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Factorizations of Rational Matrix Functions with Application to Discrete Isomonodromic Transformations and Difference Painlev\'e Equations

We study factorizations of rational matrix functions with simple poles on the Riemann sphere. For the quadratic case (two poles) we show, using multiplicative representations of such matrix functions, that a good coordinate system on this space is given by a mix of residue eigenvectors of the matrix and its inverse. Our approach is motivated by the theory of discrete isomonodromic transformations and their relationship with difference Painlev\'e equations. In particular, in these coordinates, basic isomonodromic transformations take the form of the discrete Euler-Lagrange equations. Secondly we show that dPV equations, previously obtained in this context by D. Arinkin and A. Borodin, can be understood as simple relationships between the residues of such matrices and their inverses.


Introduction
In this paper we are concerned with various ways of introducing coordinates on the space of rational matrix functions L(z) (with simple poles) on the Riemann sphere. Examples like this play an important role in various applications, like Yang-Baxter maps and matrix solitons [Ves03,Ves07], Lax equations and isomonodromy transformations on algebraic curves, [Kri02a,Kri02b], and others. Our main interest is related to the study of discrete isomonodromic transformations and their relationship with discrete Painlevé equations. The theory of discrete isomonodromic transformations was developed by A. Borodin for polynomial matrices in [Bor04] and adapted to the notion of local monodromy using rational matrix functions by I. Krichever in [Kri04]. In [AB06] D. Arinkin and A. Borodin used the theory of d-connections on vector bundles to explained the appearance of difference Painlevé equations, considered from the geometric point of view of H. Sakai [Sak01], in the theory of discrete isomonodromic transformations and later, in [AB07], they introduced the notion of the τ -function of such transformation. These τ -functions appear as the gap probabilities in the discrete probabilistic models of random matrix type.
I. Krichever conjectured that discrete isomonodromic transformations can be written in the Lagrangian form and that they should be related to the universal symplectic form of Krichever-Phong, [KP98]. In [Dzh08], using the methods of [MV91], we verified this conjecture for the quadratic (two-pole) case by using the multiplicative coordinates on the space of these matrix functions and finding the explicit formula for the Lagrangian. Recently F. Soloviev showed that our Lagrangian symplectic form coincides with the reduction of the quadratic symplectic form of Krichever-Phong to some symplectic leaves [Sol09].
Unfortunately, the multiplicative coordinates are not easily obtained from other characteristic properties of L(z), such as its residue matrices. This is an obstacle to the generalization of our results to the higher-order case. In this paper we argue that one way around this difficulty is to consider, in addition to the residues of L(z), the residues of the inverse matrix L −1 (z). Then good coordinates on the space of L(z), again in the quadratic case, are given by half of the residues data of L(z) and half of the residue data of L −1 (z). Such residues also explain the almost symmetric form of the expressions for the multiplicative coordinates of L(z) and allow us to recognize dPV equations as simple relations between the residues of L(z) ±1 and the residues of the transformed matricesL(z) ±1 .
The paper is organized as follows. In Section 2 we give a short overview of the additive representation of L(z). In Section 3 we study the multiplicative representation of L(z) and establish the relationship between the eigenvectors of the residues of L(z) ±1 and the eigenvectors of the left and right-divisors of L(z). We also significantly simplify many of the arguments and formulas of [Dzh08]. Finally, in Section 4 we restrict our attention to the rank-two case. It this case it is possible to introduce the so-called spectral coordinates on the space of L(z) in such a way that the equations relating the spectral coordinates of L(z) andL(z) are precisely the difference Painlevé equations. Our main observation here is the following. In addition to spectral coordinates, divisor of zeroes and poles, and some asymptotic behavior at infinity, the entries of the matrix L(z) also depend on a choice of a gauge with respect the action of the group of constant nondegenerate diagonal matrices. Understanding the change of this gauge from L(z) to L −1 (z) andL(z) allows us to easily obtain the expressions of the multiplicative representation of L(z) in the spectral coordinates and the dPV equations.

Additive form of Rational Matrix Functions
Let L(z) be a rational matrix function on the Riemann sphere, rank L(z) = m. Let z 0 be a normalization point at which L(z) is regular, lim z→z0 L(z) = L 0 , det L 0 = 0, and all eigenvalues of L 0 are distinct (we assume everything is generic). We can then do a gauge transformation to make L 0 diagonal, thus reducing the global gauge group to the group of diagonal matrices. Without any loss of generality we can assume that z 0 = ∞, and so lim z→∞ L(z) = L 0 = diag{ρ 1 , . . . , ρ m }. (2.1) Next, consider the pole structure of L(z). First, we require that L(z) has only simple poles, denote them by z 1 , . . . , z k . Next, we require that the determinant det L(z), which is a meromorphic function on the Riemann sphere that is holomorphic away from z 1 , . . . , z k , also has only simple poles at those points. This condition is equivalent to requiring that L i = res zi L(z) is a matrix of rank one. Finally, we require that the inverse matrix M(z) = L(z) −1 satisfies similar conditions. To summarize, we have: and the negative signs in M(z) are for future convenience. We call the above representations of L(z) and M(z) additive representations and the vectors a i , b † i (resp. c i , d † i ) additive eigenvectors of L(z) (resp. M(z)). By the divisor of L(z) we mean the the divisor D of its determinant, D = i z i − i ζ i . We denote the space of matrices L(z) satisfying the conditions (2.2)-(2.4) by M D r . An important question is how to choose a good coordinate system on the space M D r . For example, given L i , we can determine a i and b † i only up to a common scaling factor. This factor has to be adjusted to ensure that det L(z) has zeros at ζ i , which a complicated condition on tr(L i ). Same problem is present for the collection {c i , d † i }. Some insight for a good choice of coordinates is provided by the study of the isospectral and isomonodromic transformations, which suggests that half of the coordinates should be taken from the residues of L(z) and half from the residues of M(z). In fact, in the quadratic case, we have the following result.
Theorem 2.1. The vectors (c 2 , d † 1 ; a 2 , b † 1 ), considered up to rescaling (i.e., as points in P r−1 ) are coordinates on the space M D r . To recover L ±1 (z), consider the function (2.5) Then The proof of this theorem is based on the multiplicative representations of L(z), which we consider next.

Elementary Divisors
To change from the additive to a multiplicative representation, we first need to define the building blocks for it. We call such building blocks the elementary divisors.
A direct calculation establishes the following elementary facts.
Lemma 3.1. Let B(z) be an elementary divisor. Then In fact, for us it will be more convenient to fix the points ζ 0 and z 0 on CP 1 . Thus, we say that a pair (ζ 0 , z 0 ) corresponds to an elementary divisor B(z) with the determinant det B(z) = (z − ζ 0 )/(z − z 0 ). In fact, any generic matrix B(z) with such determinant and normalized by the condition B(z) → I as z → ∞ is of the form B(z) = I + G/(z − z 0 ) with G of rank one and det B(ζ 0 ) = 0 (more carefully, B(z) −1 should be regular at z 0 and B(z) should be regular at ζ 0 ). Thus, B(ζ 0 ) has a left null-vector q † and a right null-vector p. We then immediately get that G = pq † , where we need to normalize the vectors p and q † so that q † p = z 0 − ζ 0 . In a more invariant form this can be written as From that point of view, the formula for the inverse matrix follows from the vanishing of the residue of the identity The following easy properties of elementary divisors are very useful for what follows.
(ii) Suppose that at some point z * we have B(z * )w = v for some vectors v and w. Then Proof. Part (i) is immediate. To establish the first formula in part (ii), we solve B(z * )w = v for p/(q † p) and then use (i): The second formula is proved in the similar way.
Finally, we need the following notation. Notation: We define a twisting of an elementary divisor B(z) by a constant non-degenerate matrix A to be a new elementary divisor A B(z) such that A B(z)A = AB(z), i.e.,

Factors and Divisors
We begin with the following important remark. For the additive representation of L(z), the ordering of zeroes and poles of det L(z) is not important, but for any multiplicative representation choosing such an ordering is crucial. Thus, from now on our labeling will reflect the fact that (ζ s , z s )-pair corresponds to some elementary divisor in L(z). There are two ways to look at the multiplicative structure of L(z) -we can look at factors or at divisors.
Definition 3.3. We say that elementary divisors B r s (z) (resp. B l s (z)) corresponding to pairs (ζ s , z s ) are right (resp. left) divisors of L(z) if L(z) = L r s (z)B r s (z) (resp. L(z) = B l s (z)L l s (z)) where L r s (z) (resp. L l s (z)) is regular at z s .
The main advantage of the divisors as opposed to the factors is that they can be written explicitly in terms of the residues of L(z) and M(z). Note that C k (z) = B r k (z) and L0 C 1 (z) = B l 1 (z). In particular, in the quadratic case k = 2, there is no essential difference between divisors and factors. (3.9) Proof. To obtain the formula for the right divisor, we take the residue of L(z) = L r s (z)B r s (z) at z s to get b † s ∼ (q r s ) † . Then we take the residue of M(z) = (B r s (z)) −1 (L r s (z)) −1 at ζ s to get c s ∼ p r s . The formula then follows. The expression for the left divisors is obtained in the same way.

Refactorization Transformations
As shown in [Bor04], [Kri04], isomonodromic transformations on the space of rational matrix functions L(z) have the form where R(z) is a certain rational matrix function. A simpler version of this are isospectral transformations Such transformations are generated by elementary transformations given by the interchanging of factors in the multiplicative representation of L(z). Let us now restrict our attention to the quadratic case and consider this transformations with R(z) = B r 1 (z): (3.10) Note that the isospectral transformation fixes the divisor D, and the isomonodromic transformations shift D by an integer vector.
To study these transformation from the point of view of the discrete Euler-Lagrange equations it is necessary to construct a configuration space Q, a Lagrangian function L ∈ F(Q×Q), and a map η : Q×Q → M D r such that η maps the Lagrangian dynamics to the refactorization dynamics. That is, denoting by Q the previous and byQ the next point of the discrete dynamics, we want η(Q , Q) = L(z) and η(Q,Q) =L(z), see [MV91], [Ves91] for the general description of this approach, and [Dzh08] for our specific situation.
and using the language of divisors and Lemma 3.3, we see that , the statement of Theorem 2.1 follows from Theorem 3.1 of [Dzh08]. Below, we restate that theorem in these coordinates (we also improve those formulas by using L 0 instead of its root). (3.12) (ii) The equations of motion (Q,Q) = Φ(Q , Q) of both the isospectral and isomonodromic dynamics in these coordinates are given by the discrete Euler-Lagrange equations with the Lagrangian function where in the isomonodromic case z 1 (t) = z 1 − t, ζ 1 (t) = ζ 1 − t, and in the isospectral case z 1 (t) = z 1 , ζ 1 (t) = ζ 1 and L(X, Y ) is time-independent.
The proof is based on computing the residues ofL(z) at the points z i and residues ofL(z) −1 at the points ζ i , followed by the application of Lemma 3.2(ii) to the row and column parts of the resulting rank-one matrices.

Isomonodromic Transformations and dPV
In [AB06], Arinkin and Borodin showed that for rank-two matrices isomonodromic transformations above, when written in a special coordinate system, are given by the difference Painlevé equations. In this section, choosing the dPV case as an example, we show that these equations appear explicitly as relations between the residues of L ±1 (z) andL ±1 (z).

Spectral Coordinates
In the quadratic (two-pole) case the space of the rank-two matrices L(z) satisfying the requirements (2.2-2.3) can be described using different parameters. These parameters come in two groups. The first group, that we call the type of L(z), consists of the zeroes and poles of the determinant of L(z) and some asymptotic data at z = ∞. The space of L(z) of the fixed type is two-dimensional, and the second group of parameters is a special coordinate system on this space, called the spectral coordinates. Expressing isomonodromy transformation in those coordinates gives rise to the difference Painlevé equations.
Definition 4.1. Let L(z) be a rational 2 × 2 matrix function on the Riemann sphere satisfying the following conditions: (4.1) In addition, put We call (ρ 1 , ρ 2 , ζ 1 , ζ 2 , z 1 , z 2 , k 1 , k 2 , µ) the type of L(z). These parameters are not independent, since The conditions on ρ i k i correspond to fixing the formal type of the solution of the difference equation at infinity, and the choice of µ corresponds to fixing the gauge under the global action by constant nondegenerate diagonal matrices.
The normalization conditions in this definition are chosen to match the formulas in [AB06] and [AB07].
Lemma 4.1. Let L(z) be a rational 2 × 2 matrix function on the Riemann sphere that has the type (ρ 1 , ρ 2 , ζ 1 , ζ 2 , z 1 , z 2 , k 1 , k 2 , µ). Then, in spectral coordinates, the residues of the matrix L(z) are given by . (4.4) Then the normalization at infinity results in the equations α 1 + α 2 = µ, ρ 1 k 1 = α 1 a 1 + α 2 a 2 and ρ 2 k 2 = α 1 b 1 + α 2 b 2 . The first of these equations, together with the condition L(γ) 21 = 0 gives the expressions for α i : (4.5) The equation on L(γ) 11 is which then gives Using the equation . This allows us to find the expressions for b i : which completes the proof.
Corollary 4.2. In the same gauge, the residues M i of the inverse matrix are given by ; (4.10) Proof. Since M(z) has the same form as L(z), we only have to determine the type and spectral coordinates of M(z) in terms of those of L(z), L(z): z 1 z 2 ζ 1 ζ 2 ρ 1 ρ 2 k 1 k 2 µ γ π M(z): and take the negative sign in the definitions of M i into account.