the determinant. In doing so, we obtain an associated linear problem for both q-PIII and q-PIV. We also introduce a symmetry as one is able to naturally permute the two roots of the determinant, which we will formalize later. We turn to the particular presentation of Noumi et al. [18]. We present a version of q-PIII as{ b0 b1 b2 c : f, g } → { qb0 b1/q b2 c : f̃ , g̃ } , where q = b0b1b2 and f̃ = qc fg 1 + gb0 g + b0 , g̃ = qc2 gf̃ b1 + qf̃ q + b1f̃ . Using the same variables, a representation of q-PIV is given by{ b0 b1 b2 c : f, g } → { b0 b1 b2 qc : f̃ , g̃ } , where f̃f = c2qb1 (1 + gb0 + fgb1b0) b2 qb1b2c2 + g + fgb1 , g̃g = b0b1 ( q (gb0 + 1) b2c 2 + fg ) (gb0 (fb1 + 1) + 1) . These two equations are interesting as a pair as they have the same surface of initial conditions. If any natural correspondence was to be sought between the theory of the associated linear problems and the theory of the surfaces of initial conditions [23], these two equations should possess the same associated linear problem up to reparameterizations. It is interesting to note that the associated linear problems defined by [17] do indeed possess the same characteristic data and asymptotics. We unify them by writing them as one linear problem, given by A(x) = ( κ2((x− α)(x− y) + z1) κ2w(x− y) κ1 w (γx+ δ) κ2(x− y + z2) ) , 14 C.M. Ormerod where detA(x) = κ1κ2x(x− a1)(x− a2). This determinant and allowing A(0) to have one non-zero eigenvalue, λ1, allows us to specify the parameterization as α = −z1κ1 + (y − z2)κ2 + λ1 yκ1 , γ = a1 + a2 + z2 − 2y − α, δ = (yα+ z1) (y − z2) y . We choose z to be specified by z1 = y (y − a1) z , z2 = z (y − a2) . The formal solutions are given by Y0(x) = (( wy wyκ2 y − zy + za2 (−zy + y + za2)κ2 + λ1 ) +O(x) )( eq,λ1(x) 0 0 eq,λ2 (x) θq(x) ) , Y∞(x) = I + 1 x  q(y + α) q − 1 −wκ2 κ1 qγ w −q (y + α− a1 − a2) q − 1 +O ( 1 x2 )  eq,κ1(x) θq (x)2 0 0 eq,κ2(x) θq (x)  , which is enough to define M = { κ1 κ2 λ1 λ2 a1 a2 } , hence, the co-ordinates for A(x) may be stated as MA = { κ1 κ2 λ1 λ2 a1 a2 : w, y, z } . We notice that there is a natural symmetry introduced in the parameterization, that is that A is left invariant under the action s1 : { κ1 κ2 λ1 λ2 a1 a2 : w, y, z } → { κ1 κ2 λ1 λ2 a2 a1 : w, y, z y − a2 y − a1 } . If we include s1 as a symmetry, we need only specify Ta1,λ1 , Tκ1,λ1 and Tκ1,λ2 as we may exploit this symmetry to obtain Ta2,λ1 = s1 ◦ Ta1,λ1 ◦ s1. The matrix Ra1,λ1 is of the form (4.4) and the matrices Rκ1,λ1 and Rκ1,λ2 are of the form (4.2). We may use the compatibility conditions to obtain the following correspondences: Ta1,λ1 : w̃ = w(1− z̃), ỹy = z̃ (z̃a2κ2 + qλ1) q(z̃ − 1)κ1 , z̃z = qy (y − a1)κ1 (y − a2)κ2 , Tκ1,λ1 : w̃ = w (qyκ1 − zκ2) (z̃ − 1) zκ1 , ỹy = z (qλ1 + a2κ2z̃) qκ1(z̃ − 1) , z̃ = qya1κ1 + qzλ1 qy2κ1 − yzκ2 , Tκ1,λ2 : w̃ = w ( q (a1 − y) z + κ2 κ1 ) , ỹy = (za2κ2 + λ1) z̃ qκ1(z̃ − 1) , z̃z = −qκ1 (za1a2κ2 + (a1 − y)λ1) κ2 (a2 (zκ2 − qyκ1) + λ1) . Lattice Structure of Connection Preserving Deformations 15 We may now specify Ta2,λ1 = s1 ◦ Ta1,λ1 ◦ s1. Ta2,λ1 : w̃ = w ( 1− qyκ1 zκ2 ) , ỹ = qya1κ1 + qzλ1 qyzκ1 − z2κ2 , z̃z = qyκ1 (ỹ − a1) κ2 (ỹ − qa2) . This gives us the full lattice Z5 ∼= 〈Ta1,λ1 , Ta2,λ1 , Tκ1,λ1 , Tκ1,λ2 , Tκ1,κ2,λ1,λ2〉 = GA. We identify the evolution of q-PIII with Ta1,λ1 under the change of variables y = qb0b 2 1λ1 gκ2 , z = − f qb1 , so long as the following relations hold: b20 = a1 a2 , λ2 = c2κ2, κ2a2 = −qb21λ1. The symmetry between a1 and a2 alternates between the two translational components of the lattice of translations. With the above identification, we also deduce q-PVI may be identified with Ta1,λ1◦ Ta2,λ1◦ Tκ1,λ2◦ T −1 κ1,λ1 : w̃ = w − qw (y − a1)κ1 zκ2 , ỹ = q (y − a1)λ1 − qza1a2κ2 y (q (y − a1)κ1 − zκ2) , z̃ = −qκ1 (za1a2κ2 + (a1 − y)λ1) zκ2 (a2 (zκ2 − qyκ1) + λ1) . We also note that in addition to Tκ1,κ2,λ1,λ2 being trivial, we have that according to the variab- les f and g, the transformation Tκ1,λ−1 1 ,a−1 1 ,a−1 2 = T−1a1,λ1 ◦ Tκ1,λ1 ◦ T −1 a2,λ1 is also trivial. 4.5 q-P ( A (1) 4 ) The next logical step in the progression of associated linear problems is to assume that the determinant has three non-zero roots. This brings us to the case of q-PV:{ b0 b1 b2 b3 b4 : f, g } → { qb0 b1 b2 b3 b4/q : f̃ , g̃ } where ff̃ = b0 + gb2 b0 (gb2 + 1) (gb1b2 + 1) b3 , gg̃ = (f̃ − 1)b0 ( f̃ b3 − 1 ) f̃ b2 ( f̃ b0b1b2b3 − 1 ) . The associated linear problem was introduced by Murata [17], and the particular group has been explored in connection with the big q-Laguerre polynomials [20]. This case introduces two new symmetries; the first is that we have the symmetric group on the three roots of detA, and the other is that the two solutions at x = 0 are of the same order. The result is that they are interchangeable on the level of solutions of a single second order q-difference equation. The offshoot of this is, by including extra symmetries, we need only compute two non-trivial connection preserving deformations. Let us proceed by first giving a parameterization of the associated linear problem. We let A(x) = ( κ1((x− α)(x− y) + z1) κ2w(x− y) κ1 w (γx+ δ) κ2(x− y + z2) ) , 16 C.M. Ormerod where detA(x) = κ1κ2(x− a1)(x− a2)(x− a3). Differently to previous case, the determinant of A(0) is non-zero, hence, we require that the eigenvalues of A(0), λ1 and λ2, are both not necessarily zero. This determines that the parame- ters in A(x) are specified by α = −z1κ1 + yκ2 − z2κ2 + λ1 + λ2 yκ1 , γ = −2y − α+ a1 + a2 + a3 + z2, δ = (yα+ z1) (y − z2)− a1a2a3 y , where we choose our parameter z to be specified by z1 = (y − a1) (y − a2) z , z2 = z (y − a3) . Our constraint is now κ1κ2a1a2a3 = −λ1λ2. The fundamental solutions are of the form Y0(x) =  −wy −wyκ2 y(z − 1)− za3 − λ2 κ2 (y(z − 1)− za3)κ2 − λ1 +O(x)  × ( eq,λ1(x) 0 0 eq,λ2(x) ) , Y∞(x) = I + 1 x  q(y + α) q − 1 −wκ2 κ1 qγ w −q (y + α− a1 − a2 − a3) q − 1 +O ( 1 x2 ) ×  eq,κ1(x) θq (x)2 0 0 eq,κ2(x) θq (x)  . This specifies that our characteristic data may be taken to be M = { a1 a2 a3 κ1 κ2 λ1 λ2 } , and hence, our co-ordinate space for A(x) is given by MA = { a1 a2 a3 κ1 κ2 λ1 λ2 : w, y, z } . We identify immediately that the parametrization is invariant under the following transforma- tions s0 : { a1 a2 a3 κ1 κ2 λ1 λ2 : w, y, z } → { a1 a2 a3 κ1 κ2 λ2 λ1 : w, y, z } , Lattice Structure of Connection Preserving Deformations 17 s1 : { a1 a2 a3 κ1 κ2 λ1 λ2 : w, y, z } → { a2 a1 a3 κ1 κ2 λ1 λ2 : w, y, z } , s2 : { a1 a2 a3 κ1 κ2 λ1 λ2 : w, y, z } → { a1 a3 a2 κ1 κ2 λ1 λ2 : w, y, z y − a3 y − a2 } . If we now specify Tκ1,λ1 , the trivial transformation, Tκ1,κ2,λ1,λ2 and Ta1,λ1 , we may obtain a ge- nerating set using the symmetries to obtain Ta2,λ1 = s1 ◦ Ta1,λ1 ◦ s1, Ta3,λ1 = s2 ◦ s1 ◦ Ta1,λ1 ◦ s1 ◦ s2, Tκ1,λ2 = s0 ◦ Tκ1,λ1 ◦ s0. The transformations Tκ1,λ1 and Ta1,λ1 are specified by the relations Ta1,λ1 : w̃ = w ((y − a1) a2a3κ1κ2 + (qy (y − a1)κ1 + z (a3 − y)κ2)λ1) κ2 ((y − a1) a2a3κ1 + z (a3 − y)λ1) , ỹ = −a2(w − w̃) (qλ1 + a3κ2z̃) qλ1(w̃ + w(z̃ − 1)) , z̃ = zλ1(w − w̃) + a2κ1 (w (y − a1) + (y(z − 1) + a1) w̃) w (y − a1) a2κ1 + wzλ1 , Tκ1,λ1 : w̃ = w ( −qκ1y2 + (q (a1 + a2)κ1 + zκ2) y − qa1a2κ1 + qzλ1 ) yzκ1 , ỹ = qλ1 + a3κ2z̃ qψκ1λ1 − qψκ1λ1z̃ , z̃ = q (ψa1κ1 − 1) (ψa2κ1 − 1) zψ (κ2 + qψκ1λ1) , where we have introduced ψ = y a1a2κ1 − zλ1 , for convenience. We may now identify the lattice of connection preserving deformations as Z6 ∼= 〈Tκ1,λ1 , Tκ1,λ2 , Tκ1,κ2,λ1,λ2 , Ta1,λ1 , Ta2,λ1 , Ta3,λ1〉 = GA. We may identify the evolution of q-PV with the Schlesinger transformation q-PV = Ta1,a2,λ1,λ2 : w̃ = w(1− z̃), ỹy = (qλ1 + a3κ2z̃) (λ1z̃ − qa1a2κ1) qκ1λ1(z̃ − 1) , z̃z = q (y − a1) (y − a2)κ1 (y − a3)κ2 . where the correspondence between y and z with f and g is given by y = a2 f , z = −b2 b0 g, under the relations a1 = a2b3, a2 = a3b4, qa2b0κ1 = −b2κ2, b0b1b4λ1 = a2κ2. In addition to Tκ1,κ2,λ1,λ2 being trivial, we also have that the transformation Ta1,a2,a3,κ−1 1 ,λ1,λ2 is trivial. 18 C.M. Ormerod 4.6 q-P ( A (1) 3 ) This particular case was originally the subject of the article by Jimbo and Sakai [14], which in- troduced the idea of the preservation of a connection matrix [3]. The article was also responsible for introducing a discrete analogue of the sixth Painlevé equation [14], given by{ b1 b2 b3 b4 b5 b6 b7 b8 : f, g } → { qb1 qb2 b3 b4 qb5 qb6 b7 b8 : f̃ , g̃ } , where f̃f b7b8 = g̃ − qb1 g̃ − b3 g̃ − qb2 g̃ − b4 , g̃g b3b4 = f − b5 f − b7 f − b6 f − b8 , (4.5) where q = b3b4b5b6 b1b2b7b8 . The way in which q-PVI, or q-P ( A (1) 3 ) , was derived by Jimbo and Sakai [14] mirrors the theory that led to the formulation of PVI by Fuchs via the classical theory of monodromy preserving deformations [9, 8]. The associated linear problem originally derived by Jimbo and Sakai [14] is given by A(x) = ( κ1((x− y)(x− α) + z1) κ2w(x− y) κ1 w (γx+ δ) κ2((x− y)(x− β) + z2) ) , where letting the eigenvalues of A(0) be λ1 and λ2, and the determinant of A(x) be detA(x) = κ1κ2(x− a1)(x− a2)(x− a3)(x− a4), leads to α = −z1κ1 − (y (−2y + a1 + a2 + a3 + a4) + z2)κ2 + λ1 + λ2 y (κ1 − κ2) , β = (y (−2y + a1 + a2 + a3 + a4) + z1)κ1 + z2κ2 − λ1 − λ2 y (κ1 − κ2) , γ = y2 + 2(α+ β)y + αβ − a2a3 − (a2 + a3) a4 − a1 (a2 + a3 + a4) + z1 + z2, δ = −(yα+ z1) (yβ + z2)− a1a2a3a4 y . We need to choose how to define z, which we take to be specified by z1 = (y − a1) (y − a2) z , z2 = z (y − a3) (y − a4) . The theory regarding the existence and convergence of Y0(x) and Y∞(x) for this case, where the asymptotics of the leading terms in the expansion of A(x) around x = 0 and x = ∞ are invertible, was determined very early by Birkhoff et al. [3]. In light of this, we solve for the first terms of Y0 and Y∞ so that we may take the expansion to be Y0(x) =  w ((yα+ z1)κ1 − (yβ + z2)κ2 + λ1 − λ2) 2κ1 δ Lattice Structure of Connection Preserving Deformations 19 w ((yα+ z1)κ1 − (yβ + z2)κ2 − λ1 + λ2) 2κ1 δ +O(x) (eq,λ1(x) 0 0 eq,λ2(x) ) , Y∞(x) = I + 1 x  q(y + α) q − 1 − qw qκ1 − κ2 qγκ1 w (κ1 − qκ2) q(y + β) q − 1 +O ( 1 x2 )  eq,κ1(x) θq (x)2 0 0 eq,κ2(x) θq (x)2  . It is this parameterization in which we have a number of symmetries being introduced. We have the group of permutations on the ai generated by the elements s1 : { κ1 κ2 λ1 λ2 a1 a2 a3 a4 ;w, y, z } → { κ1 κ2 λ1 λ2 a2 a1 a3 a4 ;w, y, z } , s2 : { κ1 κ2 λ1 λ2 a1 a2 a3 a4 ;w, y, z } → { κ1 κ2 λ1 λ2 a1 a3 a2 a4 ;w, y, z y − a3 y − a2 } , s3 : { κ1 κ2 λ1 λ2 a1 a2 a3 a4 ;w, y, z } → { κ1 κ2 λ1 λ2 a1 a2 a4 a3 ;w, y, z } , which are all elementary Bäcklund transformations, as is the elementary transformation swap- ping the eigenvalues/asymptotic behavior at x = 0, given by s4 : { κ1 κ2 λ1 λ2 a1 a2 a3 a4 ;w, y, z } → { κ1 κ2 λ2 λ1 a1 a2 a3 a4 ;w, y, z } . To obtain the corresponding permutation of κ1 and κ2 we multiply Y (x) on the left by R(x) = ( 0 1 1 0 ) , in which the resulting transformation is most cleanly represented in terms of the parameteriza- tion of the linear problem, given as s5 : { κ1 κ2 λ1 λ2 a1 a2 a3 a4 ;w, y, z } → { κ2 κ1 λ1 λ2 a1 a2 a3 a4 ; κ1γ w ,− δ γ , z (δ + γa1)(δ + γa2) (yγ + δ)(βγ + δ) + γ2z2 } . Since the operator, Tκ1,κ2,λ1,λ2 , which is induced by Rκ1,κ2,λ1,λ2(x) = xI is still a simple shift, if we include Tκ1,κ2,λ1,λ2 and the symmetries s1, . . . , s5, we still need two non-trivial translations, Ta1,λ1 and Tκ1,λ1 . These are specified by Ta1,λ1 : w̃ = w − wy (y − a1) a1 (qκ1 − κ2) a1 ((z − 1)y2 + (a1 − z (a3 + a4)) y + za3a4)κ2 + (y − a1)λ2 , ỹ = a2κ1(w − w̃) (a3a4κ2z̃ − qλ1) λ1 (qκ1(w̃ + w(z̃ − 1))− κ2w̃z̃) , z̃ = κ1 (a2 (q (w (y − a1) + w̃ (y(z − 1) + a1))κ1 − wyzκ2) + qz (w − w̃)λ1) a2κ1 (qw (y − a1)κ1 + (w̃ (y(z − 1) + a1)− wyz)κ2) + z (qwκ1 − w̃κ2)λ1 , Tκ1,λ1 : w̃ = q2w (ψa1 + 1) (ψa2 + 1)κ1 (qκ1 − zκ2) (z̃ − 1) zψ (κ2 − qκ1)2 z̃ , ỹ = a3a4κ2z̃ − qλ1 qψλ1 − qψλ1z̃ , z̃z = q2 (ψa1 + 1) (ψa2 + 1)κ1λ1 (a3κ2 + qψλ1) (a4κ2 + qψλ1) , 20 C.M. Ormerod where ψ = y (zκ2 − qκ1) q (a1a2κ1 − zλ1) . The remaining elementary Schlesinger transformations are obtained from the symmetries and the above shifts. This gives Z7 ∼= 〈Tκ1,κ2,λ1,λ2 , Tκ1,λ1 , Tκ1,λ2 , Ta1,λ1 , Ta2,λ1 , Ta3,λ1 , Ta4,λ1〉 = GA. We may identify the evolution of q-PVI with the connection preserving deformation q-PVI : { a1 a2 a3 a4 κ1 κ2 λ1 λ2 : w, y, z } → { qa1 qa2 a3 a4 κ1 κ2 qλ2 qλ1 : w̃, ỹ, z̃ } , or in terms of Schlesinger transformations q-PVI = Ta1,λ1,a2,λ2 : w̃ = qwκ1(z̃ − 1) κ2z̃ − qκ1 , ỹy = −(a3a4κ2z̃ − qλ1) (λ1z̃ − qa1a2κ1) λ1(z̃ − 1) (qκ1 − κ2z̃) , z̃z = q (y − a1) (y − a2)κ1 (y − a3) (y − a4)κ2 . We make a the correspondence with (4.5) by letting y = f and z = g/b3, and the relations between parameters are given by a1 = b5, a2 = b6, a3 = b7, a4 = b8, κ1 = b4λ1 b2b7b8 , κ1 = b3λ1 b2b7b8 , which completes the correspondences. We remark that in the representation of (4.5) the trans- formation b5 → qb5, b6 → qb6, b7 → qb7, b8 → qb8 and f → qf is a trivial transformation and corresponds to Ta1,a2,a3,a4,λ21,λ22 , which may also be regarded as trivial. 5 Conclusion and discussion The above list comprises of 2×2 associated linear problems for q-Painlevé equations in which the associated linear problem is quadratic in x. The way in which we have increased the dimension of the underlying lattice of connection preserving deformations has been to successively increase the number of non-zero roots of the determinant. If we were to consider a natural extension of the q-P ( A (1) 3 ) case by adding a root making the determinant of order five in x then we would necessarily obtain a matrix characteristically different from the Lax pair of Sakai [25] and Yamada [31]. Sakai adjoins two roots of the determinant while introducing a relation between κ1 and κ2 while Yamada [31] introduces four roots on top of the q-P ( A (1) 3 ) case and fixes the asymptotic behaviors at x = 0 and x =∞. This seems to skip the cases in which the determinant of A is five. Of course, the above theory could be applied to those Lax pairs of Sakai and Yamada [25, 31], yet, the jump between the given characteristic data for the associated linear problems of q-P ( A (1) 3 ) and q-P ( A (1) 2 ) is intriguing, and worth exploring in greater depth. Asides from the exceptional cases, where the symmetry of the underlying discrete Painlevé equations are of type E (1) 6 , E (1) 7 or E (1) 8 , there are a few issues that need to be addressed. We have outlined a very natural lattice structure underlying the connection preserving deformations, however, it is unknown whether the full set of symmetries may be derived from the underlying connection preserving deformation setting. It seems that in order to fully address this, one would require a certain duality between roots of the determinant and the leading behavior at 0 and ∞. This could be explained from the point of view of rational matrices for example. We Lattice Structure of Connection Preserving Deformations 21 expect a framework which gives rise to a lift of the entire group of Bäcklund transformations to the level of Schlesinger transformations. This work, and the work of Yamada [31] may provide valuable insight into this. We note the exceptional work of Noumi and Yamada in this case for a continuous analogue in their treatment of PVI [19]. They are able to extend the work of Jimbo and Miwa [11] for PVI in a manner that includes all the symmetries of PVI. We remark however that the more geometric approach to Lax pairs explored by Yamada [32] gives us a greater insight into a possible fundamental link between the geometry and Lax integrability of the discrete Painlevé equations. 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[32] Yamada Y., A Lax formalism for the elliptic difference Painlevé equation, SIGMA 5 (2009), 042, 15 pages, arXiv:0811.1796. http://dx.doi.org/10.1007/s002200100446 http://dx.doi.org/10.1619/fesi.48.273 http://dx.doi.org/10.1088/0305-4470/39/39/S13 http://dx.doi.org/10.1016/j.ansens.2002.10.001 http://arxiv.org/abs/math.QA/0210221 http://dx.doi.org/10.1007/BF02547785 http://arxiv.org/abs/math.QA/0507098 http://dx.doi.org/10.1093/imrn/rnq232 http://dx.doi.org/10.1093/imrn/rnq232 http://arxiv.org/abs/1004.1687 http://dx.doi.org/10.3842/SIGMA.2009.042 http://arxiv.org/abs/0811.1796 1 Introduction and outline 2 q-special functions 3 Connection preserving deformations 4 q-Painlevé equations 4.1 q-P(to.A7(1))to. 4.2 q-P(to.A7(1)')to. 4.3 q-P(to.A6(1))to. 4.4 q-P(to.A5(1))to. 4.5 q-P(to.A4(1))to. 4.6 q-P(to.A3(1))to. 5 Conclusion and discussion References

The Lattice Structure of Connection Preserving Deformations for q-Painlevé Equations I

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