Complete classification of rational solutions of A 2n -Painlevé systems
aa r X i v : . [ m a t h - ph ] S e p COMPLETE CLASSIFICATION OF RATIONAL SOLUTIONS OF A n -PAINLEV´E SYSTEMS. DAVID G ´OMEZ-ULLATE, YVES GRANDATI, AND ROBERT MILSON
Abstract.
We provide a complete classification and an explicit representation of rationalsolutions to the fourth Painlev´e equation P IV and its higher order generalizations known asthe A n -Painlev´e or Noumi-Yamada systems. The construction of solutions makes use ofthe theory of cyclic dressing chains of Schr¨odinger operators. Studying the local expansionsof the solutions around their singularities we find that some coefficients in their Laurentexpansion must vanish, which express precisely the conditions of trivial monodromy of theassociated potentials. The characterization of trivial monodromy potentials with quadraticgrowth implies that all rational solutions can be expressed as Wronskian determinants ofsuitably chosen sequences of Hermite polynomials. The main classification result states thatevery rational solution to the A n -Painlev´e system corresponds to a cycle of Maya diagrams,which can be indexed by an oddly coloured integer sequence. Finally, we establish thelink with the standard approach to building rational solutions, based on applying B¨acklundtransformations on seed solutions, by providing a representation for the symmetry groupaction on coloured sequences and Maya cycles. Keywords.
Painlev´e equations, Noumi-Yamada systems, rational solutions, Darboux dress-ing chains, Maya diagrams, Wronskian determinants, Hermite polynomials.
Contents
1. Introduction 22. Higher order Painlev´e equations and dressing chains 42.1. Factorization chains of Schr¨odinger operators 63. Maya diagrams and Trivial monodromy potentials 73.1. Maya diagrams 73.2. Trivial monodromy 94. Characterization of rational solutions to odd-cyclic dressing chains 125. Cyclic Maya diagrams 185.1. Modular decomposition and colouring. 216. Classification of rational solutions to A n -Painlev´e 246.1. Coloured sequences. 256.2. Enumeration and construction of explicit examples 277. Connection with the symmetry group approach 297.1. Symmetries on Maya cycles 307.2. Seed solutions 327.3. Orbits of the extended affine Weyl group ˜ A (1)2 n Introduction
The solutions of Painlev´e equations are considered to be the nonlinear analogues of specialfunctions, [11, 22, 32]. In general, they are transcendental functions, but for special valuesof the parameters, Painlev´e equations (except the first one) possess solutions that can beexpressed via rational or special functions. For a review of rational solutions to Painlev´eequations, see the recent book by Van Assche, [51].In this paper we focus on the rational solutions of Painlev´e’s fourth equation (P IV ) (1.1)(1.1) P IV : u ′′ = ( u ′ ) u + 32 u + 4 zu + 2( z − α ) u + βu , α, β ∈ C , and its higher order generalizations, known as the A n -Painlev´e or Noumi-Yamada systems.Lukasevich [35] found by direct inspection the first few rational solutions of P IV . Okamoto[46] developed the theory of symmetry transformations of this equation, finding a Hamiltonianstructure, birational canonical transformations, parameters for which rational solutions existand some special solutions that now bear his name.The scalar 2nd order equation P IV is equivalent to the following system of three first orderODEs f ′ + f ( f − f ) = α , sP IV : f ′ + f ( f − f ) = α , (1.2) f ′ + f ( f − f ) = α , with ′ ≡ d / d z and α j , j = 0 , , f + f + f = z, α + α + α = 1 . System (1.2) possesses a symmetry group of B¨acklund transformations acting on the tu-ple of solutions and parameters ( f , f , f | α , α , α ), [43]. This symmetry group is theaffine Weyl group A (1)2 , generated by the operators { π , s , s , s } whose action on the tu-ple ( f , f , f | α , α , α ) is given by:(1.4) s ( f ) = f , s ( f ) = f − α f , s ( f ) = f + α f s ( α ) = − α , s ( α ) = α + α , s ( α ) = α + α π ( f ) = f , π ( f ) = f , π ( f ) = f π ( α ) = α , π ( α ) = α , π ( α ) = α with the action of s , s obtained by cyclically permuting the indices in the action of s .Noumi and Yamada soon realized that the structure of (1.2) can be generalized to anynumber of equations [42], leading to the A N -Painlev´e or the Noumi-Yamada system. Systemswith an even or odd number of equations have a rather different behaviour, and we restrictin this paper to the analysis of the A n -Painlev´e system, whose equations are given by(1.5) A n -Painlev´e: f ′ i + f i n X j =1 f i +2 j − − n X j =1 f i +2 j = α i , i = 0 , . . . , n mod (2 n + 1) ATIONAL SOLUTIONS OF A n -PAINLEV´E SYSTEMS 3 subject to the normalization conditions(1.6) f + · · · + f n = z, α + · · · + α n = 1 . This system can be considered the natural higher order generalization of sP IV (which corre-sponds to n = 1), since its symmetry group is the extended affine Weyl group ˜ A (1)2 n , acting byB¨acklund transformations as in (1.4). The system passes the Painlev´e-Kowalevskaya test, [53].The standard technique to construct rational solutions of (1.5) is to start from a numberof very simple rational seed solutions , and successively apply the B¨acklund transformations(1.4) to generate new solutions, which are rational by construction. However, this methoddoes not produce per se explicit representations of the solutions. For this reason, other moreexplicit representations have been investigated, most notably via recursion relations [24, 46],determinantal representations [33,43] or Schur functions, exploiting suitable reductions of theKP hierarchy in Sato’s theory of integrable systems, [48, 50]. Perhaps the simplest represen-tation of the rational solutions of P IV is via Wronskian determinants of certain sequences ofHermite polynomials: H m,n ( z ) = Wr( H m , H m +1 , . . . , H m + n − ) , (1.7) Q m,n ( z ) = Wr( H , H , . . . , H m − , H , H , . . . , H n − ) , (1.8)which are known as generalized Hermite and generalized Okamoto polynomials, respectively.Regarding higher order systems, rational solutions of A -Painlev´e have been investigatedin [21,39] and classified recently in [15], which lays the ground for the construction of solutionsin this paper. For systems of arbitrary order N , Tsuda [49] has described one special familyof solutions in terms of Schur functions associated to N -reduced partitions, which can beregarded as a generalization of (1.8). Indeed, the families (1.7) and (1.8) can be generalizedto the higher order system (1.5), but they represent only a small part of all the solutions,those corresponding to the minimal and maximal shifts.The special polynomials associated with rational solutions of Painlev´e equations haveattracted much interest for various reasons. First, they appear in a number of applica-tions, in connection with random matrix theory [10, 23], supersymmetric quantum mechan-ics [6,7,36,44], vortex dynamics with quadrupole background flow [14], recurrence relations fororthogonal polynomials [16,51], exceptional orthogonal polynomials [25] or rational-oscillatorysolutions of the defocusing nonlinear Schr¨odinger equation [13].Second, the complex zeros of these special polynomials form remarkably regular patterns inthe complex plane, as it has been mostly studied by Clarkson, [12]. The zeros of generalizedHermite polynomials (1.7) form rectangular patterns, and for large m, n with m/n fixed theyfill densely a curvilinear rectangle whose boundary is described by Buckingham [8] using thesteepest descent method for a Riemann-Hilbert problem. The distribution of these zeros isalso studied recently by Masoero and Raffolsen in other asymptotic regimes, [37,38]. The rootsof generalized Okamoto polynomials form patterns that combine rectangular and triangularfilled regions, and recently Buckingham and Miller [9] have extended their analysis to providea rigorous description of the boundaries. Remarkably, the zeros and poles of rational solutionsto higher order A n -Painlev´e systems show much richer structures, which so far have only beeninvestigated numerically, [15].Construction methods are able to prove existence of rational solutions and equivalence ofdifferent representations, but the question of establishing that all of the rational solutions areobtained is much harder, and has only been addressed in very few papers. Parameters for DAVID G ´OMEZ-ULLATE, YVES GRANDATI, AND ROBERT MILSON which rational solutions exist have been identified by Murata [40] for P IV and by Kitaev, Lawand McLeod [34] for P V . These results are obtained by direct computation on local expansions,and they do not scale well to higher order systems due to increasing complexity and branching.By contrast, Veselov was able to establish that rational solutions of A n -Painlev´e are in one-to-one correspondence with Schr¨odinger operators whose potentials have quadratic growthat infinity and trivial monodromy. His paper [52], which has received comparatively lessattention, is the basis for the characterization of rational solutions performed in this work,together with [18, 45].As mentioned in [54], our aim in this paper is to combine the strength of the τ -functionand geometric approach of the japanese school [24, 41, 43, 46, 49, 50] with that of the dressingchains and trivial monodromy approach of the russian school [2, 45, 52, 53], to attain our goalof giving a complete classification of the rational solutions to higher order Painlev´e systems.Given the breadth of both points of view, beyond this goal there is much to be learnt fromtheir common interplay.The paper is organized as follows: in Section 2 we recall the equivalence between the A n -Painlev´e system and cyclic dressing chains of Darboux transformations obtained as factoriza-tions of Schr¨odinger operators, [2, 53]. Section 3 introduces the class of rational extensions ofthe harmonic oscillator and identifies them as the only potentials with quadratic growth atinfinity and trivial monodromy [45]. It also introduces Hermite pseudo-Wronskians indexedby Maya diagrams, and recalls some of their basic properties [29]. The main result in thissection is Proposition 3.12 that provides all quasi-rational eigenfunctions of Schr¨odinger op-erators belonging to that class of potentials. In Section 4 we follow the work of Veselov [52]on rational solutions to odd-cyclic dressing chains. Studying the Laurent expansions of thesesolutions, the constraints imposed on the coefficients of the expansion are identified preciselyas the conditions that express trivial monodromy of the associated potentials of the chain.The main result in this section is Theorem 4.12, which establishes that all rational solutionsof an odd-cyclic dressing chain must necessarily be expressible as Wronskian determinantsof Hermite polynomials. Section 5 studies cycles of Maya diagrams and introduces all thenecessary concepts (genus, interlacing, block coordinates) to achieve a complete classification,which is described in Proposition 5.9. Section 6 uses all the previously derived results tostate and prove the main Theorem 6.10 on the classification of rational solutions of odd-cyclicdressing chains. The result not only provides a full classification, but also allows for an explicitrepresentation of all solutions in terms of oddly coloured sequences. To illustrate this, someexplicit examples are given in § Higher order Painlev´e equations and dressing chains
The A n -Painlev´e system is the following set of 2 n + 1 nonlinear differential equations forthe functions f i = f i ( z ) and parameters α i ∈ C (2.1) f ′ i + f i n X j =1 f i +2 j − − n X j =1 f i +2 j = α i , i = 0 , . . . , n mod (2 n + 1) ATIONAL SOLUTIONS OF A n -PAINLEV´E SYSTEMS 5 subject to the normalization conditions(2.2) f + · · · + f n = z, α + · · · + α n = 1 . Definition 2.1. A rational solution of the A n -Painlev´e system (2.1) is a tuple of functionsand parameters ( f , . . . , f n | α , . . . , α n ) where f i = f i ( z ) are rational functions of z . Remark 2.2.
As we shall see later, for every solution of an A m -Painlev´e system, one canbuild an infinite number of degenerate solutions of a higher order A n -Painlev´e system, with n > m . One of them corresponds to trivially setting some of the f i and α i to zero, but manyother non-trivial embeddings also exist.It will be convenient throughout the paper to work with a different set of functions andparameters, namely the set of functions that satisfy a Darboux dressing chain, which we definenext. Definition 2.3.
A (2 n + 1)-cyclic dressing chain with shift ∆ is a sequence of 2 n + 1 functions w , . . . , w n and complex numbers a , . . . , a n that satisfy the following coupled system of2 n + 1 Riccati-like ordinary differential equations(2.3) ( w i + w i +1 ) ′ + w i +1 − w i = a i , i = 0 , , . . . , n mod (2 n + 1)subject to the condition(2.4) a + · · · + a n = − ∆ . Note that by adding the 2 n + 1 equations (2.3) we immediately obtain a first integral ofthe system(2.5) n X j =0 w j = z n X j =0 a j = − ∆ z. The system (2.3) has a a group of symmetries that will be discussed in Section 7. For now,we will just observe that it is invariant under two obvious transformations of functions andparameters:i) reversal symmetry(2.6) w i
7→ − w − i , a i
7→ − a − i , ∆
7→ − ∆ii) cyclic symmetry(2.7) w i w i +1 , a i a i +1 , ∆ ∆for i = 0 , . . . n mod (2 n + 1).The equivalence between the A n -Painlev´e system (2.1) and the (2 n + 1)-cyclic dressing chain(2.3) is given by the following proposition. Proposition 2.4.
The tuple of functions and complex numbers ( w , . . . , w n | a , . . . , a n ) sat-isfy (2.3) (2.5) , the relations of a (2 n + 1) -cyclic Darboux dressing chain with shift ∆ , if andonly if the tuple (cid:0) f , . . . , f n (cid:12)(cid:12) α , . . . , α n (cid:1) defined by (2.8) f i ( z ) = c ( w i + w i +1 ) ( cz ) , i = 0 , . . . , n mod (2 n + 1) ,α i = c a i ,c = − DAVID G ´OMEZ-ULLATE, YVES GRANDATI, AND ROBERT MILSON satisfies the A n -Painlev´e system (2.1) subject to the normalization (2.2) .Proof. It suffices to invert the linear transformation(2.9) f i = w i + w i +1 , i = 0 , . . . , n mod (2 n + 1)to obtain(2.10) w i =
12 2 n X j =0 ( − j f i + j , i = 0 , . . . , n mod (2 n + 1) , which imply the relations(2.11) w i +1 − w i = n − X j =0 ( − j f i + j +1 , i = 0 , . . . , n mod (2 n + 1) . Inserting (2.9) and (2.11) into the equations of the cyclic dressing chain (2.3) leads to the A n -Painlev´e system (2.1). For any constant c ∈ C , the scaling transformation f i cf i , z cz, α i c α i preserves the form of the equations (2.1). The choice c = − ensures that the normalization(2.2) always holds, for dressing chains with different shifts ∆. (cid:3) Factorization chains of Schr¨odinger operators.
We next recall the relation betweendressing chains and sequences of Schr¨odinger operators related by Darboux transformations,following the theory developed by Veselov and Shabat [53] and Adler [2].Consider the following sequence of Schr¨odinger operators(2.12) L i = − D z + U i , D z = dd z , U i = U i ( z ) , i ∈ Z where each operator is related to the next by a Darboux transformation, i.e. by the followingfactorization(2.13) L i = ( D z + w i )( − D z + w i ) + λ i , w i = w i ( z ) ,L i +1 = ( − D z + w i )( D z + w i ) + λ i . Eliminating the derivative terms we see that (2.13) is equivalent to(2.14) w ′ i + w i = U i − λ i , − w ′ i + w i = U i +1 − λ i . Equivalently, we can characterize w i as the log-derivative of ψ i , the seed function of theDarboux transformation that maps L i to L i +1 (2.15) L i ψ i = λ i ψ i , where w i = ψ ′ i ψ i . Using (2.12) and (2.13), the potentials of the dressing chain are then related by U i − U i +1 = 2 w ′ i , (2.16) U i + U i +1 = 2 w i + 2 λ i . (2.17)It follows that if (2.13) holds with non-constant U i , then the corresponding w i , λ i are deter-mined uniquely by the potentials U i , i ∈ Z . ATIONAL SOLUTIONS OF A n -PAINLEV´E SYSTEMS 7 Definition 2.5.
We say that a sequence of Schrodinger operators L i , i ∈ Z forms a (2 n + 1)-cyclic factorization chain with shift ∆ ∈ C if in addition to (2.13) we also have(2.18) U i +2 n +1 = U i + ∆ , i ∈ Z . Proposition 2.6.
Suppose that the Schr¨odinger operators L i , i ∈ Z form a factorizationchain with shift ∆ . Then, the corresponding w , . . . , w n and (2.19) a i = λ i − λ i +1 , i = 0 , . . . , n form a (2 n + 1) -cyclic Darboux dressing chain with shift ∆ .Proof. Eliminating the potentials in (2.14) and setting (2.19), we obtain the system of coupledequations ( w i + w i +1 ) ′ + w i +1 − w i = a i , i ∈ N whose form coincides with (2.3). Relation (2.18) implies that w i +2 n +1 = w i and that λ i +2 n +1 = λ i +∆. The latter implies that a i +2 n +1 = a i also. Hence the infinite chain of equations relating w i , w i +1 , a i closes onto the finite system (2.3). Since λ n +1 = λ + ∆, from (2.19) it followsthat (2.4) holds. (cid:3) Maya diagrams and Trivial monodromy potentials
In this Section we introduce the main elements and results needed for the classification ofrational solutions of odd-cyclic dressing chains. In Section 4 we will prove the main char-acterization result, namely that all rational solutions of an odd-cyclic dressing chain can beexpressed as log-derivatives of Wronskian determinants whose entries are Hermite polynomi-als. The basis for this proof lies in the theory of Schr¨odinger operators with trivial mon-odromy, for which we refer to the celebrated papers of Duistermaat and Gr¨unbaum [18] andOblomkov [45]. However, before we can state the main theorem we need to recall some basicdefinitions on Maya diagrams and Hermite pseudo-Wronskians, which will be the buildingblocks of all solutions.3.1.
Maya diagrams.
Following Noumi [41], we define a Maya diagram in the followingmanner.
Definition 3.1.
A Maya diagram is a set of integers M ⊂ Z that contains a finite number ofpositive integers, and excludes a finite number of negative integers. Definition 3.2.
Let m > m > · · · be the elements of a Maya diagram M arranged indecreasing order. We define s M ∈ Z , the index of M , as the unique integer such that m i = − i + s M for all i sufficiently large.A Maya diagram can be visually represented as a sequence of • (cid:3) and (cid:3) symbols with thefilled symbol • (cid:3) in position i indicating membership i ∈ M . A Maya diagram thus begins withan infinite sequence of filled • (cid:3) and terminates with an infinite sequence of empty (cid:3) .We next describe the various forms to label Maya diagrams. Definition 3.3.
Let M be a Maya diagram, and M − = {− m − m / ∈ M, m < } , M + = { m : m ∈ M , m ≥ } . Let s > s > · · · > s p and t > t > · · · > t q be the elements of M − and M + arrangedin descending order. The Frobenius symbol of M is defined as the double list ( s , . . . , s p | t q , . . . , t ), DAVID G ´OMEZ-ULLATE, YVES GRANDATI, AND ROBERT MILSON
If a Maya diagram M has the Frobenius symbol ( s , . . . , s p | t q , . . . , t ) , its index is givenby s M = q − p . The classical Frobenius symbol [4, 5, 47] corresponds to the zero index casewhere q = p .A natural operation in Maya diagrams is the following translation by an integer k (3.1) M + k = { m + k : m ∈ M } , k ∈ Z . The behaviour of the index s M under translation of k is given by(3.2) M ′ = M + k ⇒ s M ′ = s M + k. A Maya diagram M ⊂ Z is said to be in standard form if p = 0 and t q >
0. We visuallyrecognize a Maya diagram in standard form when all the boxes to the left of the origin arefilled • (cid:3) and the first box to the right of the origin is empty (cid:3) . Following [29], to every Mayadiagram we associate a polynomial called a Hermite pseudo-Wronskian. Definition 3.4.
Let M be a Maya diagram and ( s , . . . , s r | t q , . . . , t ) its corresponding Frobe-nius symbol. We define the polynomial(3.3) H M ( z ) = exp( − rz ) Wr[exp( z ) e H s , . . . , exp( z ) e H s r , H t q , . . . H t ] , where Wr denotes the Wronskian determinant of the indicated functions, and(3.4) e H n ( z ) = i − n H n (i z )is the n th degree conjugate Hermite polynomial. The polynomial nature of H M ( z ) becomesevident in the following determinantal representation(3.5) H M ( z ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e H s e H s +1 . . . e H s + r + q − ... ... . . . ... e H s r e H s r +1 . . . e H s r + r + q − H t q D z H t q . . . D r + q − z H t q ... ... . . . ... H t D z H t . . . D r + q − z H t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Proposition 3.5 (Theorem 1 in [29]) . For any k ∈ Z , the Hermite pseudo-Wronskians H M and H M + k coincide up to a multiplicative constant. In fact, with the following suitable rescaling of H M (3.6) b H M ( z ) = c M H M ( z ) , c M = ( − rq Q ≤ i We define a rational extension of the harmonic oscillator as the Schr¨odingeroperator L M = − D zz + U M ( z ) , (3.8) U M ( z ) = z − D z log H M ( z ) + 2 s M , (3.9)where H M ( z ) is the associated pseudo-Wronskian (3.3)–(3.5), and s M ∈ Z is the index of M .Potentials U M receive that name because they are the harmonic term plus a rational termthat vanishes for large z . It is easy to give conditions on M for U M to be regular on the realline, [1] or to count the number of real poles of the potential, [26].Let us also note that the above pseudo-Wronskians are related to the τ -functions of the KPhierarchy and that the factorization chain is equivalent to a chain of Hirota bilinear relations.For more details, see the chapter by the present authors in [20].3.2. Trivial monodromy.Definition 3.7. A Schr¨odinger operator L = − D zz + U ( z ) has trivial monodromy at ξ ∈ C if the general solution of the equation L [ ψ ] = − ψ ′′ + U ψ = λψ is meromorphic in a neighbourhood of ξ for all values of λ ∈ C . If L has trivial monodromyat every point ξ ∈ C we say that L is monodromy-free.Duistermaat and Gr¨unbaum proved that the condition that L has trivial monodromy at ξ ∈ C is equivalent to certain restrictions on the coefficients of the Laurent series expansionof the potential. Proposition 3.8 (Proposition 3.3 in [18]) . Let U ( z ) be meromorphic in a neighbourhood of z = ξ with Laurent expansion U ( z ) = X j ≥− c j ( z − ξ ) j , c − = 0 . Then the Schr¨odinger operator L = − D zz + U ( z ) has trivial monodromy at z = ξ if and onlyif there exists an integer ν ≥ such that (3.10) c − = ν ( ν + 1) , c j − = 0 , ≤ j ≤ ν. Oblomkov classified monodromy-free potentials with quadratic growth at infinity, findingthat they can all be obtained by a finite sequence of rational Darboux transformations appliedon the harmonic oscillator. Proposition 3.9 (Theorem 3 in [45]) . The rational extensions of the harmonic oscillator L M (3.8) (3.9) have trivial monodromy. Conversely, if a Schr¨odinger operator L = − D zz + U ( z ) has trivial monodromy and the potential has quadratic growth at infinity, then, up to anadditive constant, U = U M for some Maya digram M . We see thus that the class of monodromy-free potentials with quadratic growth at infinitycoincides with the class of rational extensions of the harmonic oscillator given in Definition 3.7.To every Maya diagram M there corresponds a monodromy-free Schr¨odinger operator whosepotential is a rational extension of the harmonic oscillator. The set of rational Darbouxtransformations preserves this class of operators. More specifically, a single step Darbouxtransformation (2.13) on a Schr¨odinger operator of the form (3.8)-(3.9) leads to anotherrational extension whose Maya diagram differs from the previous one by a single flip. Definition 3.10. Given a Maya diagram M , we define the flip at position m ∈ Z to be theinvolution(3.11) φ m : M ( M ∪ { m } , if m / ∈ M,M \ { m } , if m ∈ M. In the first case, we say that φ m acts on M by a state-deleting transformation ( (cid:3) → • (cid:3) ). Inthe second case, we say that φ m acts by a state-adding transformation ( • (cid:3) → (cid:3) ). Proposition 3.11 (Proposition 3.11 [15]) . Two Maya diagrams M, M ′ are related by a flip (3.11) if and only if their associated rational extensions U M , U M ′ , see (3.9) , are connected bya Darboux transformation (2.16) . Exceptional orthogonal poynomials are intimately related with Darboux transformationsof Schr¨odinger operators, [25, 31]. In fact, the bound states of operators (3.8)-(3.9) essen-tially define exceptional Hermite polynomials, [27, 28, 30]. More generally, it will be useful tocharacterize the class of quasi-rational eigenfunctions of (3.8)-(3.9). We recall that f ( z ) is quasi-rational if (log f ) ′ is a rational function of z . Proposition 3.12. Up to a scalar multiple, every quasi-rational eigenfunction of the Schr¨odingeroperator L M = − D zz + U M ( z ) with U M as in (3.9) has the form (3.12) ψ M,m = exp( σz ) H φ m ( M ) ( z ) H M ( z ) , m ∈ Z , with σ = ( − , if m / ∈ M, +1 , if m ∈ M, Explicitly, we have (3.13) L M ψ M,m = (2 m + 1) ψ M,m , m ∈ Z . Proof. We first prove that (3.12) implies (3.13). Consider a sequence of Maya diagrams M , . . . , M n , M n +1 where M = Z − is the trivial Maya diagram, M n = M, M n +1 = φ m ( M ) , and each Maya diagram M i +1 = φ µ i ( M i ) , i = 0 , . . . , n − , differs from the preceeding one M i by a single flip at position µ i ∈ Z . Let σ i = ( − , if µ i / ∈ M i +1 , otherwise . ATIONAL SOLUTIONS OF A n -PAINLEV´E SYSTEMS 11 The unique quasi-rational eigenfunctions [19] of the classical harmonic oscillator operator L = − D zz + z are L ψ m = (2 m + 1) ψ m , where ψ m ( z ) = ( H m ( z ) e − z / if m ≥ , e H − m ( z ) e z / if m < . A straightforward induction shows that L i = L M i , i = 0 , . . . , n + 1 is a factorization chainwith the corresponding w i ( z ) = σ i z + H ′ M i +1 ( z ) H M i +1 ( z ) − H ′ M i ( z ) H M i ( z ) , (3.14) λ i = 2 µ i + 1 . (3.15)Since w n is the log-derivative of ψ M,m , the eigenvalue relation (3.13) follows immediately.Conversely, suppose that L M ˆ ψ = ˆ λ ˆ ψ, and that ˆ w ( z ) = log( ˆ ψ ( z )) ′ is a rational function. As above, let M , . . . , M n = M be asequence of Maya diagrams such that M is trivial and M i +1 = φ µ i ( M i ) , i = 0 , . . . , n − L , . . . , L n = L M be the corresponding factorization chain of Schr¨odinger operators and( w i | a i ) n − i =0 as in (3.14)-(3.15) with a i = λ i +1 − λ i be the corresponding dressing chain. Wecan extend the chain by setting w n = ˆ w and a n = ˆ λ − λ n . Let(3.16) ˆ w n − = w n − + a n − w n − + w n , ˆ a n − = a n − + a n − , and observe that w ′ n − + ˆ w ′ n − + ˆ w n − − w n − = w ′ n − − a n − w n − + w n ( w n − + w n ) ′ w n − + w n + (cid:18) w n − + a n − w n − + w n (cid:19) − w n − = w ′ n − − a n − w n − + w n (cid:18) w n − − w n + a n − w n − + w n (cid:19) + (cid:18) w n − + a n − w n − + w n (cid:19) − w n − = a n − + a n − = ˆ a n − . In this way we obtain a shorter dressing chain w , . . . , w n − , ˆ w n − where all of the componentsare rational functions. Continuing this argument inductively we arrive at a rational function˜ w ( z ) that satisfies the Ricatti equation˜ w ′ + ˜ w = z − ˜ λ, and is related to ˆ w by a sequence of rational transformations (3.16). We conclude that ˜ w ( z )is the log-derivative of a quasi-rational eigenfunction of the classical harmonic oscillator L ,and hence either ˜ w = − z + H ′ m H m , m ≥ w = z + e H ′− m e H − m , m < . Successively applying the inverse of the rational transformation (3.16) we conclude thatˆ w = ± z + H ′ φ m ( M ) H φ m ( M ) − H ′ M H M . Therefore, up to a non-zero scalar multiple the corresponding ˆ ψ must have the form (3.12). (cid:3) Note that this characterization covers all quasi-rational eigenfunctions, not just the squareintegrable ones. For our purpose of classifying rational solutions to (2 n + 1)-cyclic dressingchains this is the relevant class, and square integrability of the eigenfunctions plays no role.Therefore, we employ the term eigenfunction in this formal sense, as solutions to the eigenvalueproblem.4. Characterization of rational solutions to odd-cyclic dressing chains In this section we state and prove the main result that allows the classification of rationalsolutions to the A n -Painlev´e system, namely that all of them belong to the class of rationalextensions of the harmonic oscillator. Most contents of this Section follow closely the resultsobtained by Veselov in [52], adapting the notation to our needs and providing further proofsfor intermediate results where we found it necessary.We start by proving that the only possible poles of w i are simple, and growth at infinity isat most linear. Proposition 4.1. If ( w , . . . , w n | a , . . . , a n ) is a rational solution of a (2 n + 1) -cyclic dress-ing chain, then each function w i necessarily has the form (4.1) w i = ± az + b i + N X j =1 a ij z − ζ j , a, b i , a ij , ζ j ∈ C , a = 0 i = 0 , , . . . , n. Proof. We can rewrite the dressing chain equations (2.3) as f ′ i + f i d i = α i , f ′ i = 0 , where f i = w i + w i +1 , d i = w i − w i +1 . Hence, we can write 2 w i = f i − f ′ i f i + α i f i (4.2) 2 w i +1 = f i + f ′ i f i − α i f i . (4.3)Suppose that each w i in the chain has the following behaviour for large zw i = a i z k i + O ( z k i − ) , z → ∞ , a i = 0 , i = 0 , , . . . , n. Our first claim is that k i ≥ i . This follows by inspection of (4.2). Our second claimis that that k i +1 = k i for all i . If k i = 0 for every i , then the claim follows trivially. Supposethen that k i > i . By (4.2), it is clear that as z → ∞ either f i = 12 a i z k i + O ( z k i − ) , ATIONAL SOLUTIONS OF A n -PAINLEV´E SYSTEMS 13 or α i f i = 12 a i z k i + O ( z k i − ) , α i = 0 . In any of the two cases we have(4.4) w i +1 = ± a i z k i + O ( z k i − ) , z → ∞ , thereby proving the claim. Our third claim is that k i = 1 for all i . This follows because thesum in (2.5) involves an odd number of terms. Finally, we conclude that a i +1 = ± a i by (4.4).Let ζ j , j = 1 , . . . N be the poles of w , . . . , w n . Fix a j and write the Laurent expansion w i = a ij ( z − ζ j ) − ℓ i + O (( z − ζ j ) − ℓ i +1 ) , z → ζ j , a ij = 0 , i = 0 , , . . . , n, We claim that ℓ i ≤ i . Suppose not and that ℓ i ≥ i . By (4.2), as z → ζ j either f i = 12 a ij ( z − ζ j ) − ℓ i + O (( z − ζ j ) − ℓ i +1 ) , or α i f i = 12 a ij ( z − ζ j ) − ℓ i + O (( z − ζ j ) − ℓ i +1 ) , α i = 0 . In both cases, w i +1 = ± a ij ( z − ζ j ) − ℓ i + O (( z − ζ j ) − ℓ i +1 ) , z → ζ j . Thus, ℓ i +1 = ℓ i and a i +1 ,j = ± a i,j for every i . Since the sum in (2.5) involves an odd numberof terms, this leads to a contradiction. (cid:3) Proposition 4.2. Let ( w , . . . , w n | a , . . . , a n ) be a rational solution of a (2 n + 1) -cyclicdressing chain and let ζ ∈ C be a pole of some function w i in the chain. Then we have (4.5) Res ζ w i = 0 , Res ζ w i ∈ Z , | Res ζ w i | ≤ n. We need to show that if ζ ∈ C is a pole of w i , then(4.6) w i = m i ( z − ζ ) − + O ( z − ζ ) , z → ζ, m i ∈ Z . By Proposition 4.1, ζ is a simple pole of w i so the local behaviour of w i near ζ is(4.7) w i = a i ( z − ζ ) − + b i + O ( z − ζ ) , z → ζ, i = 0 , , . . . , n. Inserting the above expansion for w i in the equations of the dressing chain (2.3) and collectingthe leading order terms at ( z − ζ ) − and ( z − ζ ) − we obtain the relations − ( a i + a i +1 ) + a i +1 − a i = 0 , (4.8) a i +1 b i +1 − a i b i = 0 , i = 0 , . . . , n mod (2 n + 1) . (4.9)These equations, together with the constraints on { a i , b i } , i = 0 , . . . , n derived from theclosure condition (2.5) are enough to prove the desired claim, which proceeds by derivingthree chained lemmas. Lemma 4.3. Let ζ be a simple pole of a rational function in a (2 n + 1) -cyclic dressing chainas per (4.7) , and let { a i } ni =0 be the sequence of residues of w i at ζ as per (4.7) . For each i = 1 , , . . . , n + 1 , there exists a k i ∈ { , . . . , i } such that (4.10) a i = ( − k i a + k i − i. Proof. The proof is by induction on i . By (4.8) we have(4.11) a i +1 = − a i , or a i +1 = a i + 1 . For i = 0, the first case corresponds to k = 1, and second case to k = 2. Suppose (4.10)holds for a given i . Hence, a i +1 = ( − k i +1 a − k i + i, or a i +1 = ( − k i a + k i − i + 1 . The first possibility corresponds to k i +1 = − k i + 2 i + 1 , and the second possibility corresponds to k i +1 = k i + 2 , but in both cases k i +1 ∈ { , . . . , i + 1) } , thus establishing the claim. (cid:3) Lemma 4.4. Let ζ be a simple pole of a rational function w i that solves a (2 n + 1) -cyclicdressing chain. Then the residue a i at ζ must be an integer a i ∈ {− n, . . . , n } .Proof. The results follows trivially from the previous lemma and the closure condition. Indeed,Lemma 4.4 for i = 2 n + 1 reads a n +1 = a = ( − k n +1 a + k n +1 − (2 n + 1) . The closure condition a n +1 = a implies that the second possibility in (4.11) occurs an evennumber of times and the first possibility an odd number of times. Since k = 0, we see that k n +1 must be an odd number, and therefore we can write k n +1 = 2 j +1 with j ∈ { , . . . , n } .Hence, 2 a = k n +1 − (2 n + 1) = 2( j − n ) , which proves the claim for the residue a . The result extends from a to any a i in the chainby cyclicity. (cid:3) Lemma 4.5. Let ζ be a simple pole of a rational function in a (2 n + 1) -cyclic dressing chainas per (4.7) , and let { a i } ni =0 be the sequence of residues of w i at ζ as per (4.7) . Then a i = 0 for some i = 0 , , . . . , n .Proof. We argue by contradiction and suppose that sgn a i ∈ {− , } for all i = 0 , , . . . , n .Hence, sgn a i +1 = ± sgn a i . From the cyclic condition a n +1 = a , it follows that in the set i ∈ { , . . . , n } there mustbe an even number of indices such that sgn a i +1 = − sgn a i , and therefore an odd number ofindices such that | a i +1 | = | a i | ± . It follows that | a n +1 | − | a | is an odd integer, which is a contradiction. (cid:3) The previous lemma implies that it is impossible that all the rational functions in thedressing chain have a common pole. We are now ready to conclude the proof of Proposition 4.2. Proof of Proposition 4.2. From Lemma 4.5 it follows that a i b i = 0 for at least one i =0 , , . . . , n . Hence, by (4.9), a i b i = 0 for all i = 0 , , . . . , n ; i.e. either a i = 0 or b i = 0for every i . Relative to form (4.7), this is equivalent to the condition that Res ζ w i = 0. Theintegrality and bounds on the possible values of the residues a i follow from Lemma 4.4. (cid:3) ATIONAL SOLUTIONS OF A n -PAINLEV´E SYSTEMS 15 By Proposition 4.2, the expansion (4.6) holds at every pole z = ζ of a (2 n + 1)-cyclicfactorization chain. In particular the residue m i = Res ζ w i is an integer and | m i | ≤ n . Theconclusions of that proposition can be strengthened in the following manner. Proposition 4.6. Let ( w , . . . , w n | a , . . . , a n ) be a rational solution of a (2 n + 1) -cyclicdressing chain. Let ζ ∈ C be a pole of a function w i in the chain and m i = Res ζ w i . Then wehave (4.12) Res ζ w ji = 0 , j = 1 , . . . , | m i | . In a similar manner, we structure the proof of this result in three simple lemmas. Considerthe local expansion of w i around the pole at z = ζ , which according to Proposition 4.2 hasthe form(4.13) w i = m i ( z − ζ ) − + ∞ X j =0 b ij ( z − ζ ) j , where m i ∈ Z is an integer. Lemma 4.7. Let S be the set of residues of all functions of the chain at z = ζ : S = { m i : i = 0 , , . . . , n } . Then, we have − S = S .Proof. Consider an arbitrary m i ∈ S . If m i = 0, then evidently − m i ∈ S . Suppose that m i > 0. If m i +1 = − m i , we are done. Otherwise let k > | m i +1 | , . . . , | m i + k | > m i . Such a k must exist because of cyclicity. By (4.11), | m i + k +1 | − | m i + k | ∈ {− , , } . Since k is as large as possible, | m i + k +1 | = m i . Suppose that m i + k +1 = m i . That wouldmean from (4.11) that either m i + k = m i − m i + k = − m i , both of which contradict thehypothesis that | m i + k | > m i . Therefore m i + k +1 = − m i , and − m i ∈ S also. The case m i < (cid:3) Lemma 4.8. If k ∈ S is positive, then k − ∈ S also.Proof. We argue by contradiction and suppose that there exists a k > k ∈ S but k − / ∈ S . By Lemma 4.7, 1 − k / ∈ S also. From (4.11) we have | m i +1 | − | m i | ∈ {− , , } soit would follow that | m i | ≥ k for all i = 0 , . . . , n, which contradicts Lemma 4.5. (cid:3) Lemma 4.9. Let m = max { m i } ni =0 . Then, S = {− m, − m + 1 , . . . , m − , m } . Proof. This follows directly from Lemmas 4.7 and 4.8. (cid:3) We see that the set of residues at a given pole z = ζ along the chain contains all integervalues between − m and m , with m ≤ n . We are now ready to prove Proposition 4.6. Proof of Proposition 4.6. Given the expansion in (4.13), the claim (4.12) is equivalent to show-ing that b i, j − = 0 , for all j = 1 , . . . , | m i | . The argument proceeds by induction on j . Proposition 4.2 established (4.12) for j = 1.Suppose that (4.12) holds for all j ≤ k for a given k ∈ N . Suppose that | m i | > k . We willshow that b i, k = 0.Let p < i be the largest integer such that m p = k and q > i be the smallest integer such that m q = − k . If k < | m i | , such p, q are guaranteed to exist by Lemma 4.9. Thus, by construction(4.14) | m ℓ | ≥ k + 1 , ℓ = p + 1 , . . . , i, . . . , q − . By the inductive assumption b i, j − = 0 , i = p, . . . , q, j = 1 , . . . k. Hence, the vanishing of the coefficient of z k − in (2.3), implies that(4.15) ( k + m ℓ +1 ) b ℓ +1 , k + ( k − m ℓ ) b ℓ, k = 0 , ℓ = p, . . . , q − . Since k + m q = 0 and k − m p = 0, from (4.14) and (4.15) it follows that b q − , k = b p +1 , k = 0 . which in turn imply by cascade that b ℓ, k = 0 for all ℓ = p + 1 , . . . , i . . . q − , and in particular b i, k = 0 as was to be shown. (cid:3) Definition 4.10. Given two sets A and B we define its symmetric difference as the union ofthe set of elements of A that are not in B with the set of elements of B that are not in A (4.16) A ⊖ B = ( A \ B ) ∪ ( B \ A ) . The characterization of rational solutions of the A n system can be done in terms of Mayadiagrams and Maya cycles, a new concept that we introduce below. Definition 4.11. A ( p, k ) Maya cycle is a sequence of Maya diagrams M = ( M , M , . . . , M p )such that M i is related to M i +1 by a single flip, and such that M p = M + k, k ∈ Z . Thesequence µ ∈ Z p where(4.17) { µ i } = M i +1 ⊖ M i , i = 0 , . . . , p − flip sequence of the Maya cycle, because, by construction, µ is the uniquesequence such that M i +1 = φ µ i ( M i ) , i = 0 , . . . , p − . For each i = 0 , . . . , p − σ i = ( − , if µ i / ∈ M i , +1 , if µ i ∈ M i , and refer to the sequence σ = ( σ , . . . , σ p − ) as the sign sequence of the Maya cycle.Observe that if M is a Maya cycle, then so is(4.19) M + j = ( M + j, M + j, . . . , M p + j ) , j ∈ Z . We use M / Z to denote the equivalence class of a Maya cycle M modulo integer translations. ATIONAL SOLUTIONS OF A n -PAINLEV´E SYSTEMS 17 We are now able to formulate the main theorem of this Section that characterizes rationalsolutions of an odd-cyclic dressing chain, and therefore rational solutions of the A n system. Theorem 4.12. Let M = ( M , . . . , M n +1 ) be a (2 n +1 , k ) Maya cycle with flip sequence µ =( µ , . . . , µ n ) ∈ Z n +1 and sign sequence σ = ( σ , . . . , σ n ) ∈ {− , } n +1 . For i = 0 , . . . , n ,set w i ( z ) = σ i z + H ′ M i +1 ( z ) H M i +1 ( z ) − H ′ M i ( z ) H M i ( z ) , (4.20) a i = 2( µ i − µ i +1 ) , µ n +1 = µ + k, (4.21) where H M i ( z ) and H M i +1 ( z ) are the corresponding Hermite pseudo-Wronskians (3.5) . Then, ( w | a ) = ( w , . . . , w n | a , . . . , a n ) is a rational solution of a (2 n + 1) -cyclic dressing chainwith shift ∆ = 2 k . Conversely, every rational solution ( w | a ) of a (2 n + 1) -cyclic dressingchain is determined in this fashion by a unique Maya cycle class M / Z .Proof. Let M be a (2 n + 1 , k ) Maya cycle and L i = L M i = − D zz + U i ( z ) , i = 0 , . . . , n + 1 , the corresponding sequence of rational extensions defined by (3.8) (3.9). The cyclicity condi-tion M n +1 = M + k , together with (3.2), (3.7), and (3.9) imply that U n +1 = U + 2 k, so the sequence L , . . . , L n +1 is factorization chain with shift ∆ = 2 k . Using definition (3.12),set ψ i = ψ M i ,µ i , i = 0 , . . . , n, so that L i ψ i = (2 µ i + 1) ψ i . Let w i , a i , i = 0 , . . . , n be defined by (4.20) (4.21). Then, by Proposition 2.6 and byProposition 3.12, the tuple ( w , . . . , w n | a , . . . , a n ) is a rational solution of a (2 n + 1)-cyclicdressing chain with shift ∆ = 2 k .We now prove the converse statement. We first show that the conditions satisfied by eachrational solution w i at a pole ζ , as expressed by Propositions 4.2 and 4.6 are precisely theconditions that express local trivial monodromy of the corresponding potential U i . Denote by w any of the rational functions of a tuple ( w , . . . , w n | a , . . . , a n ) that satisfies a (2 n + 1)-cyclic dressing chain and let ζ be a pole of w . By Propositions 4.2 and 4.6, the Laurentexpansion of w at z = ζ is(4.22) w = ∞ X j = − b j ( z − ζ ) j , b − = m ∈ Z , b j = 0 for j = 0 , . . . , | m | − . Since U = w ′ + w + λ by (2.14), the Laurent expansion at ζ of U is: U = X j ≥− c j ( z − ξ ) j where c − = m ( m − , c − = 2 mb , c = b + (2 m + 1) b + λ and c j − = 2 jb j + 2 j − X i = − b i b j − i − , j ≥ c j = (2 j + 1) b j +1 + b j + 2 j − X i = − b i b j − i − , j ≥ U has trivial monodromy at z = ζ if and only if there exists an integer ν ≥ m , we choose ν as ν = ( − m if m < ,m − m > , Note that if w has pole at z = ζ with residue m = − m = 0, the potential U is regularin a neighbourhood of ζ . For other integer values of m , the conditions (4.22) on the evencoefficients of w imply that the precise number of odd coefficients (4.23) of U vanish, asrequired by Proposition 3.8. We conclude that U has trivial monodromy at z = ζ , and since ζ is arbitrary, U is a monodromy-free potential. From Proposition 4.1 and (2.14) it followsthat U is a monodromy-free potential with quadratic growth at infinity, so Proposition 3.9implies that U is a rational extension of the harmonic oscillator, i.e. it has the form (3.9) forsome Maya diagram M . Recalling that w is the log-derivative of the seed function for theDarboux transformation (see (2.15)), and that all quasi-rational seed functions of potentials(3.9) are characterized by Proposition 3.12, it suffices to take the log-derivative of (3.12) toachieve the desired result (4.20). This argument was applied on an arbitrary element of thedressing chain, and therefore it applies to all such elements.The cyclicity of the dressing chain implies that the corresponding Schr¨odinger opera-tors in the factorization chain given by Proposition 2.6 are rational extensions of the har-monic oscillator (3.8)-(3.9), and the closure condition (2.18) defines a Maya cycle M =( M . . . , M n , M n +1 ) where M n +1 = M + k . From Proposition 6.7 we see that all Mayacycles in the equivalence class M / Z lead to the same rational solution. (cid:3) Cyclic Maya diagrams In Section 4 we saw that the rational solutions to a (2 n + 1)-cyclic dressing chain must berational extensions of the harmonic oscillator and they are indexed by Maya diagrams. Morespecifically, components of the solution have the form (4.20) and are essentially determined bytwo Maya diagrams connected by a flip operation (or equivalently, by two potentials relatedby a Darboux transformation). The periodicity of the dresing chain thus translates into acyclicity condition on the Maya diagrams. A sequence of flip operations encodes multi-stepDarboux transformations (also called Crum transformation [17]) at the level of Maya diagrams Definition 5.1. Let ˆ Z p , p ∈ N denote set of integer sets of cardinality p , and let Z p denotethe set of integer multisets of cardinality p . We identify a set β ∈ ˆ Z p with the strictlyincreasing integer sequence β < β < · · · < β p − that enumerates β , and identify a multi-set β ∈ Z p with a non-decreasing integer sequence β ≤ β ≤ · · · ≤ β p − . In this way, we regardˆ Z p as a subset of Z p . ATIONAL SOLUTIONS OF A n -PAINLEV´E SYSTEMS 19 We will use curly braces to denote both sets and multi-sets, letting the context resolve theambiguity, and round parentheses to denote sequences/tuples. Definition 5.2. For a set β ∈ ˆ Z p let φ β denote the multi-flip (5.1) φ β = φ β ◦ · · · ◦ φ β p − . where the action of each single flip on a Maya diagram M is given by (3.11).We also use relation (5.1) to define φ β where β ∈ Z p is a multiset. Since flips are involutions,we have φ β = φ β ′ , β ∈ Z p where β ′ is the set consisting of the elements of β with an odd cardinality. Definition 5.3. A Maya diagram M is p -cyclic with shift k , or simply ( p, k )-cyclic, if thereexists a multi-flip φ β , β ∈ Z p such that(5.2) φ β ( M ) = M + k. More generally, a Maya diagram M will be said to be p -cyclic if it is ( p, k )-cyclic for someshift k ∈ Z .For a fixed shift k , every Maya diagram M is ( p, k )-cyclic for some value of the period p (Proposition 5.3 in [15]). However, the relevant problem we need to address is the converse:that of enumerating and classifying all cyclic Maya diagrams for a fixed odd period p = 2 n + 1.The desired classification for cyclic Maya diagrams of a fixed period can be achieved byemploying the key concepts of genus and interlacing . The genus of a Maya diagram countsesentially the number of blocks of filled boxes • (cid:3) in the finite part of M , and the initial andending position of each block are called the block coordinates . Specifying its block coordinatesdetermines a Maya diagram uniquely, and this will be the most convenient representation forour purpose of classifying cyclic Maya diagrams. Let us make all these notions more precise. Definition 5.4. Let β ∈ Z ℓ +1 be an integer multiset of odd cardinality with non-decreasingenumeration β ≤ β ≤ · · · ≤ β ℓ . Let Ξ( β ) be the Maya diagram defined by(5.3) Ξ( β ) = ( −∞ , β ) ∪ [ β , β ) ∪ · · · ∪ [ β ℓ − , β ℓ ) , where(5.4) [ m, n ) = { j ∈ Z : m ≤ j < n } . We refer to the set β as the block coordinates of the Maya diagram M = Ξ( β ).It is important to note that if β has repeated elements then the same Maya diagramadmits a representation in terms of a smaller number of block coordinates. We make thisnotion precise in the following proposition. Proposition 5.5. For every Maya diagram M , there exists a unique g ∈ N and a unique set β ∈ ˆ Z g +1 such that M = Ξ( β ) .Proof. Every Maya diagram M has a unique description M = ( −∞ , β ) ∪ [ β , β ) ∪ · · · ∪ [ β g − , β g ) where β < β < · · · < β g is an increasing integer sequence. Observe that β = { β , β , . . . , β g } is precisely the the set of integers that are in M but are not in M + 1 and viceversa. Thus,the desired set β ∈ ˆ Z g +1 can be given as(5.5) β = ( M + 1) ⊖ M, where ⊖ denotes the symmetric set difference defined in (4.16). (cid:3) Given a multiset β ′ = { β n , . . . , β n p p } ∈ Z ℓ +1 with elements β i and multiplicities n i ∈ N ,such that n + · · · + n p = 2 ℓ + 1, the corresponding set of block coordinates described byProposition 5.5 is given by β = { β m , . . . , β m p p } ∈ ˆ Z g +1 , where m i = n i mod (2) , i = 0 , . . . , p, and m + · · · + m p = 2 g + 1. It is clear from (5.3) that Ξ( β ′ ) = Ξ( β ). Definition 5.6. Given a Maya diagram M , let β be the set of cardinality 2 g + 1 , g ∈ N described in the above Proposition. We say that g is the genus of M . Proposition 5.7. A genus g Maya diagram is (2 g + 1 , -cyclic.Proof. As a direct consequence of (5.5) we have φ β ( M ) = M + 1 . (cid:3) Let Z odd denote the set of multisets of odd cardinality, and let ˆ Z odd denote the set ofsets of odd cardinality. The mapping (5.3) defines a bijection Ξ : ˆ Z odd → M , which mapscardinality to genus. A multi-set β ∈ Z (2 g +1) corresponds to a non-degreasing sequence β ≤ β ≤ · · · ≤ β g and can also be used to define a Maya diagram using (5.3). However, if β has some repeated elements, then some of the blocks in (5.3) will coalesce and result in aMaya diagram whose genus is strictly smaller than g . This observation may be encapsulatedby saying that the extended mapping Ξ : Z odd → M is onto, but not one-to-one, and thatthe cardinality of the multiset dominates the genus of the corresponding Maya diagram.The visual explanation of the genus concept is clear in Figure 5.1. Removing the infiniteinitial • (cid:3) and trailing (cid:3) segments, a Maya diagram consists of alternating empty (cid:3) and filled • (cid:3) segments of finite variable length. The genus g counts the number of such pairs. The blockcoordinates β i indicate the starting positions of the empty segments, and β i +1 signal thestarting positions of the filled segments. Finally, note that M is in standard form if and onlyif β = 0.With the well known corespondence between Maya diagrams and partitions, it is worthnoting that the genus of a Maya diagram coincides with the number of distinct parts of thepartition, [15]. This feature has been studied earlier in [3], in connection with some identitiesin the theory of q -series. ... ... M = ( −∞ , β ) ∪ [ β , β ) ∪ [ β , β ) − − − β β β β β Figure 5.1. Block coordinates β = { , , , , } of a genus 2 Maya diagram. ATIONAL SOLUTIONS OF A n -PAINLEV´E SYSTEMS 21 Modular decomposition and colouring. We have just seen how to characterize cyclicMaya diagrams M ∈ M g with shift k = 1: one needs 2 g +1 flips given by the block coordinatesof M . The following dual notions of interlacing and modular decomposition allow us toleverage this result to arbitrary shifts k . Note that, due to the reversal symmetry (2.6), wecan restrict the analysis to positive shifts k > k Maya diagrams M (0) , . . . , M ( k − . To describe M using the block coordinates of each M ( j ) , j = 0 , . . . , k − 1, we introduce the notion of colouredmultisets. In the following section we will extend this idea to the notion of coloured sequences,which will furnish us with a combinatorial representation of rational solutions compatible withthe action of the extended affine Weyl group ˜ A (1)2 n . Definition 5.8. Fix a k ∈ N and let M (0) , M (1) , . . . M ( k − ⊂ Z be sets of integers. We definethe interlacing of k sets to be the set(5.6) Θ k (cid:16) M (0) , M (1) , . . . M ( k − (cid:17) = k − [ i =0 ( kM ( i ) + i ) , where kM + j = { km + j : m ∈ M } , M ⊂ Z . Conversely, given a set of integers M ⊂ Z and a positive integer k ∈ N , we can define the k -modular decomposition of M as the k -tuple of sets (cid:0) M (0) , M (1) , . . . M ( k − (cid:1) , where M ( i ) = { m ∈ Z : km + i ∈ M } , i = 0 , , . . . , k − . These operations are clearly the inverse of each other, in the sense that (cid:0) M (0) , M (1) , . . . M ( k − (cid:1) is the k -modular decomposition of M if and only if M = Θ k (cid:0) M (0) , M (1) , . . . M ( k − (cid:1) .Although interlacing and modular decomposition apply to general sets, they have a welldefined restriction to Maya diagrams. Indeed, if M = Θ k (cid:0) M (0) , M (1) , . . . M ( k − (cid:1) and M isa Maya diagram, then M (0) , M (1) , . . . M ( k − are also Maya diagrams. The converse is alsotrue.The notions of interlacing and modular decomposition also apply to the setting of finiteinteger multisets. Let A ⊔ B denote the disjoint union of multisets A, B ; i.e. ⊔ is the operationthat adds element multiplicities. It is clear that if γ ( i ) ∈ Z p i , i = 0 , , . . . , k − 1, then γ [ k ] = Θ k ( γ (0) , . . . , γ ( k − ) = k − G i =0 ( k γ ( i ) + i ) , (5.7)is an integer multiset of cardinality p = p + · · · + p k − , and that ( γ (0) , . . . , γ ( k − ) serve as the k -modular decomposition of γ [ k ] .It should be noted that modular decompositions of Maya diagrams have been consideredpreviously by Noumi (Proposition 7.12 in [41]), although in a slightly different context: thatof studying the effect of B¨acklund transformations on Maya diagrams. We shall address thismatter further in Section 7. After introducing the notions of genus and interlacing, we recallwithout proof the main result to characterize cyclic Maya diagrams. Proposition 5.9 (Theorem 4.8 in [15]) . Consider an arbitrary Maya diagram M , let M =Θ k (cid:0) M (0) , M (1) , . . . M ( k − (cid:1) be its k -modular decomposition, and g i the genus of M ( i ) for i =0 , , . . . , k − . Then, M is ( p, k ) -cyclic where (5.8) p = p + p + · · · + p k − , p i = 2 g i + 1 . The proof of this Proposition essentially states that a shift of M by k can only be doneif each of the M ( i ) is shifted by one, for which precisely p flip operations at locations (5.7)are needed [15]. Proposition 5.9 establishes a link between the shift k , the period p and thegenera of the Maya diagrams that form the k -modular decomposition of M . Applying thisProposition for a fixed period p = 2 n + 1, one can enumerate all possible cyclic Maya diagramswith that period, so it is a key element towards the full classification of rational solutions tothe dressing chain. From (2.6) and (5.8) we see the only possible values of the shift ∆ for anodd-cyclic dressing chain. Corollary 5.10. For a fixed period p = 2 n + 1 ∈ N , there exist (2 n + 1) -cyclic Maya diagramswith shifts k = ± (2 n + 1) , ± (2 n − , . . . , , and no other shifts are possible. Remark 5.11. The highest shift k = p corresponds to the interlacing of p trivial (genus0) Maya diagrams. This class of solutions has been described already by Tsuda [49], wherethe interlacing of p genus-0 Maya diagrams correspond to p -reduced partitions. For p = 3,these solutions are known as Okamoto polynomials, so in general the highest shift k = 2 n + 1dressing chains generalize the Okamoto class.We now introduce the notion of colouring, a useful visual representation of modular decom-position. Colouring also plays an essential role in the formulation of the classification resultsthat follow. Definition 5.12. A k -coloured multiset is the assignment of one of k colours to the elementsof a given integer multiset. Formally, we represent a k -coloured multiset by γ = { ( γ i , C i ) } pi =1 where γ i ∈ Z and C i ∈ Z /k Z = { , , . . . , k − } is the “colour” of the i -th element. A k -coloured multiset defines the following multiset decomposition(5.9) γ = γ (0) ⊔ · · · ⊔ γ ( k − , where γ ( j ) , j ∈ Z /k Z is the sub-multiset of elements having colour j . We use Z pk to denotethe set of all k -coloured multisets of cardinality p . Let p j , j ∈ Z /k Z denote the cardinalityof γ ( j ) . We will call the sequence p = ( p , . . . , p k − ) the signature of γ . Observe that, bydefinition, p serves as a composition of p , namely p = p + · · · + p k − .We may now express the interlacing operator Θ k defined in (5.7) as the bijection Θ k : Z pk → Z p with action given by (5.7) Definition 5.13. Fix a k ∈ N and let M be a Maya diagram. We refer to(5.10) γ [ k ] = ( M + k ) ⊖ M as the k th order flip set of M . We will call γ = Θ − k ( γ [ k ] ) the k th order block coordinates of M , and refer to the corresponding p = ( p , . . . , p k − ) as the k th order signature of M .Observe that (5.10) entails(5.11) φ γ [ k ] ( M ) = M + k ATIONAL SOLUTIONS OF A n -PAINLEV´E SYSTEMS 23 Thus, the k th order flip set γ [ k ] is the minimum set of flips that turns M into M + k . Thisset coincides with the block coordinates β of M when k = 1, but otherwise the two sets aredifferent. Definition 5.14. We say that that γ ∈ Z pk is an oddly coloured multiset if the entries ofthe corresponding signature p are odd, that is if each colour occurs an odd number of times .We say that γ ∈ Z pk is a k -coloured set if the corresponding γ [ k ] = Θ k ( γ, C ) is a set, orequivalently if each of the γ ( j ) in the decomposition (5.9) do not contain repeated elements.Note that for a given oddly coloured multiset γ ∈ Z pk , γ [ k ] ∈ Z p defined by (5.7) is ingeneral a multi-set such that (5.11) holds. However, the k th order flip set defined by (5.10) isalways a set as it contains no repeated integers. Proposition 5.15 (Proposition 4.13 in [15]) . Fix a k ∈ N . For every Maya diagram M ,the corresponding k th order block coordinates are an oddly k -coloured set. Conversely, for anoddly coloured multiset γ ∈ Z pk , define (5.12) Ξ k ( γ ) = Θ k (Ξ( γ (0) ) , . . . , Ξ( γ ( k − )) . Then, M = Ξ k ( γ ) is a ( p, k ) -cyclic Maya diagram.Proof. Let M be a Maya diagram and γ [ k ] = ( M + k ) ⊖ M its k th order flip set. Set γ =Θ − k ( γ [ k ] ) and observe that φ γ [ k ] ( M ) = M + k = Θ k ( φ γ (0) M (0) , . . . , φ γ ( k − M ( k − )= Θ k ( M (0) + 1 , . . . , M ( k − + 1)It follows that M ( j ) = Ξ( γ ( j ) ) , j ∈ Z /k Z , and hence that each γ ( j ) , j ∈ Z /k Z has odd cardinality.We turn to the proof of the converse. Suppose that γ ∈ Z pk is an oddly coloured multiset,and let M = Ξ k ( γ ). Set γ [ k ] = Θ k ( γ ) and observe that, by construction, φ γ [ k ] ( M ) = M + k .This proves the second assertion. (cid:3) Example 5.16. Figure 5.2 provides a visual interpretation of the modular decomposition ofa Maya diagram M into Maya diagrams M (0) , M (1) , M (2) of genus 1 , , 0, respectively. Eachof these Maya diagrams is dilated by a factor of 3, shifted by one unit with respect to theprevious one and superimposed.The block coordinates of each of the three diagrams are given by: γ (0) = { , , } , M (0) = Ξ( γ (0) ) = ( −∞ , ∪ [1 , γ (1) = {− , , , , } , M (1) = Ξ( γ (1) ) = ( −∞ , − ∪ [1 , ∪ [5 , γ (2) = { } , M (2) = Ξ( γ (2) ) = ( −∞ , Z / Z = { , , } . The interlacing of these three Maya diagrams isdescribed by the 3 rd order block coordinates γ = γ (0) ⊔ γ (1) ⊔ γ (0) = { , , , − , , , , , } , If p is odd, then the number of colours k in an odd colouring must also be odd. which form a 3-coloured set of cardinality p = p + p + p = 3 + 5 + 1 = 9. The signature istherefore p = (3 , , rd order flip set is given by γ [3] = Θ ( { , , , − , , , , , } ) = { , , , − , , , , , } ∈ Z It is straightforward to verify in this example that (5.11) holds. Note that the interlaceddiagram M has genus 5 and its block coordinates β are given by β = {− , − , , , , , , , , , } ∈ ˆ Z In general, there are no simple expressions to derive β from γ ( i ) or to connect the genera g i of the coloured Maya diagrams M ( i ) with the genus g of the resulting interlaced Mayadiagram M . However, the block coordinates β of interlaced Maya diagrams M do not playany significant role in the construction of Maya cycles and rational solutions. ... ... M = Ξ( { , , } ) , g = 1 − − − − ... ... M = Ξ( {− , , , , } ) , g = 2 ... ... M = Ξ( { } ) , g = 0 − − − − − M = Θ ( M , M , M ) = Ξ ( { , , , − , , , , , } ) Figure 5.2. Interlacing of three Maya diagrams with genus 1 , Classification of rational solutions to A n -Painlev´e Definition 4.11 above introduced the concept of Maya cycles : sequences of Maya diagramsconnected by flip operations that close into a cycle. In order to build all rational solutions ofa (2 n + 1) cyclic dressing chain, all we need to specify is how to build all (2 n + 1 , k ) Mayacycles for k = ± , . . . , ± n + 1.Evidently, every M i in a ( p, k ) Maya cycle is ( p, k )-cyclic as per Definition 5.3. The followingProposition elucidates the relationship between cyclic Maya diagrams and Maya cycles. Proposition 6.1. Let M be a ( p, k ) -cyclic Maya diagram, φ γ [ k ] , γ [ k ] ∈ Z p a multi-flip suchthat φ γ [ k ] ( M ) = M + k , and µ ∈ Z p an arbitrary enumeration of γ [ k ] . Then (6.1) M i +1 = φ µ i ( M i ) , i = 0 , . . . , p − , defines a ( p, k ) Maya cycle. ATIONAL SOLUTIONS OF A n -PAINLEV´E SYSTEMS 25 Coloured sequences. In the same way that ( p, k ) cyclic Maya diagrams can be indexedby k -coloured sets, ( p, k ) Maya cycles will be indexed by k -coloured sequences, a concept thatwe introduce next. Definition 6.2. A k -colouring of a sequence is the assignment of one of k colours to eachcomponent of that sequence. Setting Z pk = Z p × ( Z /k Z ) p , we formally represent a k -colouredinteger sequence of length p as a pair( ν , C ) = (cid:0) ( ν , . . . , ν p − ) , ( C , . . . , C p − ) (cid:1) ∈ Z pk where C i is the colour of ν i , for i = 0 , . . . , p − 1. As before, we say that ( ν , C ) is oddlycoloured if each colour occurs an odd number of times.Given a coloured sequence ( ν , C ) ∈ Z pk , let [ ν , C ] ∈ Z pk be the corresponding k -colouredmultiset whose elements are the components of the sequence in question. Formally,(6.2) [ ν , C ] = ( γ (0) , . . . , γ ( k − ) , where(6.3) γ ( j ) := { ν i : C i = j } , j ∈ Z /k Z . Definition 6.3. Define the shift operator π : Z pk → Z pk with action(6.4) π ( ν , C ) = ( L ( ν ) + e p − , L ( C )) , ν ∈ Z p , C ∈ ( Z /k Z ) p , where L is the circular permutation(6.5) L ( C ) = ( C , . . . , C p − , C ) , and e i ∈ Z p , i = 0 , . . . , p − i th unit vector. Thus, π ( ν , C ) = π (cid:0) ( ν , . . . , ν p − ) , ( C , . . . , C p − ) (cid:1) = (cid:0) ( ν , ν , . . . , ν p − , ν + 1) , ( C , . . . , C p − , C ) (cid:1) . The next Proposition describes the correspondence between coloured sequences and Mayacycles. It makes use of the following auxilliary notation. Consider(6.6) Ξ k ( ν , C ) := Ξ k ([ ν , C ]) , ( ν , C ) ∈ Z pk . as the generalization of (5.12) from coloured multisets to coloured sequences, and let π i = i times z }| { π ◦ · · · ◦ π denote the iterated action of π as defined by (6.4). Proposition 6.4. Let ( ν , C ) ∈ Z pk be an oddly coloured integer sequence, and set µ i = kν i + C i , i = 0 , . . . , p − M = Ξ k ( ν , C ) , (6.8) M i +1 = φ µ i ( M i ) , i = 0 , . . . , p − . (6.9) Then, M = ( M , . . . , M p ) is a ( p, k ) Maya cycle with flip sequence µ that satisfies M i = Ξ k ( π i ( ν , C )) , i = 0 , . . . , p, (6.10) The above mapping ( ν , C ) M , ( ν , C ) ∈ Z pk constitutes a bijection between the set ofoddly k -coloured sequences of length p and the set of ( p, k ) Maya cycles. The inverse mapping M → ( ν , C ) is given by taking the k -modular decomposition of the flip sequence correspondingto M .Proof. The proof floows by a straightforward application of the relevant definitions. (cid:3) Definition 6.5. In parallel to the terminology introduced above, we will refer to the colouredinteger sequence ( ν , C ) as the block coordinates of the Maya cycle generated by (6.9).We next describe the effect of translations on a Maya cycle at the level of the colouredsequences, in order to define an equivalence class under translations. Let T : Z pk → Z pk be theinvertible mapping defined by T : ( ν , C ) (ˆ ν , L − ( C )) , ν ∈ Z p , C ∈ ( Z /k Z ) p , where(6.11) ˆ ν i = ( ν i + 1 if C i = k − ν i otherwise,and where L is the circular permutation (6.5). Proposition 6.6. Let ( ν , C ) ∈ Z pk be the block coordinates of a ( p, k ) Maya cycle M . Then T ( ν , C ) are the block coordinates of the Maya cycle M + 1 = ( M + 1 , M + 1 , . . . , M p + 1) . Proposition 6.7. Maya cycles M = ( M , . . . , M n +1 ) and M + 1 = ( M + 1 , . . . , M n +1 + 1) generate the same rational solution ( w , . . . , w n | a , . . . , a n ) of a n + 1 -cyclic dressing chain.Proof. The proof comes a straightforward application on the construction formulas (4.20)-(4.21). We recall that for any Maya diagram M , the pseudo-Wronskians H M and H M +1 onlydiffer by a multiplicative constant, as seen in Proposition 3.5. Since only log-derivatives ofHermite pseudo-Wronskians enter in the rational solution (4.20) and the parameters (4.21)only involve differences of the components of the flip sequence, an overall translation of theMaya cycle has no effect in the rational solution. (cid:3) The last two Propositions imply that there is an equivalence class of Maya cycles relatedby translations that generate the same rational solution of a dressing chain. The correspon-dence between coloured sequences and rational solutions is thus many to one. A one-to-onecorrespondence can be achieved by fixing a canonical representative in each equivalence class. Definition 6.8. A ( p, k ) Maya cycle with k ∈ N is in standard form if and only if its firstdiagram M is in standard form. A coloured sequence ( ν , C ) is in standard form if the Mayacycle it defines by (6.7)-(6.9) in standard form too.It is obvious that in each equivalence class M / Z of Maya cycles related by translations,only one of them is in standard form. Example 6.9. The (5 , 3) Maya cycle M defined by the coloured sequence (4 , , , , 0) is instandard form. The action of a unit translation gives a Maya cycle M + 1 which is not instandard form. Both of them are shown in Figure 6.1, where the action of T on the blockcoordinates described by Proposition 6.6 can be verified.We can finally state the main theorem that expresses a good indexing scheme for rationalsolutions to an odd cyclic dressing chain. ATIONAL SOLUTIONS OF A n -PAINLEV´E SYSTEMS 27 (4 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Figure 6.1. The effect of a unit translation T on a (5 , 3) Maya cycle. Theorem 6.10. The set of rational solutions to a (2 n + 1) -cyclic dressing chain with shift ∆ = 2 k bijectively corresponds to the set of oddly k -coloured sequences ( ν , C ) ∈ Z n +1 k instandard form.Proof. Proposition 6.4 establishes a bijection between (2 n + 1 , k ) Maya cycles and oddlycoloured sequences ( ν , C ) ∈ Z n +1 k . Theorem 4.12 establishes a bijection between rationalsolutions of a (2 n + 1)-cyclic dressing chain with shift ∆ = 2 k and (2 n + 1 , k ) Maya cycles,up to a translation of the cycle. If the Maya cycle is required to be in standard form, thecorrespondence between rational solutions and oddly k -coloured sequences is one-to-one. (cid:3) Enumeration and construction of explicit examples. We shall describe now howto enumerate and construct explicitly all rational solutions to the (2 n + 1)-cyclic dressingchain system (2.3), and therefore, by the equivalence described in Proposition 2.4, also allrational solutions of the A n -Painlev´e system (2.1).For a given cyclicity of the chain 2 n +1, by Corollary 5.10 we see that the only possible shiftsare k = ± , . . . , ± (2 n + 1). The reversal symmetry (2.6) allows to invert the sign of the shift,so we can focus without loss of generality on solutions with positive shifts k = 1 , . . . , n + 1.Next, we fix a given k in that range, and ask ourselves how many different k -signaturesmust be considered. This is the number of different compositions of length k of an odd number2 n + 1 with odd parts, which is precisely(6.12) a (2 n + 1 , k ) = (cid:18) n + k − k − (cid:19) , k = 1 , . . . , n + 1 . The total number of possible signatures for a given period 2 n + 1 is(6.13) n +1 X k =1 a (2 n + 1 , k ) = F n +18 DAVID G ´OMEZ-ULLATE, YVES GRANDATI, AND ROBERT MILSON where F j is the j th Fibonacci number. As an example, an enumeration of all the possiblesignatures for 5-periodic chains is: k = 1 : a (5 , 1) = 1 , (5) k = 3 : a (5 , 3) = 3 , (3 , , , (1 , , , (1 , , k = 5 : a (5 , 5) = 1 , (1 , , , , F =5. Example 6.11. In order to construct a given rational solution, pick a shift and a signature,say k = 3 and 5 = 1 + 1 + 3. This means that the coloured block coordinates are given by aninteger 5-tuple grouped into 3 colours as per the above composition. We assume, without lossof generality, that M is in standard form. Let us choose for instance, ( ν , C ) = (4 , , , , Z / Z = { , , } as in Example 5.16.Following (5.7), the k th order flip set corresponds to γ [3] = Θ ( { , , , , } ) = { , , , , } . As ( ν , C ) does not contain repeated integers with the same colour, this leads to a non-degenerate cycle (see Section) and γ [3] ∈ ˆ Z is a set. In this non-degenerate setting, everypermutation of γ [3] yields a different flip sequence µ and correspondingly a different (5 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Figure 6.2. The (5 , 3) Maya cycle for the coloured sequence (4 , , , , 0) andfor the Maya cycle (3 , , , , 0) = s (4 , , , , ν , C ) = (4 , , , , 0) determines the flip sequence µ = (14 , , , , M i +1 in the cycle is obtained from the coloured setthat defines M i by applying π as described in Proposition 6.4 and (6.4) . The flip sequence µ determines the values of the parameters ( a , . . . , a ), which according to (4.21) become( a , a , a , a , a ) = (8 , , − , , − ATIONAL SOLUTIONS OF A n -PAINLEV´E SYSTEMS 29 In principle, H M would be pseudo-Wronskians (3.5) for an arbitrary Maya diagram, buthaving normalized ( ν , C ) to standard form, all the rational solutions can be expressed interms of ordinary Hermite Wronskians, and no generality is lost. In the case of the choicesmade above, the sequence of Wronskians is H M ( z ) = Wr( H , H , H , H , H , H ) ,H M ( z ) = Wr( H , H , H , H , H , H , H ) ,H M ( z ) = Wr( H , H , H , H , H , H , H , H ) ,H M ( z ) = Wr( H , H , H , H , H , H , H , H , H ) ,H M ( z ) = Wr( H , H , H , H , H , H , H , H ) , where H n = H n ( z ) is the n -th Hermite polynomial. The rational solution to the dressingchain is given by the tuple ( w , w , w , w , w | a , a , a , a , a ), where a i and w i are given by(4.20)–(4.21) as: w ( z ) = − z + dd z h log H M ( z ) − log H M ( z ) i , a = 8 ,w ( z ) = − z + dd z h log H M ( z ) − log H M ( z ) i , a = 10 ,w ( z ) = − z + dd z h log H M ( z ) − log H M ( z ) i , a = − ,w ( z ) = z + dd z h log H M ( z ) − log H M ( z ) i , a = 16 ,w ( z ) = − z + dd z h log H M ( z ) − log H M ( z ) i , a = − . Finally, Proposition 2.4 implies that the corresponding rational solution to the A -Painlev´esystem (2.1) is given by the tuple ( f , f , f , f , f | α , α , α , α , α ), where f ( z ) = z + dd z h log H M ( c z ) − log H M ( cz ) i , α = − ,f ( z ) = z + dd z h log H M ( cz ) − log H M ( cz ) i , α = − ,f ( z ) = dd z h log H M ( cz ) − log H M ( cz ) i , α = 1 ,f ( z ) = dd z h log H M ( cz ) − log H M ( cz ) i , α = − ,f ( z ) = z + dd z h log H M ( cz ) − log H M ( cz ) i , α = , with c = − . 7. Connection with the symmetry group approach Noumi and Yamada showed [42] that system (2.1) is invariant under a symmetry group,which acts by B¨acklund transformations on a tuple of functions and parameters. This sym-metry group is the extended affine Weyl group ˜ A (1)2 n , generated by the operators π , s , . . . , s n whose action on the tuple ( f , . . . , f n | α , . . . , α n ) is given by: s i ( f i ) = f i , s i ( f j ) = f j ∓ α i f i ( j = i ± , s i ( f j ) = f j ( j = i, i ± s i ( α i ) = − α i , s i ( α j ) = α j + α i ( j = i ± , s i ( α j ) = α j ( j = i, i ± π ( f j ) = f j +1 , (7.3) π ( α j ) = α j +1 (7.4)where i, j = 0 , . . . , n mod (2 n + 1). A direct calculation and inspection of (2.8) serve toestablish that the above B¨acklund transformations correspond to the following transformationof the dressing chain (2.3): s i ( w i ) = w i + a i w i + w i +1 , s i ( w i +1 ) = w i +1 − a i w i + w i +1 , s i ( w j ) = w j ( n = i, i + 1)(7.5) s i ( a i ) = − a i , s i ( a j ) = a j + a i ( j = i ± , s i ( a j ) = a j ( j = i, i ± π ( w j ) = w j +1 , (7.7) π ( a j ) = a j +1 (7.8)The two realizations are equivalent, but we will focus mostly on the latter realization in termsof dressing chains.7.1. Symmetries on Maya cycles. Above, we showed that an oddly coloured sequence( ν , C ) specifies a rational solution to the A n -Painlev´e system (2.1), and that, up to integertranslations, every rational solution can be represented by one such sequence. Since thesymmetry group of transformations (7.1)-(7.4) preserves the rational character of the solutions,it must have a well defined action on Maya cycles M and coloured sequences ( ν , C ). Wedescribe this group action below.Let M = ( M , . . . , M n , M n +1 ) be a (2 n + 1 , k ) Maya cycle with flip sequence µ =( µ , . . . , µ n ). Define π ( M ) = ( M , . . . , M n , M n +1 , M + k ) , (7.9) s i ( M ) = ( M , . . . , ˆ M i +1 , . . . , M n +1 ) , i = 0 , . . . , n − , (7.10) s n ( M ) = ( ˆ M , M , . . . , M n , ˆ M + k ) , (7.11)where ˆ M i = φ µ i ( M i +1 ) , i = 0 , . . . , , . . . , n It is clear by inspection that π ( M ) and s i ( M ) , i = 0 , . . . , n are Maya cycles with flipsequences given by ,respectively, by: π ( µ ) = L ( µ ) s i ( µ ) = K i,i +1 ( µ ) , i = 0 , . . . , n − s n ( µ ) = K , n ( µ ) , where L is the circular permutation (6.5), and K i,j denotes a transposition of the indicatedelements. ATIONAL SOLUTIONS OF A n -PAINLEV´E SYSTEMS 31 Proposition 7.1. The action of ˜ A (1)2 n on Maya cycles described in (7.9) - (7.11) and theaction of ˜ A (1)2 n by B¨acklund transformations (7.5) - (7.8) is compatible with the transformation M → ( w | a ) defined in (4.20) (4.21) .Proof. The compatibility of π follows by a direct inspection. We demonstrate the compatibil-ity of s ; the compatibility of the other actions is argued similarly. Let M = ( M , . . . , M n +1 )be a (2 n + 1 , k ) Maya cycle, and let ˆ M = s ( M ) as per (7.10). Let ( w , a ) and ( ˆ w , ˆ a ) bethe corresponding rational solutions of the 2 n + 1-cyclic dressing chain as per (4.20) and(4.21). Let µ = ( µ , . . . , µ n ) be the flip sequence corresponding to M . By construction,ˆ M = ( M , ˆ M , M , . . . , M n +1 ) has flip sequence ( µ , µ , µ , . . . , µ n ) with M = φ µ ( M ) , ˆ M = φ µ ( M ) . Hence, ˆ w i = w i , ˆ a i = a i for i = 2 , . . . , n . Applying (2.19) with λ = 2 µ + 1 , λ = 2 µ + 1 , ˆ λ = 2 µ + 1 , ˆ λ = 2 µ + 1)establishes that ˆ a = 2( µ − µ ) − a = s ( a ) , ˆ a = 2( µ − µ ) = a + a = s ( a ) , ˆ a n = 2( µ − µ n ) = a + a n = s ( a n )Observe that L ˆ M − ˆ w −→ L M w −→ L M w −→ L M − ˆ w −→ L ˆ M forms a cyclic factorization chain with shift 0. Hence,(7.12) ( − ˆ w + w ) ′ + w − ˆ w = 2( µ − µ ) , ( w + w ) ′ + w − w = 2( µ − µ ) , ( w − ˆ w ) ′ + ˆ w − w = 2( µ − µ ) , ( − ˆ w − ˆ w ) ′ + ˆ w − ˆ w = 2( µ − µ ) , − ˆ w + w + w − ˆ w = 0It follows by a straightforward elimination that( − ˆ w + w )( w + w ) = 2( µ − µ ) = − a Therefore, s ( w ) = ˆ w = w + a w + w . A similar elimination in (7.12) serves to show that s ( w ) = ˆ w . (cid:3) It is also clear that the action of π shown in (7.9) is compatible with the B¨acklund trans-formation (7.3) (7.4). Note that the action of s i on the cycle only changes one Maya diagram M i +1 , and consequently the functions f i +1 and f i − are in agreement with (7.1). We next describe the corresponding action of the symmetry operators π, s , . . . , s n on theset of coloured sequences. Let ( ν , C ) ∈ Z n +1 k be a coloured sequence. Define π ( ν , c ) as in(6.4). Define s i ( ν , C ) = ( K i,i +1 ( ν ) , K i,i +1 ( C )) , ν ∈ Z p , C ∈ ( Z /k Z ) p , (7.13) s n ( ν , C ) = ( K n, ( ν ) − e + e n , K n, ( C ))(7.14) = ( ν n − , ν , . . . , ν n − , ν + 1 , K n, ( C ))where K i,j denotes the transposition of components in positions i and j . Proposition 7.2. The actions (7.9) - (7.10) of π , s , . . . , s n on Maya cycles and the corre-sponding actions on coloured seqences (6.4) (7.13)(7.14) satisfy the defining relations of theextended affine Weyl group of type ˜ A (1)2 n : (7.15) s i ≡ , ( s i s i +1 ) n +1 ≡ , πs i ≡ s i +1 π , π n +1 ≡ , where ≡ indicates equality modulo translations.Proof. The above relations follow directly from the relevant definitions. (cid:3) Proposition 7.3. The action of ˜ A (1)2 n on coloured sequences given by (6.4) (7.13) (7.14) andthe action of ˜ A (1)2 n on Maya cycles described in (7.9) (7.10) (7.11) are compatible with thetransformation ( ν , C ) → M defined in (6.9) .Proof. Because both sets of actions satisfy (7.15) it suffices to establish the compatibility of π and s . By inspection of (6.9), the compatibility of π follows from the relation π n +1 ( ν , C ) = T n +1 ( ν , C ) = ( ν + 1 , C ) . The compatibility of s follows directly from Proposition 6.4 and the definitions of s ( M )and s ( ν , C ). (cid:3) Note that the above actions preserve the colouring signature. It follows that π , s , . . . , s n preserve the set of oddly coloured sequences.According to Proposition 6.7 both M and M + j for any j ∈ Z describe the same rationalsolution of the dressing chain, so the relations (7.15) are strict identities for the group actionon rational solutions (7.1)-(7.4). The coincidence of the two representations given by (7.1)and (7.10) (with (4.20) and (2.8)), entails interesting identities between Hermite Wronskians,that shall be further explored elsewhere.7.2. Seed solutions. The usual approach to constructing rational solutions to the A n -Painlev´e system (2.1) is to let the symmetry group act on a number of very simple seedsolutions , to construct the rest of the rational solutions. Definition 7.4. Fix a k ∈ N and n ∈ N and let 2 n + 1 = p + · · · + p k − be a compositionof 2 n + 1 into k odd parts. A seed sequence is a coloured sequence ( , C p ) where ∈ Z n +1 is the zero vector and where C p = (0 p , p , . . . , ) . All the Maya diagrams in the cycle corresponding to a seed coloured sequence ( , C p )have genus zero. The corresponding seed solutions to the A n -Painlev´e system (2.1) for eachsignature p are given by the following Proposition. ATIONAL SOLUTIONS OF A n -PAINLEV´E SYSTEMS 33 Proposition 7.5. Let n + 1 = p + · · · + p k − be a composition of an odd number into k odd parts. Let ( f , . . . , f n | α , . . . , α n ) be the rational solution generated by the correspondingseed solution ( , C p ) . Then, for every i = 0 , . . . , n we have f i = ( k − z if i + 1 ∈ Q, otherwise , (7.16) α i = ( k − if i + 1 ∈ Q, otherwise , (7.17) where Q = { q , . . . , q k } is the set of corresponding partial sums q j = j − X r =0 p r , j = 1 , . . . , k. Proof. The proof follows by a straightforward application of the construction rules (4.20)-(4.18) with Proposition 2.4 to build the solution ( f , . . . , f n | α , . . . , α n ) corresponding tothe Maya cycle specified by ( , C p ). (cid:3) Example 7.6. Consider the (5 , 3) seed solution of the A -Painlev´e system corresponding tothe composition 5 = 1 + 3 + 1, i.e. with signature p = (1 , , , , , , 0) , and the Maya cycle generated by this sequence is shown in Figure 7.1.Note that all the Maya diagrams M i in the cycle have genus zero. For signature p = (1 , , Q = { , , } and the corresponding seed solution given by (7.16)-(7.17) is shown inFigure 7.1. (0 , , , , , , , , , , , , , , , , , , , , , , , , (cid:18) z , , , z , z (cid:12)(cid:12)(cid:12) , , , , (cid:19) Figure 7.1. The Maya cycle for the (1 , , 1) seed solution.Applying the symmetry operators s and s on the seed colour sequence (0 , , , , 0) leadto s (0 , , , , 0) = (0 , , , , , s (0 , , , , 0) = ( − , , , , A -Painlev´e system are shown in Figure 7.2. It can be readily verified that the rationalsolutions are the same that result from the action of the B¨acklund transformations (7.1)-(7.4)on the seed solution. (0 , , , , , , , , , , , , , , , , , , , , , , , , (cid:18) z , − z , , z , z + z (cid:12)(cid:12)(cid:12) − , , , , (cid:19) ( − , , , , , , , , , , , , , , , , , , , , , , , , − (cid:18) z − z , , , z z , z (cid:12)(cid:12)(cid:12) , , , , − (cid:19) Figure 7.2. Applying s and s to the seed solution for signature p = (1 , , Orbits of the extended affine Weyl group ˜ A (1)2 n . In this section we address thetransitivity problem, namely to establish that all rational solutions of the A n -Painlev´e systemcan be obtained by applying the symmetry group to one of the seed solutions (7.16)-(7.17).We would like to represent all the rational solutions described by Theorem 6.10 as thedifferent orbits of the seed solutions under the action of the symmetry group.For this purpose, it should be first noted that the action of the symmetry group (7.9) -(7.11) preserves the signature composition p = p + · · · + p k − . Next, we try to build operatorswhose action on block coordinates has a particularly simple action. Proposition 7.7. Consider the operators (7.18) E i = s i s i +1 · · · s i +2 n − π, i = 0 , . . . , n mod (2 n + 1) . The action of these operators on a coloured sequence is given by (7.19) E i ( ν , C ) = ( ν + e i , C ) , ν ∈ Z n +1 , C ∈ ( Z /k Z ) n +1 , where e i ∈ Z n +1 is the i th unit vector.Proof. The proof follows immediately from the action of the operators π , s , . . . , s n on thecoloured sequence ( ν , C ) as shown in (6.4), (7.13), (7.14) . For i = 0 , . . . , n , we have K i,i +1 · · · K n, K , · · · K i − ,i L = 1 , where K i,j is the transposition of components i, j and where L is the circular permutation(6.5). Hence, for i = 1 , . . . , n we have E i ( ν , C ) = s i · · · s n · · · s i − ( L ( ν ) + e n , L ( C ))= s i · · · s n ( K , · · · K i − ,i L ( ν ) + e n , K , · · · K i − ,i L ( C ))= s i · · · s n − ( K n, K , · · · K i − ,i L ( ν ) + e n , K n, K , · · · K i − ,i L ( C ))= ( ν + e i , C ) ATIONAL SOLUTIONS OF A n -PAINLEV´E SYSTEMS 35 For i = 0 we have E ( ν , C ) = s · · · s n − ( L ( ν ) + e n , L ( C ))= (( K , · · · K n − , n L )( ν ) + ( K , · · · K n − , n ) e n , K , · · · K n − , n L ( C ))= ( ν + e , C ) (cid:3) Example 7.8. Consider the action of E = s s π on the coloured sequence (2 , , , , π −→ (3 , , s −→ (2 , , s −→ (2 , , 0) = (2 , , 0) + e . Next, we describe the symmetry operators that act on coloured sequences by direct per-mutations. Proposition 7.9. The operators s , . . . , s n − generate the action of the permutation group S n +1 on the set of coloured sequences ( ν , C ) of length n + 1 .Proof. This follows directly from the definition (7.13). (cid:3) Theorem 7.10. Every rational solution of a (2 n + 1) -cyclic dressing chain can be obtainedby the action of the symmetry group ˜ A (1)2 n on a seed solution.Proof. Assume ( w | a ) is rational solution of a (2 n + 1)-cyclic dressing chain with shift ∆ = 2 k .By Theorem 6.10, this solution can be indexed by a coloured sequence ( ν , C ) ∈ Z n +1 k instandard form. Let p = ( p , . . . , p k − ) be its signature and consider the seed solution ( , C p )corresponding to that signature. Since ( ν , C ) is in standard form, all the components are ν i ≥ i = 0 , . . . , n . There clearly exists a sequence of operators E i defined in (7.18) thatmap the seed sequence ( , C p ) into a seed sequence ( ν ′ , C ′ ) that differs from ( ν , C ) at mostby a permutation of its elements, i.e. such that [( ν ′ , C ′ )] = [( ν , C )]. By Proposition7.9, thereis a sequence of symmetry operators that map ( ν ′ , C ′ ) into ( ν , C ). (cid:3) Note, however, that given a rational solution ( w | a ), the seed solution and sequence ofsymmetry transformations is not unique. If ( ν , C ) is the coloured sequence in standard formthat corresponds to ( w | a ), so does the coloured sequence T ( ν , C ) by Propositions 6.6 and6.7. But T ( ν , C ) has signature L − ( p ) = ( p k − , p , . . . , p k − ). We see thus that seed solutions(7.16)-(7.17) corresponding to circular permutations of a given signature belong to the sameorbit under the symmetry group ˜ A (1)2 n . Remark 7.11. Similar formulas for the action of the generators of the extended affine Weylgroup ˜ A (1)2 n on Maya diagrams have been given by Noumi in his book (see [41] Ch. 7.6).However, the main difference between Proposition 7.2 in [41] and Proposition 7.1 in this workis that in the former the symmetry group acts on a single Maya diagram, while the lattertreats the group action on a Maya cycle. Since a rational solution corresponds to a Maya cyclerather than to single Maya diagram (see Theorem 4.12), we believe that Propositions 7.1 and7.3 provide the correct corespondence with the action of ˜ A (1)2 n on rational solutions (7.1)-(7.4)via B¨acklund transformations. Degenerate solutions Degenerate solutions describe certain embeddings of lower order systems A m -Painlev´e intohigher order systems A n -Painlev´e for n > m . Also, they single out special properties withrespect to the isotropy of the group action. Definition 8.1. Let ( w | a ) = ( w , . . . , w n | a , . . . , a n ) be a rational solution to a (2 n + 1)-dressing chain with shift ∆ = 2 k and let ( ν , C ) ∈ Z n +1 k be its standard coloured sequence.The flip sequence µ = ( µ , . . . , µ n ) ∈ Z n +1 is given by µ i = kν i + C i , i = 0 , . . . , n. The rational solution ( w | a ) is said to be a degenerate solution if at least one of the followingconditions hold(1) The flip sequence µ contains repeated elements.(2) µ n = µ + k .From a degenerate solution one can immediately build a solution to a lower order systemin the following manner. Consider for simplicity that condition (1) above holds and µ hasrepeated elements. This means that [( ν , C )] is a multiset but not a set. Let ( ν i , C i ) and( ν j , C j ) be two elements of the coloured sequence ( ν , C ) whose value and colour coincide.Then the coloured sequence (˜ ν , ˜ C ) obtained by dropping these two elements is an oddly k -coloured sequence of length 2 n − A n − -Painlev´e system.If the two equal elements in the coloured sequence are consecutive, e.g. ( ν i , C i ) = ( ν i +1 , C i +1 ),then two consecutive flips µ i = µ i +1 happen at the same site and the Maya cycle M =( M , . . . , M n +1 ) contains two identical Maya diagrams M i = M i +2 . Correspondingly, theMaya cycle ˜ M obtained by dropping M i +1 and M i +2 is (2 n − , k )-cyclic. M = ( M , . . . , M i , M i +1 , M i +2 , . . . , M n +1 ) → ˜ M = ( M , . . . , M i , M i +3 , . . . , M n +1 )If ( f , . . . , f n | α , . . . , α n ) is the corresponding degenerate solution to the A n -Painlev´e sys-tem, then we can build a solution ( ˜ f , . . . , ˜ f n − | ˜ α , . . . , ˜ α n − ) to the A n − -Painlev´e systemby setting(8.1) ˜ f j = f j if j ≤ i − ,f j + f j +2 if j = i − ,f j +2 if j ≥ i, ˜ α j = α j if j ≤ i − ,α j + α j +2 if j = i − ,α j +2 if j ≥ i, where the indices for f i , α i are taken mod 2 n + 1 and those for ˜ f i , ˜ α i are taken mod 2 n − f i ( z ) = 0 and α i = 0 in the degenerate solution.We see thus that in the case of the consecutive flips at the same site leads to a rather trivialembedding of the lower order system into the higher order one. However, there could be tworepeated elements in the flip sequence which are not consecutive, i.e ( ν i , C i ) = ( ν j , C j ) for j > i + 1. It is still true that the coloured sequence (˜ ν , ˜ C ) obtained by dropping thesetwo repeated elements will define a solution to the A n − -Painlev´e system, but it is nolonger true that one can simply eliminate two Maya diagrams from the (2 n + 1 , k ) Mayacycle M = ( M , . . . , M n +1 ) to obtain a (2 n − , k ) Maya cycle ˜ M . This case leads tonontrivial embeddings, in which the reduced solution ( ˜ f , . . . , ˜ f n − | ˜ α , . . . , ˜ α n − ) cannotbe obtained by a linear combination of the degenerate solution of the higher order system( f , . . . , f n | α , . . . , α n ). ATIONAL SOLUTIONS OF A n -PAINLEV´E SYSTEMS 37 Finally, the following Proposition states that degenerate solutions have a nontrivial isotropygroup. Proposition 8.2. If ( f , . . . , f n | α , . . . , α n ) is a degenerate rational solution of the A n system, then there is an element of the symmetry group ˜ A (1)2 n that leaves it invariant.Proof. This follows directly from Theorem 6.10 and the action (7.13)-(7.14) of the symmetrygroup ˜ A (1)2 n on coloured sequences. In the case where ( ν i , C i ) = ( ν i +1 , C i +1 ), the rationalsolution is a fixed point of the generator s i . Otherwise, there is a sequence of symmetrytransformations that performs the transposition between the two identical elements of thecoloured sequence. (cid:3) Example 8.3. We illustrate here the occurence of degenerate solutions and the differencebetween trivial and non-trivial embeddings. Consider the degenerate solutions of the A -Painlev´e system corresponding to the following coloured sequences:( ν (1) , C (1) ) = (0 , , , , µ (1) = (0 , , , , , (8.2) ( ν (2) , C (2) ) = (0 , , , , , µ (2) = (0 , , , , . (8.3)Both solutions are degenerate because the flip sequence contains repeated elements, but inthe first case they are consecutive while in the second the are not. The corresponding rationalsolutions are ( f (1) | α (1) ) = z − z + 2 zz + 3 , z zz − , , z − zz − , z − zz + 3 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , − , , − , ! , ( f (2) | α (2) ) = z , z zz − , − z + 2 zz + 3 , − z − zz − , z − zz + 3 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , , − , − , ! If we drop the two repeated elements of the coloured sequence, we obtain in both cases thefollowing solution of the A -Painlev´e system.(˜ ν , ˜ C ) = (0 , , , ˜ µ = (0 , , , (˜ f | ˜ α ) = z − z + 2 zz + 3 , z z , z − zz + 3 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , − , ! This solution to the A -Painlev´e system is clearly obtained from ( f (1) | α (1) ) by applying(8.1). It is also clear from (7.1)-(7.2) that s ( f (1) | α (1) ) = ( f (1) | α (1) ) , as explained in Proposition 8.2.9. Summary and Outlook This paper provides a complete classification of the rational solutions of Painlev´e P IV andits higher order hierarchy known as the A n -Painlev´e or Noumi-Yamada system. First, werecall the equivalence between the Noumi-Yamada system (2.1) and a cyclic dressing chainof Schr¨odinger operators. Then, we show by a careful investigation of the local expansionsof the rational solutions around their poles, that the solutions have trivial monodromy, andtherefore they must be expressible in terms of Wronskian determinants whose entries areHermite polynomials. Next, we use Maya diagrams to classify all the (2 n + 1)-cyclic dressing chains and therefore achieve a complete classification. Finally, we connect our results withthe geometric approach mastered by the japanese school, showing a representation for theaction of the symmetry group of B¨acklund transformations in terms of Maya cycles and oddlycoloured integer sequences.The natural extension of this work is to tackle the full classification of the rational solutionsto the A n +1 -Painlev´e systems, which include P V and its higher order extensions. Althoughthe analysis is considerably more involved, it should be possible to extend this approach fromthe odd-cyclic to the even-cyclic case. In this regard, we would like to formulate the following Conjecture 9.1. All rational solutions of P V (and the A n +1 -Painlev´e system) can be ex-pressed as Wronskian determinants whose entries involve Laguerre polynomials. We have solid evidence to believe that this conjecture is true. 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