Discrete Hamiltonian Structure of Schlesinger Transformations
Abstract.
Schlesinger transformations are algebraic transformations of a Fuchsian system that preserve its monodromy representation and act on the characteristic indices of the system by integral shifts. One of the important reasons to study such transformations is the relationship between Schlesinger transformations and discrete Painlevé equations; this is also the main theme behind our work. In this paper we show how to write an elementary Schlesinger transformation as a discrete Hamiltonian system w.r.t. the standard symplectic structure on the space of Fuchsian systems. We then show how Schlesinger transformations reduce to discrete Painlevé equations by considering two explicit examples, d (or difference Painlevé V) and d.
Key words and phrases:
Integrable systems, Painlevé equations, difference equations, isomonodromic transformations, birational transformations2010 Mathematics Subject Classification:
34M55, 34M56, 14E071. Introduction
In the theory of ordinary linear differential equations on a complex domain, and in particular in the theory of Fuchsian systems, an important characteristic of the equation is its configuration of singularities and the characteristic indices at these singular points. Associated to this data is the notion of the monodromy representation of the equation. Roughly speaking, this representation describes how the fundamental solution matrix of the equation changes under analytic continuation along the closed paths around the singular points and it gives a significant insight into the global behavior of solutions of the equation.
The fruitful idea of deforming the equation by moving the location of critical points into different configurations, or by changing the characteristic indices, without changing its monodromy representation, goes back to B. Riemann, but the actual foundations of this theory of isomonodromic deformations in the Fuchsian case were laid down in the works of R. Fuchs [Fuc07], L. Schlesinger [Sch12], and R. Garnier [Gar26]. An extension of the theory to the nonFuchsian case was done relatively recently in the series of papers by M. Jimbo, T. Miwa, and K. Ueno, [JMU81, JM82, JM81]. At present, the theory of isomonodromic deformations is a very active research field with deep connections to other areas such as the theory of integrable systems, classical theory of differential equations, and differential and algebraic geometry.
In this work we focus on the relationship between isomonodromic deformations and Painlevé equations. We need to distinguish between continuous and discrete isomonodromic deformations. In the continuous case we move the location of singular points and the resulting Schlesinger equations reduce to Painlevétype nonlinear differential equations. On the other hand, if we deform the characteristic indices, the isomonodromy condition requires that the indices change by integral shifts, and so the resulting dynamic is discrete and is expressed in the form of difference equations called Schlesinger transformations. Similarly to the continuous case, these transformations reduce to discrete analogues of Painlevétype equations and the study of this correspondence has been a major research topic in the field of discrete integrable systems over the last twenty years, [RGH91, PNGR92]. We need to remark here that Schlesinger transformations correspond to difference Painlevé equations, but the other two types of discrete Painlevé equations, difference and ellipticdifference Painlevé equations can also be considered in a modification of this approach.
Discrete Painlevé equations share many properties with the differential Painlevé equations, e.g., the existence of special solutions such as algebraic solutions or solutions that can be expressed in terms of special functions, affine Weyl group symmetries, and the geometric classification of equations in terms of rational surfaces. However, while it is wellknown how to write differential Painlevé equations in the Hamiltonian form [Oka80], the discrete Hamiltonian structure for discrete Painlevé equations is at present missing.
The aim of the present paper is to study such discrete Hamiltonian structure for Schlesinger transformations. In particular, we present an explicit formula for the discrete Hamiltonian function of an elementary Schlesinger transformation expressed in the same coordinates as the Hamiltonian function of the continuous Schlesinger equations considered by Jimbo, Miwa, Môri, and Sato in [JMMS80]. Note that this explicit formula gives a convenient tool for computing Schlesinger transformations and consequently for the derivation of discrete Painlevé equations, since the usual computation of a Schlesinger transformation from the compatibility condition between the Fuchsian equation and the deformation equation is often quite complicated.
We also present two explicit examples of discrete Painlevé equation computed in this framework. The first example that we consider is an isomonodromic deformation of a twodimensional Fuchsian system with three finite singular points. A nice feature of this example is that it allows us to compare, in the same setting and using the same coordinates, continuous deformations that are described by the PainlevéVI equation , and discrete deformations that are described by the difference PainlevéV equation d, which corresponds to the Bäcklund transformations of . Recall that in terms of the surface of initial conditions d is also denoted as d. In the second example we consider a discrete Painlevé equation of the type d that does not have a continuous Painlevé counterpart. In [Sak07] one of the authors posed a problem of representing this equation using Schlesinger transformations of some Fuchsian system. In [Boa09] P. Boalch described the Fuchsian system whose Schlesinger transformations are described by d without doing the explicit coordinate computation. Also note that D. Arinkin and A. Borodin clarified the correspondence between d and the difference Fuchsian equation in [AB06]. Here we compute the d example explicitly using the discrete Hamiltonian and use it to illustrate some interesting features, as well as some difficulties, that occur when we study discrete Painlevé equations using this approach.
The text is organized as follows. In the next section we briefly set up the context of the problem and describe in more details some of the main objects. In Section 3 we derive the equations of the elementary Schlesinger transformations and in Section 4 we show how to write them in the Hamiltonian form. Section 5 is dedicated to the d and d examples, and the final section is conclusions and discussion of the resul.
Acknowledgements: This work started at the Discrete Integrable Systems program at the Isaac Newton Institute for Mathematical Sciences in Cambridge, UK. The authors are very grateful to the Institute for its hospitality and the stimulating working environment. H.S. was supported by GrantinAid no. 24540205 of the Japan Society for the Promotion of Science. Part of this work was done when A.D. visited H.S. and T.T in Tokyo, and he thanks the Tokyo University of Marine Science and Technology and the University of Tokyo for their hospitality. A.D. would also like to acknowledge the generous travel support by the University of Northern Colorado Provost Fund which was very important for the success of this collaboration.
2. Preliminaries
2.1. Fuchsian Equations
Schlesinger transformations that we study originate from the deformation theory of Fuchsian equations. Recall that a Fuchsian equation (or a Fuchsian system) is a matrix linear differential equation on the Riemann sphere such that all of its singular points are regular singular points. We consider a generic case when a Fuchsian equation can be written in the the Schlesinger normal form, i.e., when the coefficient matrix is a rational function with at most simple poles at some (distinct) points (and possibly at the point ),
(2.1) 
Here are constant matrices. We also put and so this equation is regular at infinity iff .
Assumption 2.1.
From now on we make an additional semisimplicity assumption that the coefficient matrices are diagonalizable.
2.2. Spectral Type and Accessory Parameters
Geometrically, Schlesinger (or isomonodromic) dynamic takes place on the space of coefficients of the Fuchsian equation (2.1), considered modulo gauge transformations. Here we briefly outline the description of this space following [Sak10].
First, given , we separate the data about locations of singular points (that we think of as parameters of the dynamic) and the residue matrices at those points. Thus, we define
to be the set of all Fuchsian equations with possible singularities at the points . Further, since Schlesinger dynamic either preserves (in the continuous case) or shifts (in the discrete case) the eigenvalues of , we treat the eigenvalues as the parameters of the dynamic as well. Thus, the appropriate phase space is a quotient of the fiber of the eigenvalue map by gauge transformations. Local coordinates on the phase space are called accessory parameters. Note that the space of accessory parameters is quite complicated and its dimension depends on the spectral type of that we define next.
Spectral type of encodes the degeneracy of eigenvalues of the coefficient matrices via partitions , , where is the matrix size and denotes the multiplicities of the eigenvalues of . The spectral type of is then defined to be the collection of these partitions for all indices (including ):
It can be used to classify Fuchsian systems up to isomorphisms and the operations of addition and middle convolution introduced by N. Katz [Kat96], as in the recent work by T. Oshima [Osh08].
Assumption 2.2.
Since for a Fuchsian system we can use scalar local gauge transformation of the form , where is a solution of the scalar equation
to change the residue matrices by , we can assume, without loss of generality, that one of the eigenvalues . Thus, we always assume that the eigenvalue of the highest multiplicity is zero and we denote the corresponding subset of of reduced Fuchsian equations by . This amounts to choosing a representative in the quotient space of all Fuchsian equations by the group of local scalar gauge transformations.
In view of Assumptions 2.1 and 2.2, is similar to a diagonal matrix , where . Omitting the zero eigenvalues, we put
Denote by the set of all possible diagonal matrices of the spectral type with the highest degeneracy eigenvalues at finite points set to and omitted, as above. Thus, and . Then we have an eigenvalue map from the set of reduced Fuchsian equations to the set of characteristic indices, where is the subset of satisfying the Fuchs relation (or the trace condition)
With these definitions, given , the fiber of the eigenvalue map is .
We still need to take into account the global similarity transformations. It is convenient to do this in two steps. First, we use such transformations to normalize our equation at infinity by reducing to a particular form, e.g., make it diagonal, and then take the further quotient by the stabilizer subgroup of the group of global similarity transformations. Thus, for a given , we fix such that and denote by the subset of all Fuchsian equations in satisfying the condition . Then, finally, our phase space is
When is diagonal, which is the case that we are mostly interested in, the dimension of this space of accessory parameters of the spectral type is given by the formula
2.3. The Decomposition Space
To study the Hamiltonian structure of Schlesinger equations, and also of Schlesinger transformations, it is convenient to consider a larger decomposition space that is defined as follows.
In view of Assumption 2.1, there exist full sets of right eigenvectors , , and left eigenvectors , , (here we use the symbol to indicate a rowvector). In the matrix form, omitting vectors with indices that are in the kernel of , we can write
with defined by (2.2). But this means that we have a decomposition provided that , where the last condition is related to the normalization ambiguity of the eigenvectors.
Thus, given , we can construct, in a nonunique fashion, a corresponding decomposition pair . The space of all such pairs for all finite indices , without any additional conditions, we call the decomposition space. We denote it as
since it is convenient to write an element of this space as a list of pairs . This space can be equipped with the natural symplectic structure where we take the matrix elements of and as canonical coordinates, i.e., we take our symplectic form to be
Remark 2.3.
To avoid symbol overcrowding we slightly abuse the notation and use the same symbols, e.g., or , to denote both the canonical coordinate system in the decomposition space and an actual point in the space; the exact meaning of the symbol is always clear from the context.
The idea of using the decomposition space to study the Hamiltonian structure of isomonodromic deformations goes back to the paper by M. Jimbo, T. Miwa, Y. Môri, and M. Sato [JMMS80]. Since then it has been used by many other researchers studying isomonodromic deformations, most notably by J. Harnad, see, e.g., [Har94].
Remark 2.4.
There are two natural actions on the decomposition space . First, the group of global gauge transformations of the Fuchsian system induces the following action. Given , we have the action which translates into the action . We refer to such transformations as similarity transformations. It is often necessary to restrict this action to the subgroup preserving the form of . Second, for any pair the pair determines the same matrix for . The condition restricts to the stabilizer subgroup of . In particular, when all are distinct, has to be a diagonal matrix. We refer to such transformations as trivial transformations. These two actions obviously commute with each other.
We are now ready to give an alternative description of the space of accessory parameters . Given the pair as in the previous section, let
Then  
The following diagram illustrates the relationship between different spaces defined in these two sections:
2.4. Schlesinger Equations and Schlesinger Transformations
Consider now isomonodromic deformations of a Fuchsian system. For the continuous case, the most interesting situation is when we take the deformation parameters to be locations of critical points. Thus, we let (i.e., ), and consider a family of Fuchsian equations on the function ,
In [Sch12] L. Schlesinger showed that the monodromy preserving condition is equivalent to the set of deformation equations  
The compatibility conditions for these deformation equations are called Schlesinger equations. These are partial differential equations on the coefficient matrices and they have the form
In [JMMS80] Jimbo et.al showed that Schlesinger equations can be written as a Hamiltonian system on the decomposition space ,
with the Hamiltonian
Schlesinger equations admit reductions to Painlevétype nonlinear differential equations, which remains valid on the level of the Hamiltonians as well; we review an example of in Section 5. The main question that we consider is what happens to this picture in the discrete case.
The discrete counterparts of Schlesinger equations are Schlesinger transformations. Those are rational transformations preserving the singularity structure and the monodromy data of the Fuchsian system (2.1), except for the integral shifts in the characteristic indices . The study of such transformations again goes back to Schlesinger [Sch12], and this and the more general case of irregular singular points was considered in great detail in [JM81]. Schlesinger transformations have the form , where is a specially chosen rational matrix function called the multiplier of the transformation. The coefficient matrix of our Fuchsian system then transforms to that is related to by the equation
(2.2) 
Similarly to the continuous case, Schlesinger transformations induce discrete Painlevétype dynamic on the space of accessory parameters. This dynamic is very complicated, but it becomes much simpler when considered on the decomposition space, which allows us to represent it as a discrete Hamiltonian system using the same canonical coordinates as [JMMS80], as in Figure 2:
Unfortunately, at present we do not have a discrete version of the Hamiltonian reduction procedure from the decomposition space to the space of accessory parameters, this is an important problem for future research.
In this paper we focus on the elementary Schlesinger transformations that only change two of the characteristic indices by unit shifts, i.e., and . For the examples section, we also assume and have no multiplicities. In [JM81] such elementary transformation are denoted by 4 we construct a function of the decomposition space that is a discrete Hamiltonian function, in the sense explained in the next section, for such elementary Schlesinger transformation. . In Section
2.5. Discrete Lagrangian and Hamiltonian Formalism
In developing a discrete version of the Hamiltonian formalism, one approach is to discretize the continuous dynamic preserving its integrability properties, see the recent encyclopedic book by Y. Suris, [Sur03], but it requires underlying continuous dynamical system. An alternative and more direct procedure is to develop a discrete version based on the variational principles. In this case, the Lagrangian formalism is more natural, but it is possible to extend it to include the discrete version of the Hamiltonian formalism as well. This approach has its origin in the optimal control theory (see, e.g., B. Jordan and E. Polak [JP64], and J. Cadzow [Cad70]) and mechanics (see e.g., J. Logan [Log73] and S. Maeda [Mae82]). In the theory of integrable systems it was first used in the foundational works by A. Veselov [Ves91, Ves88] and A. Veselov and J. Moser [MV91]. A lot of recent work in this field is motivated by the development of very effective symplectic integrators, see an excellent review paper by J. Marsden and M. West [MW01] (and the references therein), as well as more recent works by S. Lall and M. West [LW06] and A. Bloch, M. Leok, and T. Oshawa [OBL11] that emphasize the Hamiltonian aspects of the theory. Below we give a very brief outline of this approach closely following [OBL11], we refer to this and other publications above for details.
Let be our configuration space. Then, in the discrete case, the Lagrangian is a function on the state space . For the discrete time parameter , a trajectory of motion is a map , , or, equivalently, a sequence . The action functional on the space of such sequences is given by . Using the variational principle we obtain discrete EulerLagrange equations that have the form
where (resp. ) denote vectors of partial derivatives w.r.t. the first (resp. second) sets of local coordinates on the state space . These equations then implicitly determine the map (or more precisely, the correspondence) , which then defines the discrete Lagrangian flow on the state space via . This flow is symplectic w.r.t. the discrete Lagrangian symplectic form
, where oneforms are defined by
If we introduce right and left discrete Legendre transforms and the momenta variables , by
we see that and , where and are the standard Liouville and symplectic forms on respectively.
We can then define the discrete Hamiltonian flow by , i.e., . But then the equations
mean that is just the generating function (of type one, see [GPS00] for the terminology) of the canonical transformation . We can then define the right and left discrete Hamiltonian functions as generating functions of this canonical transformation of type two and three respectively. Namely, right discrete Hamiltonian is
and then the map is given (implicitly) by the right discrete Hamiltonian equations
Similarly, left discrete Hamiltonian
gives left discrete Hamiltonian equations
For completeness, we mention that the generating function of type four,
is nothing but the Lagrangian function on the space of momenta satisfying the discrete EulerLagrange equations
2.6. Elementary Divisors
For elementary Schlesinger transformations has a very special simple form, and in this section we describe some useful properties of such matrices that we need for our computations. Consider a rational matrix function the multiplier matrix
Because of their role in representing general matrix functions as a product of factors of this form, such matrices are sometimes called elementary divisors [Dzh09]. Assuming that , which is the case we need, we can consider instead of a corresponding rankone projector , .
Lemma 2.5.
Let
Then we have the following

basic properties,

the Vanishing Rule,

and the Exchange Rule,
Proof.
Part (i) is a simple direct computation, see also [Dzh09].
Next, notice that for a rankone projector ,
and taking the trace proves the Vanishing Rule. Finally,
and  
which proves the Exchange Rule. ∎
3. Schlesinger Transformations
In this section we derive equations of an elementary Schlesinger transformation . We now take to be of the form considered in Lemma 2.5. Then we have the following result. in terms of the coordinates on the decomposition space
Lemma 3.1.
Consider transformation (2.2) with
(3.1) 
Then

the poles of coincide with the poles of if and only if, for some choice of indices , , , and , we have
(3.2) where means proportional. Thus, such defines the elementary transformation .

For satisfying (3.2), the residue matrices of and are connected by the following equations:
(3.3) (3.4)
Proof.
From the Schlesinger transformation equations
(3.5) 
and
we see that
we see that . Let and . Then and .
Since both is regular and invertible at for , taking residues of (3.5) at gives (3.3). Since is regular (but degenerate) at , we can take a residue at of the first equation in (3.5) to get the second equation in (3.4). Similarly, taking the residue of the second equation in (3.5) at we get the second equation in (3.4).
Further, at the first equation in (3.5) becomes
The terms give
and so we see that has to be an eigenvector of and has to be an eigenvector of . Choose an index and let and , then . Taking the residue at gives
but this equation follows from and
which is an immediate consequence of (3.3). Repeating this argument for and the second equation in (3.5) implies that , , and , for some choice of the index , which completes the proof. ∎
We can extend elementary Schlesinger transformations to the decomposition space as follows.
Corollary 3.2.
Proof.
Multiplying first equation in (3.3) on the right by gives , and so the matrix is the matrix of eigenvectors of , . The remaining equations are proved similarly. ∎
4. The Generating Function
The main objective of this section is to obtain an explicit formula for the discrete Hamiltonian function that generates Schlesinger transformations dynamic on the decomposition space. In this case the equations (2.5) take the form (recall Remark 2.3)
where we use the identifications .
First we introduce some notation. Let
and for a multiindex , we put
(4.1) 
Theorem 4.1.
Let
(4.2)  
(4.3) 