Periodicity, linearizability and integrability in seed mutations of type A (1) N
aa r X i v : . [ n li n . S I] S e p Periodicity, linearizability and integrability in seed mutations of type A (1) N Atsushi Nobe a) and Junta Matsukidaira b)1) Faculty of Education, Chiba University,1-33 Yayoi-cho Inage-ku, Chiba 263-8522, Japan Department of Applied Mathematics and Informatics,Ryukoku University,1-5 Yokotani, Seta Oe-cho, Otsu, Shiga 520-2194, Japan (Dated: 21 September 2020)
In the network of seed mutations arising from a certain initial seed, an appropriatepath emanating from the initial seed is intendedly chosen, noticing periodicity of theexchange matrices in the path each of which is assigned to the generalized Cartanmatrix of type A (1) N . Then dynamical property of the seed mutations along the path,which is referred to as of type A (1) N , is intensively investigated. The coefficients as-signed to the path form certain N monomials that posses periodicity with period N under the seed mutations and enable to obtain the general terms of the coefficients.The cluster variables assigned to the path of type A (1) N also form certain N Laurentpolynomials possessing the same periodicity as the monomials generated by the coeffi-cients. These Laurent polynomials lead to sufficiently number of conserved quantitiesof the dynamical system derived from the cluster mutations along the path. Further-more, by virtue of the Laurent polynomials with periodicity, the dynamical system isnon-autonomously linearized and its general solution is concretely constructed. Thusthe seed mutations along the path of type A (1) N exhibit discrete integrability.PACS numbers: 02.10.Hh, 02.30.Ik, 05.45.YvKeywords: cluster algebra, integrable system, linearization a) Electronic mail: [email protected] b) Electronic mail: [email protected] . INTRODUCTION Seed mutations in a cluster algebra produce new seeds from old ones, each of which is atuple of the exchange matrix, the coefficients and the cluster variables, via their birationalequations called the exchange relations . A cluster algebra of rank r has the seed mutationsin r directions, hence the network of seeds generated by iteration of the seed mutations fromthe initial one forms an r -regular tree in which every vertex (seed) is connected with exactly r vertices respectively by an edge (mutation). The rank of a cluster algebra is defined to be thenumber of cluster variables in the initial seed. In order to find significant paths (sequences ofseeds) in such huge network consisting of infinitely many seeds, we often use the periodicityas an important indicator. Fordy and Marsh defined the cluster mutation-periodic quiversconcerning periodicity of the exchange matrices and discussed their dynamical propertiesunder the seed mutations . By using the notion of cluster mutation-periodic quivers, werelate seed mutations in cluster algebras with dynamical systems governed by birationalmaps and investigate the seed mutations in terms of the methods of dynamical systems.Unfortunately, almost all of the infinitely many dynamical systems thus related with clusteralgebras do not have integrable structures; nevertheless, we can find abundant integrablesystems among them. In fact, since the introduction of cluster algebras by Fomin andZelevinsky in 2002 we have found plenty of cluster algebras related with discrete/quantumintegrable systems such as discrete soliton equations, integrable maps on algebraic curves,discrete/ q - Painlev´e equations and Y -systems . Thus we see that appropriate paths in thenetwork of seeds in adequate cluster algebras are strongly related with integrable systems,and hence it is expected that we find unknown integrable systems among cluster algebras.It is well known that appropriate paths in the network of seeds in a cluster algebra canbe assigned to the generalized Cartan matrices (GCMs) via the exchange matrices in thepaths . Especially, a rank 2 cluster algebra itself is assigned to a 2 × , the authors investigated a certainfamily of rank 2 cluster algebras from the viewpoint of discrete integrability. The familyconsists of infinitely many cluster algebras some of which have integrable structures and arerespectively assigned to the GCMs of finite and affine types. The remaining infinitely manymembers in the family, however, are non-integrable and are assigned to the GCMs of strictly2yperbolic type. We saw the integrability via conserved quantities of the dynamical systemsgoverned by birational maps derived from the seed mutations. Moreover, the integrablesystems associated with the rank 2 cluster algebras of affine types, A (1)1 and A (2)2 , have lineardegree growth of the map iteration , and hence the systems are linearizable . Basedon these results, in this paper, we consider the seed mutations of rank N + 1 assigned tothe GCM of type A (1) N for N ≥
2. It should be noted that, in general, a sequence of seedmutations in a cluster algebra of higher rank is not assigned to any GCM in contrast to thecase of rank 2 mentioned above because the GCMs of the exchange matrices in the sequenceof higher rank seed mutations are not unique. Nevertheless, the sequence of seed mutationsconsidered in this paper is so carefully chosen from the network of seeds in a certain clusteralgebra of rank N + 1 that it can be assigned to the unique GCM of type A (1) N . We notethat the sequence of seed mutations thus chosen has several remarkable periodicities; thequivers associated with the exchange matrices are the cluster mutation-periodic quivers withperiod 1 , the coefficients generate certain N monomials periodic with period N and thecluster variables also generate certain N Laurent polynomials periodic with the same period N under iteration of the seed mutations. By using the periodicity of exchange matrices, werespectively obtain dynamical systems of the coefficients and of the cluster variables from thesequence of seed mutations assigned to the GCM of type A (1) N . Sufficiently many conservedquantities of each dynamical system naturally follow from the monomials and the Laurentpolynomials, both of which have the same periodicity, respectively. Moreover, the dynamicalsystem of cluster variables is non-autonomously linearized by virtue of the periodic Laurentpolynomials similar to the rank 2 cases investigated in the previous papers . Due to thelinearizability, the general solution to the dynamical system is concretely constructed, andit gives the general terms of the cluster variables.This paper is organized as follows. In § II, we briefly review cluster algebras. We thenintroduce the sub-cluster pattern which assigns the sequence of seed mutations to the pathreferred to as of type A (1) N in the ( N + 1)-regular tree. In § III, we deduce periodicity ofcertain N monomials generated by the coefficients assigned to the path of type A (1) N . Byusing the monomials with periodcity, we obtain the general terms of the coefficients. Then,in § IV, we introduce the dynamical system of cluster variables assigned to the path oftype A (1) N . We also deduce periodicity of certain N Laurent polynomials generated by thecluster variables. Since the N Laurent polynomials generate the N functionally independent3onserved quantities of the dynamical system of N + 1 variables, it suggests the system to beintegrable. Moreover, the Laurent polynomials non-autonomously linearize the dynamicalsystem and its general solution follows immediately. § V is devoted to concluding remarks.In Appendix A, we consider the dynamical system investigated in § IV in the projective space P N +1 ( C ), and induce the invariant curve of the system. II. SEED MUTATIONS OF TYPE A (1) N Let us introduce the seed ( x , y , B ), where we refer to x = ( x , x , . . . , x n ) as the clusterof the seed, to y = ( y , y , . . . , y n ) as the coefficient tuple and to B = ( b ij ) as the exchangematrix. The number n of variables in x is called the rank of the seed. The field F = QP ( x )generated by the cluster x is referred to as the ambient field, where P = ( P , · , ⊕ ) is asemifield endowed with multiplication · and auxiliary addition ⊕ and QP is the group ringof P over Q . The coefficient tuple y is taken from P n and the exchange matrix B is an n × n skew-symmetrizable integral matrix .Next we introduce the seed mutations. For an integer k ∈ { , , . . . , n } , the seed muta-tion µ k transforms a seed ( x , y , B ) into the seed ( x ′ , y ′ , B ′ ) := µ k ( x , y , B ) defined by thefollowing birational equations called the exchange relations: b ′ ij = − b ij i = k or j = k,b ij + [ − b ik ] + b kj + b ik [ b kj ] + otherwise , (1) y ′ j = ( y k ) − j = k,y j y [ b kj ] + k ( y k ⊕ − b kj j = k, (2) x ′ j = y k n Y i =1 x [ b ik ] + i + n Y i =1 x [ − b ik ] + i ( y k ⊕ x k j = k,x j j = k, (3)where we define [ a ] + := max[ a,
0] for a ∈ Z .Let T n be the n -regular tree whose edges are labeled by the integers 1 , , . . . , n so that the n edges emanating from each vertex receive different labels. We write t k t ′ to indicatethat vertices t, t ′ ∈ T n are joined by an edge labeled by k . We assign a seed Σ t = ( x t , y t , B t )to every vertex t ∈ T n so that the seeds assigned to the endpoints of any edge t k t ′ are4btained from each other by the seed mutation µ k . We refer to the assignment T n ∋ t Σ t as a cluster pattern. We write the elements of the seed Σ t as follows x t = ( x t , x t , . . . , x n ; t ) , y t = ( y t , y t , . . . , y n ; t ) , B t = ( b tij ) . Given a cluster pattern T n ∋ t Σ t , we denote the union of clusters of all seeds in thepattern by X := [ t ∈ T n x t = { x i ; t | t ∈ T n , ≤ i ≤ n } . The cluster algebra A = ZP [ X ] associated with the cluster pattern is the ZP -subalgebraof the ambient field F generated by all cluster variables. It is well known that A is alsogenerated by its initial cluster variables x , x , . . . , x n as the Laurent polynomial subring ofthe ambient field F .Now we introduce the seed mutations assigned to the GCM of type A (1) N . Let us considerthe following initial seed Σ = ( x , y , B ) of rank N + 1: x = ( x , x , · · · , x N +1 ) , y = ( y , y , · · · , y N +1 ) ,B = − · · · −
11 0 − − −
11 0 . . . , where the ( i, j )-element b ij of the skew-symmetrix matrix B is defined to be b ij = i, j ) = ( N + 1 ,
1) or ( i, j ) = ( i, i −
1) for 2 ≤ i ≤ N + 1 , − i, j ) = (1 , N + 1) or ( i, j ) = ( i, i + 1) for 1 ≤ i ≤ N , . We assume P to be the tropical semifield (Trop( y ) , · , ⊕ ) generated by y . The multi-plication · and the auxiliary addition ⊕ in P are respectively defined as follows y a y a · · · y a n n · y b y b · · · y b n n := y a + b y a + b · · · y a n + b n n ,y a y a · · · y a n n ⊕ y b y b · · · y b n n := y min[ a ,b ]1 y min[ a ,b ]2 · · · y min[ a n ,b n ] n a i , b i ∈ Z ( i = 1 , , . . . , n ).We give a cluster pattern. Let T N +1 be the ( N + 1)-regular tree whose edges are labeledby 1 , , . . . , N + 1. We label the vertices in T N +1 in the following manner. First choose anarbitrary vertex and denote it by t , which is assigned to the initial seed Σ . Next denotethe vertex connected with t by the edge labelled by 1 by t . The vertex t is assigned tothe seed Σ obtained from Σ by applying the seed mutation µ . Then inductively denotethe vertex connected with t ℓ ( N +1)+ k − by the edge labeled by k by t ℓ ( N +1)+ k for ℓ ≥ k = 1 , , . . . , N + 1. Since the vertex t ℓ ( N +1)+ k is assigned to the seed Σ ℓ ( N +1)+ k , the seedΣ ℓ ( N +1)+ k is obtained from Σ by applying the following sequence of seed mutations µ , µ , . . . , µ N +1 | {z } N +1 , µ , µ , . . . , µ N +1 | {z } N +1 , . . . , µ , µ , . . . , µ N +1 | {z } N +1 | {z } ℓ × ( N +1) , µ , µ , . . . , µ k . Thus we obtain the path ( t , t , t , . . . ) in the tree T N +1 (see figure 1) and denote it by ̟ . t t · · · t N +1 N +1 1 t ( N +1)+1 2 · · · · · · · · · t ℓ ( N +1) − N · · · t ( ℓ +1)( N +1) N +1 · · · t ℓ ( N +1)+12 t ℓ ( N +1) N +11 FIG. 1. The path ̟ in the ( N + 1)-regular tree T N +1 . Let the set of seeds be Σ := { Σ , Σ , Σ , . . . } . Then we obtain the partial assignment T N +1 ⊃ ̟ → Σ ; t ℓ ( N +1)+ k Σ ℓ ( N +1)+ k for ℓ ≥ k = 1 , , . . . , N + 1. We call the partial assignment the sub-cluster pattern,and fix it throughout this paper. Note that we need not whole cluster pattern but thesub-cluster pattern for our purpose.The quiver Q associated with the exchange matrix B assigned to the vertex t in the6ath ̟ is given as follows Q N (cid:13) rrrrrrrrrrrrrrrr (cid:13) o o · · · o o N (cid:13) o o N +1 J . f f ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ o o The vertex labeled by 1 is a sink, denoted by N , and the one by N + 1 is a source, denotedby J . The quiver Q = µ ( Q ) associated with the exchange matrix B = µ ( B ) assignedto the vertex t in ̟ is obtained by reversing the arrows connected with the vertex 1 in Q (see (1)). Note that, in Q , the vertex 2 is a sink and the vertex 1 is a source: Q N (cid:13) rrrrrrrrrrrrrrrr (cid:13) o o · · · o o N +1 (cid:13) o o J . f f ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ o o It is easy to see that we inductively obtain the quiver Q ℓ ( N +1)+ k associated with the exchangematrix B ℓ ( N +1)+ k assigned to the vertex t ℓ ( N +1)+ k in ̟ as follows Q ℓ ( N +1)+ k k +1 N k +2 (cid:13) qqqqqqqqqqqqqqqq k +3 (cid:13) o o · · · o o k − (cid:13) o o k J , f f ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ o o where the labels are reduced modulo N + 1.Thus, for ℓ ≥ k = 1 , , . . . , N + 1, we obtain the periodicity of the quivers Q ℓ ( N +1)+ k = ( σ N +1 ) k Q , where σ N +1 ∈ S N +1 is the permutation σ N +1 = · · · N N + 1 N + 1 1 · · · N − N (4)of N + 1 letters. Remark that the action τ Q of the permutation τ ∈ S N +1 on the quiver Q with N + 1 vertices is, in general, defined as follows ♯ { arrows emanating from the vertex i ∈ Q to j ∈ Q } = ♯ { arrows emanating from the vertex ( τ ) − ( i ) ∈ τ Q to ( τ ) − ( j ) ∈ τ Q } . Q , Q , Q , . . . are the cluster mutation-periodic quivers with period 1 .The exchange matrices also have the same periodicity as the quivers: B ℓ ( N +1)+ k = ( σ N +1 ) k B . The action σ N +1 B m of σ N +1 on the exchange matrix B m is defined by using the permutationmatrix so that it is compatible with the correspondence between B m and Q m .For any m ≥
0, the Cartan counterpart A ( B m ) of the exchange matrix B m is given as A ( B m ) := (cid:0) δ ij − (cid:12)(cid:12) b mij (cid:12)(cid:12)(cid:1) = − · · · − − − − − − − − . . . − . Since A ( B m ) is the GCM of type A (1) N , we refer to the path ̟ in the tree T N +1 assigned tothe set Σ of seeds as of type A (1) N . III. DYNAMICS OF COEFFICIENTS
We consider dynamics of the coefficients assigned to the path ̟ of type A (1) N . In orderto analyze the dynamics of coefficients, we first consider periodicity of certain monomialsgenerated by the coefficients. A. Periodicity
First we show a lemma concerning the coefficient tuple y N +1 = ( µ N +1 ◦ · · · ◦ µ ◦ µ ) ( y )obtained by applying the consecutive seed mutations µ N +1 ◦ · · · ◦ µ ◦ µ to the initial one y = ( y , y , . . . , y N ). Lemma 1
Let y = ( y , y , . . . , y N +1 ) be the initial coefficient tuple. Then, for any N ≥ ,we have y N +1 = ( y N +1 , y N +1 , . . . , y N +1; N +1 ) = (cid:0) ( y ) − , ( y ) − , . . . , ( y N +1 ) − (cid:1) . y = ( y , y , . . . , y N +1 ) µ ←→ y = (cid:0) ( y ) − , y ( y ⊕ , y , . . . , y N , y N +1 ( y ⊕ (cid:1) = (cid:0) ( y ) − , y , y , . . . , y N , y N +1 (cid:1) µ ←→ y = (cid:0) ( y ) − ( y ⊕ , ( y ) − , y ( y ⊕ , y , . . . , y N +1 (cid:1) = (cid:0) ( y ) − , ( y ) − , y , y , . . . , y N +1 (cid:1) · · · µ N +1 ←→ y N +1 = (cid:0) ( y ) − ( y N +1 ⊕ , ( y ) − , . . . , ( y N − ) − , ( y N ) − ( y N +1 ⊕ , ( y N +1 ) − (cid:1) = (cid:0) ( y ) − , ( y ) − , . . . , ( y N − ) − , ( y N ) − , ( y N +1 ) − (cid:1) , where we use the fact y j ⊕ y min[1 , j = 1 for j = 1 , , . . . , N + 1. (cid:3) Now we consider iteration of the consecutive seed mutations µ N +1 ◦ · · · ◦ µ ◦ µ , which de-fines the map y ℓ ( N +1) y ( ℓ +1)( N +1) for ℓ ≥
1. Let us introduce the monomials ν ℓ , ν ℓ , . . . , ν ℓN in the tropical semifield P = (Trop( y ) , · , ⊕ ) ν ℓ := y N +1; ℓ ( N +1) y ℓ ( N +1) ,ν ℓj := y j ; ℓ ( N +1) ( j = 2 , , . . . , N )generated by the coefficients in y ℓ ( N +1) = (cid:0) y ℓ ( N +1) , y ℓ ( N +1) , . . . , y N +1; ℓ ( N +1) (cid:1) . The permutation σ N = · · · N − NN · · · N − N − ∈ S N (5)of N letters acts on the monomial ν ℓj as σ N ν ℓj = ν ℓ ( σ N ) − ( j ) = ν ℓj +1 for j = 1 , , . . . , N , where the subscript is reduced modulo N . Theorem 1
For any ℓ ≥ and j = 1 , , . . . , N , we have ν ℓ +1 j = σ N ν ℓj = ν ℓj +1 , here the subscript is reduced modulo N . Therefore, every ν ℓj has period N on ℓ : ν ℓ + Nj = ν ℓj . (Proof) First note that the exchange matrix B k has the periodicity B ℓ ( N +1)+ k = B k for ℓ ≥ k = 1 , , . . . , N + 1. We then see that the k -th row of the exchange matrix B ℓ ( N +1)+ k − = B k − for k = 1 , , . . . , N + 1, which determines the seed mutation µ k , has twonon-zero elements − k, k −
1) and at ( k, k + 1): (cid:18) · · · k − k − k k + 1 k + 2 · · · N + 1 k · · · − − · · · (cid:19) . Thus the exchange relation (2) reduces to y j ;( N +1)+ k = µ k ( y j ;( N +1)+ k − ) = ( y k ;( N +1)+ k − ) − j = k,y j ;( N +1)+ k − ( y k ;( N +1)+ k − ⊕ j = k ± ,y j ;( N +1)+ k − j = k, k ± . (6)Also note that, by lemma 1, we have y N +1 = ( y N +1 , y N +1 , . . . , y N +1; N +1 ) = (cid:0) ( y ) − , ( y ) − , . . . , ( y N +1 ) − (cid:1) . Apply the mutation µ to y N +1 . We then obtain y N +1)+1 = µ ( y N +1 ) = ( y N +1 ) − ,y N +1)+1 = µ ( y N +1 ) = y N +1 ( y N +1 ⊕
1) = y N +1 y N +1 ,y j ;( N +1)+1 = µ ( y j ; N +1 ) = y j ; N +1 ( j = 3 , , . . . , N ) ,y N +1;( N +1)+1 = µ ( y N +1; N +1 ) = y N +1; N +1 ( y N +1 ⊕
1) = y N +1; N +1 y N +1 , where we use the exchange relation (6) and the fact y N +1 = ( y ) − which implies y N +1 ⊕ y N +1 . 10e inductively obtain y j ;( N +1)+ k = µ k ( y j ;( N +1)+ k − ) = y j ;( N +1)+ k − = y j +1; N +1 ( j = 1 , , . . . , k − ,y k − N +1)+ k = µ k ( y k − N +1)+ k − ) = y k − N +1)+ k − y k ;( N +1)+ k − = y k ; N +1 ,y k ;( N +1)+ k = µ k ( y k ;( N +1)+ k − ) = ( y k ;( N +1)+ k − ) − = ( y N +1 y N +1 · · · y k ; N +1 ) − ,y k +1;( N +1)+ k = µ k ( y k +1;( N +1)+ k − ) = y k +1;( N +1)+ k − y k ;( N +1)+ k − = y N +1 y N +1 · · · y k +1; N +1 ,y j ;( N +1)+ k = µ k ( y j ;( N +1)+ k − ) = y j ;( N +1)+ k − = y j ; N +1 ( j = k + 2 , k + 3 , . . . , N ) ,y N +1;( N +1)+ k = µ k ( y N +1;( N +1)+ k − ) = y N +1;( N +1)+ k − = y N +1 y N +1; N +1 by applying the consecutive seed mutations µ k ◦ · · ·◦ µ ◦ µ to y ( N +1)+1 ( k = 2 , , . . . , N − y k ;( N +1)+ k − = y N +1 y N +1 · · · y k ; N +1 = ( y ) − ( y ) − · · · ( y k ) − (7)for k = 2 , , . . . , N − y k ;( N +1)+ k − ⊕ y k ;( N +1)+ k − .Moreover, apply µ N to y ( N +1)+ N − . Then we have y j ;( N +1)+ N = µ N ( y j ;( N +1)+ N − ) = y j ;( N +1)+ N − = y j +1; N +1 ( j = 1 , , . . . , N − ,y N − N +1)+ N = µ N ( y N − N +1)+ N − ) = y N − N +1)+ N − y N ;( N +1)+ N − = y N ; N +1 ,y N ;( N +1)+ N = µ N ( y N ;( N +1)+ N − ) = ( y N ;( N +1)+ N − ) − = ( y N +1 y N +1 · · · y N ; N +1 ) − ,y N +1;( N +1)+ N = µ N ( y N +1;( N +1)+ N − ) = y N +1;( N +1)+ N − y N ;( N +1)+ N − = ( y N +1 ) y N +1 · · · y N +1; N +1 , (8)where we use (7) for k = N which implies y N ;( N +1)+ N − ⊕ y N ;( N +1)+ N − . Finally, byapplying µ N +1 to y ( N +1)+ N , we obtain y N +1) = µ N +1 ( y N +1)+ N ) = y N +1)+ N y N +1;( N +1)+ N = ( y N +1 ) ( y N +1 ) y N +1 · · · y N +1; N +1 ,y j ;2( N +1) = µ N +1 ( y j ;( N +1)+ N ) = y j ;( N +1)+ N = y j +1; N +1 ( j = 2 , , . . . , N − ,y N ;2( N +1) = µ N +1 ( y N ;( N +1)+ N ) = y N ;( N +1)+ N y N +1;( N +1)+ N = y N +1 y N +1; N +1 ,y N +1;2( N +1) = µ N +1 ( y N +1;( N +1)+ N ) = ( y N +1;( N +1)+ N ) − = (cid:0) ( y N +1 ) y N +1 · · · y N +1; N +1 (cid:1) − , y N +1;( N +1)+ N ⊕ y N +1;( N +1)+ N . It immediately follows ν = y N +1;2( N +1) y N +1) = y N +1 = ν ,ν j = y j ;2( N +1) = y j +1; N +1 = ν j +1 ( j = 2 , , . . . , N − ,ν N = y N ;2( N +1) = y N +1 y N +1; N +1 = ν . Every coefficient y j ;2( N +1) in y N +1) except for y N +1;2( N +1) is a monomial consisting ofnegative powers of the initial ones y , y , . . . , y N +1 . Thus the exchange relation (6) reducesto y j ; ℓ ( N +1)+ k = µ k ( y j ; ℓ ( N +1)+ k − ) = ( y k ; ℓ ( N +1)+ k − ) − j = k,y j ; ℓ ( N +1)+ k − y k ; ℓ ( N +1)+ k − j = k ± ,y j ; ℓ ( N +1)+ k − j = k, k ± ℓ ≥ ℓ = 1. Therefore, for any ℓ ≥
2, we inductively obtain y ℓ +1)( N +1) = ( y ℓ ( N +1) ) ( y ℓ ( N +1) ) y ℓ ( N +1) · · · y N +1; ℓ ( N +1) ,y j ;( ℓ +1)( N +1) = y j +1; ℓ ( N +1) ( j = 2 , , . . . , N − ,y N ;( ℓ +1)( N +1) , = y ℓ ( N +1) y N +1; ℓ ( N +1) ,y N +1;( ℓ +1)( N +1) = (cid:0) ( y ℓ ( N +1) ) y ℓ ( N +1) · · · y N +1; ℓ ( N +1) (cid:1) − , which implies ν ℓ +11 = y N +1;( ℓ +1)( N +1) y ℓ +1)( N +1) = y ℓ ( N +1) = ν ℓ ,ν ℓ +1 j = y j ;( ℓ +1)( N +1) = y j +1; ℓ ( N +1) = ν ℓj +1 ( j = 2 , , . . . , N − ,ν ℓ +1 N = y N ; ℓ ( N +1) = y ℓ ( N +1) y N +1; ℓ ( N +1) = ν ℓ . It immediately follows ν ℓ + Nj = ν ℓj for ℓ ≥ j = 1 , , . . . , N . (cid:3) We easily find the conserved quantities of the dynamics of coefficients via the monomials ν , ν , . . . , ν N , where we denote ν j by ν j ( j = 1 , , . . . , N ) for simplicity. Remark that wehave ν = y N +1; N +1 y N +1 = ( y N +1 ) − ( y ) − , (10) ν j = y j ; N +1 = ( y j ) − ( j = 2 , , , . . . , N ) . (11)12n the tropical semifield P , let e n ( n = 1 , , . . . , N ) be the fundamental symmetric polynomialof degree n generated by the monomials ν , ν , . . . , ν N : e n = e n ( ν , ν , . . . , ν N ) := M I ⊂{ , ,...,N }| I | = n Y j ∈ I ν j . We denote e n ( ν ℓ , ν ℓ , . . . , ν ℓN ) simply by e ℓn for ℓ ≥ ℓ ≥
1, let us introduce new variables: y ℓ := (cid:0) y ℓ , y ℓ , . . . , y ℓN +1 (cid:1) ,y ℓj := y j ; ℓ ( N +1) ( j = 1 , , . . . , N + 1) . We see from the proof of theorem 1 that the evolution of y ℓ is given as follows y ℓ +11 = (cid:0) y ℓ y ℓ (cid:1) y ℓ · · · y ℓN +1 ,y ℓ +1 j = y ℓj +1 ( j = 2 , , . . . , N − ,y ℓ +1 N = y ℓ y ℓN +1 ,y ℓ +1 N +1 = (cid:16)(cid:0) y ℓ (cid:1) y ℓ · · · y ℓN +1 (cid:17) − . (12)Thus the evolution of y ℓ defines the map ψ : P N +1 → P N +1 ; y ℓ y ℓ +1 . Corollary 1
The fundamental symmetric polynomial e ℓn ( n = 1 , , . . . , N ) of degree n gen-erated by the monomials ν ℓ , ν ℓ , . . . , ν ℓN is the conserved quantity of the dynamical system y ℓ +1 = ψ ( y ℓ ) governed by the map ψ , that is, we have e ℓn = e n for any ℓ ≥ . Moreover, all the fundamental symmetric polynomials are the same: e = e = · · · = e N = N +1 Y j =1 ( y j ) − . (Proof) The permutation σ N acts on e ℓn as follows σ N e ℓn = σ N e n ( ν ℓ , ν ℓ , . . . , ν ℓN )= e n ( ν ℓ ( σ N ) − (1) , ν ℓ ( σ N ) − (2) , . . . , ν ℓ ( σ N ) − ( N ) )= e n ( ν ℓ , ν ℓ , . . . , , ν ℓN , ν ℓ )= e n ( ν ℓ +11 , ν ℓ +12 , . . . , ν ℓ +1 N − , ν ℓ +1 N ) = e ℓ +1 n , ν ℓ +1 j = ν ℓj +1 for j = 1 , , . . . , N (see theorem 1). On the other hand, theidentity σ N e ℓn = e ℓn holds by definition.Moreover, by using (10) and (11), we compute e = ν ⊕ ν ⊕ · · · ⊕ ν N = ( y ) − ( y N +1 ) − ⊕ ( y ) − ⊕ · · · ⊕ ( y N ) − = y min[ − , ,..., y min[0 , − , ,..., · · · y min[ − , ,..., N +1 = N +1 Y j =1 ( y j ) − . Similarly, since we have M I ⊂{ , ,...,N }| I | = n Y j ∈ I ν j = M I ⊂{ , ,...,N }| I | = n Y j ∈ I ( y j ) − = N Y j =2 ( y j ) − for any n ≥
1, we obtain e n = ν M I ⊂{ , ,...,N }| I | = n − Y j ∈ I ν j ⊕ M I ⊂{ , ,...,N }| I | = n Y j ∈ I ν j = ( y ) − ( y N +1 ) − N Y j =2 ( y j ) − M N Y j =2 ( y j ) − = N +1 Y j =1 ( y j ) − . This completes the proof. (cid:3)
B. General solution
For ℓ = nN + s ≥ ≤ n , 1 ≤ s ≤ N ), we define the monomial C ℓ in P to be C ℓ := ν ν · · · ν ℓ = ( e ) n ( y N +1 ) − s Y j =1 ( y j ) − , where the subscript of ν j is reduced modulo N and we use (10) and (11). Then C N = e is the conserved quantity of the dynamical system y ℓ +1 = ψ ( y ℓ ).The dynamical system governed by the map ψ : y ℓ y ℓ +1 is easily solved by using themonomials C , C , . . . , C N . 14 heorem 2 For given N ≥ , put ℓ = nN + s , where ≤ n and ≤ s ≤ N . The generalsolution y ℓ = ( y ℓ , y ℓ , . . . , y ℓN +1 ) to the dynamical system y ℓ +1 = ψ ( y ℓ ) governed by the map ψ is given by y ℓ = ( C N ) n ( N +1)+ s − C s y N +1 ,y ℓj = ν j + s − ( j = 2 , , . . . , N ) ,y ℓN +1 = (cid:0) ( C N ) n ( N +1)+ s − C s − y N +1 (cid:1) − , (13) where the subscript of ν j is reduced module N and we assume C = 1 . (Proof) Since y ℓj = ν ℓj for j = 2 , , . . . , N , the solution y ℓj = ν j + s − ( j = 2 , , . . . , N )is a straightforward consequence of theorem 1.We compute y ℓ and y ℓN +1 . For ℓ = 1, i.e. , n = 0 and s = 1, we have y = ( y ) − = ν y N +1 = ( C N ) C y N +1 ,y N +1 = ( y N +1 ) − = (cid:0) ( C N ) C y N +1 (cid:1) − , where we use the assumption C = 1 and (10). We assume that (13) is true for ℓ = nN + s .Then, by using (12) and the fact that C N is the conserved quantity, we have y ℓ +11 = (cid:0) y ℓ y ℓ (cid:1) y ℓ · · · y ℓN +1 = C N y ℓ y ℓ = C N ( C N ) n ( N +1)+ s − C s y N +1 ν s +1 = ( C N ) n ( N +1)+ s C s +1 y N +1 ,y ℓ +1 N +1 = (cid:16)(cid:0) y ℓ (cid:1) y ℓ · · · y ℓN +1 (cid:17) − = (cid:0) C N y ℓ (cid:1) − = (cid:0) C N ( C N ) n ( N +1)+ s − C s y N +1 (cid:1) − = (cid:0) ( C N ) n ( N +1)+ s C s y N +1 (cid:1) − . Thus (13) is true for ℓ + 1 = nN + s + 1. (cid:3) In order to compute the cluster mutation x ( ℓ +1)( N +1) = ( µ N +1 ◦ · · · ◦ µ ◦ µ ) ( x ℓ ( N +1) ) ,
15e use the exchange relation (3). Throughout the consecutive mutations µ , µ , . . . , µ N +1 ,the cluster variable x k ; ℓ ( N +1) is transformed not by µ j ( j = k ) but by µ k as x k ;( ℓ +1)( N +1) = ( µ N ◦ µ N − ◦ · · · ◦ µ ) ( x k ; ℓ ( N +1) )= µ k ( x k ; ℓ ( N +1) )= x k ; ℓ ( N +1)+ k = y k ; ℓ ( N +1)+ k − x k − ℓ +1)( N +1) x k +1; ℓ ( N +1) + 1( y k ; ℓ ( N +1)+ k − ⊕ x k ; ℓ ( N +1) , (14)where we use the equalities x k − ℓ ( N +1)+ k − = ( µ N +1 ◦ · · · ◦ µ k +1 ◦ µ k ) ( x k − ℓ ( N +1)+ k − ) = x k − ℓ +1)( N +1) ,x k +1; ℓ ( N +1)+ k − = ( µ k ◦ · · · ◦ µ ◦ µ ) ( x k +1; ℓ ( N +1) ) = x k +1; ℓ ( N +1) derived from the fact that the mutations µ k , µ k +1 , . . . , µ N +1 and µ , µ , . . . , µ k do not vary x k − ℓ ( N +1)+ k − and x k +1; ℓ ( N +1)+ k − , respectively. Therefore, we need the explicit form of thecoefficient y k ; ℓ ( N +1)+ k − ( k = 1 , , . . . , N + 1) to execute the computation. Proposition 1
For ℓ = nN + s ≥ , where ≤ n and ≤ s ≤ N , the coefficient y k ; ℓ ( N +1)+ k − is explicitly given by y k ; ℓ ( N +1)+ k − = ( C N ) n ( N +1)+ s − C s + k − y N +1 for k = 1 , , . . . , N + 1 . (Proof) By using theorem 2 and the exchange relation (9), we have y ℓ ( N +1) = y ℓ = ( C N ) n ( N +1)+ s − C s y N +1 ,y ℓ ( N +1)+1 = µ ( y ℓ ( N +1) ) = y ℓ y ℓ = ( C N ) n ( N +1)+ s − C s +1 y N +1 . Thus we inductively obtain y k ; ℓ ( N +1)+ k − = µ k − ( y k ; ℓ ( N +1)+ k − ) = y k ; ℓ ( N +1)+ k − y k − ℓ ( N +1)+ k − = y ℓk y k − ℓ ( N +1)+ k − = ( C N ) n ( N +1)+ s − C s + k − y N +1 for k = 1 , , . . . , N + 1. (cid:3) V. DYNAMICS OF CLUSTER VARIABLESA. Birational map
Iteration of the consecutive seed mutations µ N +1 ◦ · · · ◦ µ ◦ µ assigned to the path ̟ oftype A (1) N induces a certain dynamical system governed by a birational map.Let Σ m = ( x m , y m , B m ) be the seed assigned to the vertex t m in the path ̟ of type A (1) N .For t ≥
0, we introduce new variables: x t := (cid:0) x t , x t , . . . , x tN +1 (cid:1) ,x ti := x i ; t ( N +1) ( i = 1 , , . . . , N + 1) . Note that x is the initial cluster x : x = (cid:0) x , x , . . . , x N +1 (cid:1) = ( x , x , . . . , x N +1 ) = x and x t is the cluster assigned to the vertex t t ( N +1) in ̟ . Theorem 3
The cluster variables assigned to the path ̟ of type A (1) N are given by using thesolutions to the following dynamical system z t +1 i = z t +1 i − z ti +1 + 1 z ti ( i = 1 , , . . . , N + 1) ,z t +10 = z tN +1 ,z tN +2 = z t +11 (15) for t ≥ . (Proof) Put t = nN + s ≥ ≤ n and 1 ≤ s ≤ N . By proposition 1, we have y i ; t ( N +1)+ i − = ( C N ) n ( N +1)+ s − C s + i − y N +1 for i = 1 , , . . . , N + 1. Then the exchange relation (14) reduces to( C N ) n ( N +1)+ s − C s + i − y N +1 x t +1 i x ti = ( C N ) n ( N +1)+ s − C s + i − y N +1 x t +1 i − x ti +1 + 1 (16)for i = 1 , , . . . , N + 1, where we assume x t +10 = x tN +1 and x tN +2 = x t +11 .17uppose that the variables z t , z t , . . . , z tN +1 for t ≥ z t +1 i z ti = ( C N ) n ( N +1)+ s − C s + i − y N +1 x t +1 i x ti , (17) z t +1 i − z ti +1 = ( C N ) n ( N +1)+ s − C s + i − y N +1 x t +1 i − x ti +1 . (18)Substitute (17) and (18) into (16). Then we see that z t , z t , . . . , z tN +1 solve (15) by setting z t +10 = z tN +1 and z tN +2 = z t +11 .We show that (17) and (18) are compatible with each other. First, by (17), we have x t +1 i = z t +1 i z ti ( C N ) n ( N +1)+ s − C s + i − y N +1 x ti . (19)Also, by (18), we have x t +1 i − = z t +1 i − z ti +1 ( C N ) n ( N +1)+ s − C s + i − y N +1 x ti +1 . (20)Thus, by successive application of (19) and (20), we obtain x t +1 i = z t +1 i z ti ( C N ) n ( N +1)+ s − C s + i − y N +1 x ti = z t +1 i z ti ( C N ) n ( N +1)+ s − C s + i − y N +1 ( C N ) n ( N +1)+ s − C s + i − y N +1 z ti z t − i +2 x t − i +2 = 1 C N z t +1 i z t − i +2 x t − i +2 . (21)Similarly, by successive application of (20) and (19), we obtain x t +1 i = z t +1 i z ti +2 ( C N ) n ( N +1)+ s − C s + i y N +1 x ti +2 = z t +1 i z ti +2 ( C N ) n ( N +1)+ s − C s + i y N +1 ( C N ) n ( N +1)+ s − C s + i y N +1 z ti +2 z t − i +2 x t − i +2 = 1 C N z t +1 i z t − i +2 x t − i +2 . This coincides with (21). Therefore, the equations (17) and (18) are compatible with eachother. Thus, if the solution (cid:0) z t , z t , . . . , z tN +1 (cid:1) to the difference equation (15) for t ≥ (cid:0) x t , x t , . . . , x tN +1 (cid:1) to the difference equation(16) for t ≥ x t , x t , . . . , x tN +1 aregiven by using x i and z i for i = 1 , , . . . , N + 1.Now we show that the cluster variables x t , x t , . . . , x tN +1 can be given by using the initialones x , x , . . . , x N +1 . Noting (17) and (18), it is clear that if z i ∝ x i , that is, z i is propor-tional to x i , for i = 1 , , . . . , N + 1 then z ti ∝ x ti for i = 1 , , . . . , N + 1 and t ≥
1. Thus if18e assume z i ∝ x i , for i = 1 , , . . . , N + 1 the cluster variables x t , x t , . . . , x tN +1 are given by x , x , . . . , x N +1 , and hence by the initial cluster variables x , x , . . . , x N +1 .Finally, we check that the proportionality z i ∝ x i , for i = 1 , , . . . , N + 1 is compatiblewith (17) and (18). Assume t = 1 ( n = 0 and s = 1) in (17) and (18): z i z i = C i y N +1 x i x i ,z i − z i +1 = C i y N +1 x i − x i +1 . Then we have z i +1 = C i y N +1 x i − x i +1 z i − = C i y N +1 x i − x i +1 C i − y N +1 x i − x i − z i − = ν i x i +1 x i − z i − (22)for i = 2 , , . . . , N . Substitute z i = α i x i ( α i ∈ P ) into (22) we obtain α i +1 = α i − ν i = α i − y − i ( i = 2 , , . . . , N ) . (23)For i = 1, we also obtain α α N +1 = y − (24)from (18), where we use the boundary conditions z = z N +1 and x = x N +1 . Thus if α , α , . . . , α N +1 satisfy (23) and (24) then z i = α i x i holds for i = 1 , , . . . , N + 1 for any α ∈ P . (cid:3) We put z t := (cid:0) z t , z t , . . . , z tN +1 (cid:1) for t ≥ z t z t +1 on F N +1 defined by (15) by ϕ . We oftenrefer to the birational map ϕ and to the dynamical system (15) governed by ϕ as of type A (1) N as well as the path ̟ from which they are arising. B. Periodicity
Let us consider the birational map ϕ given by (15). Denote the Laurent polynomial ring ZP [( x ) ± , ( x ) ± , . . . , ( x N +1 ) ± ] generated by the initial cluster variables x , x , . . . , x N +1 by ZP [ x ± ]. For simplicity, we denote the polynomial p ( x t , x t , . . . , x tN +1 ) ∈ ZP [( x t ) ± ], where p = p ( x , x , . . . , x N +1 ) ∈ ZP [( x ) ± ], by p t ( p = p ). We use the same notations for the19aurent polynomial ring ZP [( z t ) ± ] generated by z t , z t , . . . , z tN +1 . Remark that if we assume z i ∝ x i for i = 1 , , . . . , N + 1 then we have ZP [( z t ) ± ] = ZP [( x t ) ± ] for any t ≥ λ , λ , . . . , λ N ∈ ZP [( z ) ± ] to be λ i = λ i ( z , z , . . . , z N +1 ) := z i + z i +2 z i +1 ( i = 1 , , . . . , N − ,λ N = λ N ( z , z , . . . , z N +1 ) := z z N + z z N +1 + 1 z z N +1 . Lemma 2
If we assume z i ∝ x i for i = 1 , , . . . , N + 1 then we have λ i ∈ ZP [ x ± ] for i = 1 , , . . . , N . (Proof) First note that if we assume z i = α i x i ( α i ∈ P ) then α i ( i = 2 , , . . . , N + 1)satisfies (23) and (24). In addition, remark that the cluster variables x i and x i satisfy theexchange relation (see (3) and (14)) x i = y i x i − x i +1 + 1 x i (25)for i = 1 , , . . . , N + 1 and the boundary conditions x = x N +1 and x N +2 = x , where weuse the fact y i ; i − = y i (see the proof of lemma 1).Assume i = N + 1. Then (25) reduces to x N +1 x N +1 = y N +1 x N x N +2 + 1 = y N +1 x N x + 1 , where we use the boundary condition x N +2 = x . This implies that (18) reduces to z N z = y N +1 x N z for i = N + 1 and t = 0. Substitution of z i = α i x i ( i = 1 , N ) into this equation leads to α α N = y N +1 . (26)For i = 1 , , . . . , N −
2, we compute λ i = z i + z i +2 z i +1 = α i x i + α i +2 x i +2 α i +1 x i +1 = α i α i +1 y i +1 x i + x i +2 y i +1 x i +1 . λ i = α i α i +1 y i +1 x i + x i +2 y i +1 x i +1 = α i α i +1 ( y i +1 x i x i +2 + 1) + y i +2 x i +1 x i +3 y i +1 x i +1 x i +2 = α i α i +1 x i +1 x i +1 + y i +2 x i +1 x i +3 y i +1 x i +1 x i +2 = α i α i +1 x i +1 + y i +2 x i +3 y i +1 x i +2 ∈ ZP [ x ± ] . Similarly, we compute λ N − = α N − α N x N + y N +1 x y N x N +1 = α N − α N x x N + y y N +1 x x N +1 + y N +1 y N x x N +1 ∈ ZP [ x ± ] , where we use the boundary conditions x = x N +1 and x N +2 = x .Noting (24) and (26), we have λ N = α α N x x N + α α N +1 x x N +1 + 1 α α N +1 x x N +1 = y ( y N +1 x x N + 1) + x x N +1 α α N +1 y x x N +1 = y x N +1 x N +1 + x x N +1 α α N +1 y x x N +1 = y x N +1 + x α α N +1 y x = ( y x x N +1 + 1) + y x x α α N +1 y x x = x x + y x x y α α N +1 x x = x + y x α α N +1 y x ∈ ZP [ x ± ] . Thus the Laurent polynomials λ , λ , . . . , λ N ∈ ZP [( z ) ± ] are in the Laurent polynomialring ZP [ x ± ] generated by the initial cluster variables.Finally, we give the ratio α i of the variables z i and x i , explicitly. By applying (23)repeatedly, we have α α N = α α ( y ) − ( y ) − · · · ( y N − ) − ( N even) , ( α ) ( y ) − ( y ) − · · · ( y N − ) − ( N odd) ,α α N +1 = α α ( y ) − ( y ) − · · · ( y N ) − ( N even) , ( α ) ( y ) − ( y ) − · · · ( y N ) − ( N odd) . Then (24) and (26) reduce to α α = ( y ) − y y · · · y N ( N even) , ( α ) = ( y ) − y y · · · y N ( N odd) (27)and α α = y y · · · y N +1 ( N even) , ( α ) = y y · · · y N +1 ( N odd) , (28)21espectively. Therefore, the ratio α i ( i = 1 , , . . . , N + 1) is explicitly given by using theinitial coefficients y , y , . . . , y N +1 via (23), (27) and (28). Moreover, from (27) and (28), y , y , . . . , y N +1 must satisfy y y · · · y N +1 = y y · · · y N (29)for even N . (cid:3) Hereafter, we assume that the initial coefficients y , y , . . . , y N +1 satisfy (29) for even N unless otherwise stated.The action of the permutation σ N ∈ S N (see (5)) on λ i is given by σ N λ i = λ ( σ N ) − ( i ) = λ i +1 for i = 1 , , . . . , N . The Laurent polynomials λ , λ , . . . , λ N have the following periodicityunder the evolution by means of the birational map ϕ . Theorem 4
For any t ≥ and i = 1 , , . . . , N , we have λ t +1 i = σ N λ ti = λ ti +1 , where the subscript is reduced module N . Therefore, every λ ti has period N on t : λ t + Ni = λ ti . Moreover, we have λ ti ∈ ZP [ x ± ] for i = 1 , , . . . , N and t ≥ . (Proof) For i = 1 , , . . . , N −
2, by using (15), the Laurent polynomial λ t +1 i reduces to λ t +1 i = z t +1 i + z t +1 i +2 z t +1 i +1 = z t +1 i z ti +2 + z t +1 i +1 z ti +3 + 1 z t +1 i +1 z ti +2 = z t +1 i z ti +2 + z t +1 i +1 z ti +3 + 1 z t +1 i +1 z ti +2 = (cid:0) z t +1 i z ti +2 + 1 (cid:1) z ti +1 + (cid:0) z t +1 i z ti +2 + 1 (cid:1) z ti +3 (cid:0) z t +1 i z ti +2 + 1 (cid:1) z ti +2 = z ti +1 + z ti +3 z ti +2 = λ ti +1 . For i = N − i = N , we also compute λ t +1 N − = z t +1 N − + z t +1 N +1 z t +1 N = z tN + z t +11 z tN +1 = z t z tN + z t z tN +1 + 1 z t z tN +1 = λ tN λ t +1 N = z t +11 z t +1 N + z t +12 z t +1 N +1 + 1 z t +11 z t +1 N +1 = (cid:0) z t +1 N z t +11 + 1 (cid:1) z tN +1 + z t +12 (cid:0) z t +1 N z t +11 + 1 (cid:1) z t +11 (cid:0) z t +1 N z t +11 + 1 (cid:1) = z tN +1 + z t +12 z t +11 = z t z tN +1 + z t +11 z t + 1 z t +11 z t = (cid:0) z tN +1 z t + 1 (cid:1) z t + (cid:0) z tN +1 z t + 1 (cid:1) z t (cid:0) z tN +1 z t + 1 (cid:1) z t = z t + z t z t = λ t , respectively. Then it is clear that λ ti ( i = 1 , , . . . , N ) has period N on t . It is also clearfrom lemma 2 that we have λ ti = ( σ N ) t − λ i ∈ ZP [ x ± ]for i = 1 , , . . . , N and t ≥ (cid:3) We denote the (non-Laurent) polynomial subring ZP [ λ t , λ t , . . . , λ tN ] of the ambient field F generated by the Laurent polynomials λ t , λ t , . . . , λ tN simply by ZP (cid:2) λ t (cid:3) . Proposition 2
For any t ≥ , we have ZP (cid:2) λ t (cid:3) = ZP [ λ ] ⊂ ZP (cid:2) x ± (cid:3) (Proof) Let f t be a polynomial in ZP (cid:2) λ t (cid:3) . By using theorem 4, we have f t = f ( λ t , λ t , . . . , λ tN )= f (( σ N ) t − λ , ( σ N ) t − λ , . . . , ( σ N ) t − λ N ) = ( σ N ) t − f. If f ∈ ZP [ λ ] we have f t = ( σ N ) t − f ∈ ZP [ λ ]. Conversely, since f t ∈ ZP (cid:2) λ t (cid:3) , f =( σ N ) − ( t − f t ∈ ZP (cid:2) λ t (cid:3) holds. The inclusion ZP [ λ ] ⊂ ZP [ x ± ] immediately follows from thefact λ , λ , . . . , λ N ∈ ZP [ x ± ]. (cid:3) Introduce the fundamental symmetric polynomial of degree n generated by the Laurentpolynomials λ , λ , . . . , λ N and let it be q n ( n = 1 , , . . . , N ): q n = q n ( λ , λ , . . . , λ N ) := X I ⊂{ , ,...,N }| I | = n Y i ∈ I λ i ∈ ZP [ λ ] . We denote q n ( λ t , λ t , . . . , λ tN ) ∈ ZP (cid:2) λ t (cid:3) simply by q tn for t ≥ q n = q n ). We then have thefollowing corollary to theorem 4 which states conserved quantities of the dynamical systemgoverned by the map ϕ . 23 orollary 2 The fundamental symmetric polynomial q tn ( n = 1 , , . . . , N ) of degree n gen-erated by the Laurent polynomials λ t , λ t , . . . , λ tN is the conserved quantity of the dynamicalsystem z t +1 = ϕ ( z t ) governed by the birational map ϕ , that is, we have q tn = ( σ N ) t − q n = q n for any t ≥ . (Proof) For any t ≥
1, since σ N q n = q n , it immediately follows q tn = q n (cid:0) λ t , λ t , . . . , λ N (cid:1) = q n (cid:0) ( σ N ) t − λ , ( σ N ) t − λ , . . . , ( σ N ) t − λ N (cid:1) = ( σ N ) t − q n ( λ , λ , . . . , λ N )= q n from theorem 4. (cid:3) In the following subsections, we construct the general solution to the dynamical system z t +1 = ϕ ( z t ) governed by the birational map ϕ via linearization of ϕ in terms of the Laurentpolynomials λ t , λ t , . . . , λ tN . C. LinearizationProposition 3
The quadratic birational map ϕ : z t z t +1 given by (15) is equivalent tothe non-autonomous linear map z t z t +1 defined by z t +11 = λ tN z tN +1 − z tN ,z t +12 = λ t z t +11 − z tN +1 ,z t +1 i +2 = λ ti +1 z t +1 i +1 − z t +1 i ( i = 1 , , . . . , N −
1) (30) for t ≥ . (Proof) By using the Laurent polynomial λ tN , the difference equation in (15) for i = 1reduces to z t +11 = z tN +1 z t + 1 z t = λ tN z t z tN +1 − z t z tN z t = λ tN z tN +1 − z tN . λ t +1 N , the one for i = N + 1 reduces to z t +1 N +1 = z t +1 N z t +11 + 1 z tN +1 = λ t +1 N z t +11 z t +1 N +1 − z t +12 z t +1 N +1 z tN +1 = z t +1 N +1 λ t +1 N z t +11 − z t +12 z tN +1 . It follows that we have z t +12 = λ t +1 N z t +11 − z tN +1 = λ t z t +11 − z tN +1 , where we use the fact λ t +1 N = λ t . Moreover, by noticing λ t +1 i = λ ti +1 , the Laurent polynomial λ t +1 i = ( z t +1 i + z t +1 i +2 ) /z t +1 i +1 can be written as z t +1 i +2 = λ t +1 i z t +1 i +1 − z t +1 i = λ ti +1 z t +1 i +1 − z t +1 i for i = 1 , , . . . , N − (cid:3) Thus, noting proposition 2, we see that the quadratic map ϕ is linearized by using theLaurent polynomials λ , λ , . . . , λ N in ZP [ x ± ]. Remark 1
Hone, Lampe and Kouloukas showed that iteration of the cluster mutations oftype A (1) N has linear degree growth . In two dimension, it is well known that linear degreegrowth of a map iteration leads to linearization of the map ; however, as far as the authorsknow, it is not clear whether the map exhibiting linear degree growth is linearizable or notin higher dimensions. The simultaneous system (30) of linear equations can be written by using a matrix andvectors as follows · · · · · · − λ t − λ t . . . . . . ...... . . . . . . 1 00 · · · − λ tN z t +11 z t +12 ...... z t +1 N +1 = z tN − + z tN +1 λ tN − (31)Let us denote the coefficient matrix of (31) by A t . Also denote the vectors ( − , , . . . , T and ( λ tN , − , , . . . , T in the right hand side of (31) by b and b , respectively. Let thematrix obtained from A t by replacing its i -th column with b be A ti, . Also, let the oneby replacing the i -th column with b be A ti, . Remark that the determinants det A ti, anddet A ti, are in the polynomial ring ZP [ λ ] (see proposition 2).25ow we consider the following polynomials ξ i, , ξ i, ∈ ZP [ λ ]: ξ i, = det A i, = − ˜ a i ,ξ i, = det A i, = λ N ˜ a i − ˜ a i for i = 1 , , . . . , N + 1, where ˜ a ij is the ( i, j )-cofactor of the matrix A . Note that we have( σ N ) t − ξ i, = det A ti, , ( σ N ) t − ξ i, = det A ti, for ant t ≥ σ N ) t − ξ i, , ( σ N ) t − ξ i, ∈ ZP [ λ ] for i = 1 , , . . . , N + 1.We, moreover, introduce the 2 × M whose entries are taken from ZP [ λ ]: M := ξ N, ξ N, ξ N +1 , ξ N +1 , . The action of the permutation σ N ∈ S N on M is defined to be σ N M := σ N ξ N, σ N ξ N, σ N ξ N +1 , σ N ξ N +1 , . The entries of ( σ N ) t M are in ZP [ λ ] for any t ≥ D. General solution
Noting det A t = 1, we solve the system (31) of linear equations by using the Cramerformula. Then we obtain z t +1 i = (cid:0) ( σ N ) t − ξ i, (cid:1) z tN + (cid:0) ( σ N ) t − ξ i, (cid:1) z tN +1 (32)for i = 1 , , . . . , N + 1.For m ≥
2, we put M m := (cid:0) ( σ N ) m − M (cid:1) (cid:0) ( σ N ) m − M (cid:1) · · · ( σ N M ) M. With imposing i = N and N + 1 to (32), we obtain z t +1 N z t +1 N +1 = (cid:0) ( σ N ) t − M (cid:1) z tN z tN +1 = M t z N z N +1 . (33)Then a theorem which states the general solution to the linear system (30), hence to thedynamical system (15), follows. 26 heorem 5 For given N ≥ , denote t ≥ by t = nN + s , where ≤ n and ≤ s ≤ N .Then the general solution to the dynamical system z t +1 = ϕ ( z t ) governed by the birationalmap ϕ is given by z ti = (cid:0) ( σ N ) s − ξ i, , ( σ N ) s − ξ i, (cid:1) M s ( M N ) n z N z N +1 (34) for i = 1 , , . . . , N + 1 . (Proof) Since every ξ i,j has period N , ( σ N ) N ξ i,j = ξ i,j , we have ( σ N ) N M = M . Thus(33) reduces to z tN z tN +1 = M nN + s z N z N +1 = M s ( M N ) n z N z N +1 . Similarly, we obtain z t +1 i = (cid:0) ( σ N ) s − ξ i, (cid:1) z tN + (cid:0) ( σ N ) s − ξ i, (cid:1) z tN +1 from (32), which leads to (34). (cid:3) Thus we see that the variables z t , z t , . . . , z tN +1 have the form z ti = h i, ( λ , λ , . . . , λ N ) z N + h i, ( λ , λ , . . . , λ N ) z N +1 for any t ≥
1, where h i, ( λ , λ , . . . , λ N ) , h i, ( λ , λ , . . . , λ N ) ∈ ZP [ λ ]. The assumption z i ∝ x i for i = 1 , , . . . , N + 1 leads to ZP [ λ ] ⊂ ZP [ x ± ], hence we have z t , z t , . . . , z tN +1 ∈ ZP [ x ± ]for any t ≥
1. Since the assumption also leads to z ti ∝ x ti for i = 1 , , . . . , N + 1 and t ≥ x t , x t , . . . , x tN +1 assigned to the path oftype A (1) N exhibit the Laurent phenomenon , that is, they are in ZP [ x ± ] for any t ≥ V. CONCLUDING REMARKS
In the enomous network consisting of infinitely many seeds generated by the mutations µ , µ , . . . , µ N +1 from the initial seed Σ , we consider the sequence Σ , Σ , Σ , . . . of seedswhose exchange matrices B , B , B , . . . respectively correspond to the cluster mutation-periodic quivers Q , Q , Q , . . . with period 1. The sequence of seeds is assigned to the path ̟ in the ( N + 1)-regular tree T N +1 by the sub-cluster pattern ̟ → Σ . In the sequence27f seeds assigned to the path ̟ , every exchange matrix has periodicity with period N + 1and has the Cartan counterpart of type A (1) N . Due to the periodicity of exchange matrices,iteration of the consecutive seed mutations µ N +1 ◦ · · · ◦ µ ◦ µ induces dynamical systemsof the coefficients and of the cluster variables, respectively. In the dynamics of coefficients,we find that the N monomials ν ℓ , ν ℓ , . . . , ν ℓN generated by the coefficients y ℓ , y ℓ , . . . , y ℓN +1 have the periodicity with period N on ℓ , and they induce the conserved quantity C N ofthe dynamics. By using these monomials, we obtain the general terms of the coefficients.Similarly, in the dynamics of cluster variables, we also find the N Laurent polynomials λ t , λ t , . . . λ tN generated by the variables z t , z t , . . . , z tN +1 associated with the cluster variables x t , x t , . . . , x tN +1 have the same periodicity on t with N . The Laurent polynomials also inducethe conserved quantities of the dynamics. The dynamics of z ti governed by the quadraticbirational map ϕ is non-autonomously linearized by virtue of the Laurent polynomials. Viathe linearization of the map ϕ , we obtain the general solution to the dynamical systemgoverned by ϕ . Thus the seed Σ m ( m = 0 , , , . . . ) assigned to the path ̟ is completelysolved, that is, the elements of the seed are explicitly given by using the initial ones. Itimmediately follows the very well known fact that the cluster variables assigned to theGCM of type A (1) N via the path ̟ exhibit the Laurent phenomenon.In the preceding papers , we considered two kinds of rank 2 seed mutations respectivelyassigned to the GCMs of types A (1)1 and A (2)2 , and showed the integrability of the dynamicalsystems respectively associated with them. It followed that the general terms of the clustervariables were concretely constructed by using the conserved quantities. Moreover, we foundthat the two dynamical systems are mutually commutative on the conic, which is theircommon invariant curve, and are linearizable as well as the A (1) N case which is a generalizationof the A (1)1 case. Therefore, it is natural to expect that the generalized cases of A (2)2 , the A (2)2 N and A (2)2 N +1 cases, are integrable and linearizable. It is also expected that the A (1) N caseis commutative with the A (2)2 N and A (2)2 N +1 cases, respectively. We will report on this subjectin a forthcoming paper. ACKNOWLEDGMENTS
This work is partially supported by JSPS KAKENHI Grant No. 20K03692.28 ppendix A: Birational map ϕ on the projective space1. Homogeneous map Let us consider the birational map ϕ given by (15) on the projective space P N +1 ( C ).It is equivalent to assume the initial point z = ( z , z , . . . , z N +1 ) to be in P N +1 ( C ). Weintroduce the homogeneous coordinate ( z , z , . . . , z N +1 ) [ W : Z : Z : · · · : Z N +1 ] = [1 : z , z , . . . , z N +1 ] of the projective space P N +1 ( C ). Proposition 4
In the homogeneous coordinate ( z , z , . . . , z N +1 ) [ W : Z : Z : · · · : Z N +1 ] = [1 : z , z , . . . , z N +1 ] of the projective space P N +1 ( C ) , the birational map ϕ given by (15) reduces to the following homogeneous map of degree N + 3 : W t +1 = W t Z t N +1 Y j =1 Z tj ,Z t +1 i = Z t N +1 Y j = i +1 Z tj Z tN +1 i +1 Y j =2 Z tj + ( W t ) X j ,...,j i − i − Y ℓ =1 Z tj ℓ ( i = 1 , , . . . , N ) ,Z t +1 N +1 = ( W t ) Z t N Y j =1 Z tj + Z tN +1 N +1 Y j =2 Z tj + ( W t ) X j ,...,j N − N − Y ℓ =1 Z tj ℓ (cid:2) Z tN +1 Z t + ( W t ) (cid:3) , (A1) where if i = 2 , , . . . , N the sum X j ,...,j i − in Z t +1 i ranges over the set { , , . . . , i +1 }\{ k, k +1 } for k = 1 , , . . . , i , i.e., { j , . . . , j i − } = { , , . . . , i + 1 } , { , , , . . . , i + 1 } , { , , , , . . . , i + 1 } ,. . . , { , , . . . , i − , i + 1 } , { , , . . . , i − } , and if i = 1 the sum equals . (Proof) We use induction on i . First, by substitution of the homogeneous coordinateinto (15) for i = 1, we have Z t +11 W t +1 = Z tN +1 Z t + ( W t ) W t Z t . ≤ i ≤ N −
1, we assume that the following holds Z t +1 i W t +1 = Z tN +1 i +1 Y j =2 Z tj + ( W t ) X j ,...,j i − i − Y ℓ =1 Z tj ℓ W t i Y j =1 Z tj . (A2)Then we have Z t +1 i +1 W t +1 = Z t +1 i /W t +1 × Z ti +2 /W t + 1 Z ti +1 /W t = Z tN +1 i +1 Y j =2 Z tj + ( W t ) X j ,...,j i − i − Y ℓ =1 Z tj ℓ Z ti +2 + ( W t ) i Y j =1 Z tj " W t i Y j =1 Z tj Z ti +1 = Z tN +1 i +2 Y j =2 Z tj + ( W t ) X j ,...,j i i Y ℓ =1 Z tj ℓ W t i +1 Y j =1 Z tj . Moreover, for i = N + 1, we have Z t +1 N +1 W t +1 = Z t +1 N /W t +1 × Z t +11 /W t +1 + 1 Z tN +1 /W t = Z tN +1 N +1 Y j =2 Z tj + ( W t ) X j ,...,j N − N − Y ℓ =1 Z tj ℓ (cid:2) Z tN +1 Z t + ( W t ) (cid:3) + ( W t ) Z t N Y j =1 Z tj W t Z t N +1 Y j =1 Z tj . This implies that the homogeneous degree of the map must be N + 3. Thus, by respectivelymultiplying the numerator and the denominator of (A2) by Z t Q N +1 j = i +1 Z tj , we obtain theequations in (A1). (cid:3) . Invariant curve Consider the dynamical system z t +1 = ϕ ( z t ) governed by the map ϕ in the inhomoge-neous coordinate P N +1 ( C ) ∋ [ W : Z : Z : · · · : Z N +1 ] (cid:18) Z W , Z W , . . . , Z N +1 W (cid:19) = ( z , z , . . . , z N +1 ) ∈ C N +1 (A3)for W = 0. Then the invariant curve of the dynamical system is obtained via the Laurentpolynomials λ , λ , . . . , λ N in the following manner.For s = 0 , , . . . , N −
1, let the intersection of the hyperplanes( λ i ( z , z , . . . , z N +1 ) − λ si = 0) ( i = 1 , , . . . , N − λ N ( z , z , . . . , z N +1 ) − λ sN = 0)in C N +1 be γ s : γ s := N \ i =1 ( λ i ( z , z , . . . , z N +1 ) − λ si = 0) , where λ si = λ i ( z s , z s , . . . , z sN +1 )for ( z s , z s , . . . , z sN +1 ) ∈ C N +1 . Also let the compactification of γ s in P N +1 ( C ) be e γ s . Denotethe union of the compact curves e γ s by Γ:Γ := N − [ s =0 e γ s . Proposition 5
The compact curves e γ , e γ , . . . , e γ N − are mutually disjoint quadratic curveseach of which is on a 2-dimensional subspace of the projective space P N +1 ( C ) . Moreover,the point z t = ( z t , z t , . . . , z tN +1 ) = ϕ t ( z ) is on the curve e γ s for t ≡ s (mod N ) ( s =0 , , . . . , N − ). Thus the invariant curve of the dynamical system z t +1 = ϕ ( z t ) governedby the birational ϕ is the union Γ of the quadratic curves e γ s . λ i ( z , z , . . . , z N +1 ) − λ si = 0) for i =1 , , . . . , N − z − λ s z + z = 0 ,z − λ s z + z = 0 , · · · z N − − λ sN − z N + z N +1 = 0 . (A4)Since the coefficient matrix of the above system has rank N −
1, the space of solutions is ofdimension 2. We denote the plane in C N +1 given by (A4) by p s .The simultaneous system (A4) of equations reduces to − λ s · · ·
00 1 − λ s . . . ...... . . . . . . . . . 1... . . . 1 − λ sN − · · · · · · z z ...... z N − = z N − λ sN − + z N +1 − . Then, by solving this, we obtain z j = (cid:16) λ sN − ˜ b N − ,j − ˜ b N − ,j (cid:17) z N − ˜ b N − ,j z N +1 for j = 1 , , . . . , N −
1, where ˜ b ij is the ( i, j )-cofactor of the coefficients matrix. Thus a pointon the plane p s is given by using two parameters α s and β s : z ... z N − z N z N +1 = λ sN − ˜ b N − , − ˜ b N − , ... λ sN − ˜ b N − ,N − − ˜ b N − ,N − α s − ˜ b N − , ...˜ b N − ,N − β s . (A5)The hypersurface ( λ N ( z , z , . . . , z N +1 ) − λ sN = 0) is the quadratic surface given by z z N − λ sN z z N +1 + z z N +1 + 1 = 0 . (A6)32herefore, the intersection γ s of the plane p s and the hypersurface given by (A6) is aquadratic curve on the plane p s . Substituting z and z into the equation (A6), we obtain1 + (cid:16) λ sN − ˜ b N − , − ˜ b N − , (cid:17) ( z N ) + (cid:16) λ sN ˜ b N − , − ˜ b N − , (cid:17) ( z N +1 ) − h ˜ b N − , + ˜ b N − , − λ sN ˜ b N − , + λ sN − (cid:16) λ sN ˜ b N − , − ˜ b N − , (cid:17)i z N z N +1 = 0 . (A7)Denote the left hand side of (A7) by f ( z N , z N +1 ). Then a point on the quadratic curve γ s is given by (A5) with imposing f ( α s , β s ) = 0.Substitute (A3) into (A7), and remove the denominators by multiplying W . If we put W = 0 then we obtain (cid:16) λ sN − ˜ b N − , − ˜ b N − , (cid:17) ( Z N ) + (cid:16) λ sN ˜ b N − , − ˜ b N − , (cid:17) ( Z N +1 ) − h ˜ b N − , + ˜ b N − , − λ sN ˜ b N − , + λ sN − (cid:16) λ sN ˜ b N − , − ˜ b N − , (cid:17)i Z N Z N +1 = 0 . Thus we see that the compactification e γ s of the affine curve γ s has two points at infinitycounting with multiplicity.We assume that the curves γ s and γ s ( s , s ∈ { , , . . . , N − } , s = s ) have a point P = ( ζ , ζ , . . . , ζ N +1 ) in common. If N ≥ s = s and s = s reduce to ( λ s − λ s ) ζ = 0 , ( λ s − λ s ) ζ = 0 , · · · (cid:0) λ s N − − λ s N − (cid:1) ζ N = 0 . Hence ζ = ζ = · · · = ζ N = 0. It immediately follows ζ = ζ N +1 = 0 from (A4). Therefore,the two planes p s and p s respectively given by (A4) for s = s and s = s intersect only atthe origin (0 , , . . . , P of the curves γ s and γ s . It is clear that the points at infinity of these curves aregenerically different. Therefore, the compact curves e γ s and e γ s do not intersect each other.If N = 2 we have ζ = 0 and ζ + ζ = 0 from (A4). Thus the two planes p and p ,respectively given by λ = ( z + z ) /z = λ and λ = λ , meet at the line ( z + z = 0) ∩ ( z = 0). Substituting ζ = 0 into (A6) for s = 0 and s = 1, we obtain( λ − λ ) ζ ζ = 0 .
33t follows that ζ = ζ = 0. Therefore, there is no intersection point P of the curves γ and γ , and hence is of the compact curves e γ and e γ (see figure 2).Since λ ti has the periodicity (see theorem 4) λ t + Ni = λ ti , the point z t = ( z t , z t , . . . , z tN +1 ) = ϕ t ( z ) is on the curve e γ s for t ≡ s (mod N ). (cid:3) FIG. 2. The orbit of the dynamical system z t +1 = ϕ ( z t ) for N = 2 in R with imposing the initialvalues ( z , z , z ) = (4 , − , ∈ R . The invariant curve Γ consists of the 2 disjoint quadratic curves e γ and e γ whose affine parts are on the planes ( z + 4 z + z = 0) and (16 z + 15 z + 16 z = 0),respectively. These two planes meet at the line ( z + z = 0) ∩ ( z = 0). The ( N + 1)-dimensional dynamical system governed by the birational map ϕ has N con-served quantities q , q , . . . , q N which are functionally independent (see corollary 2). There-fore, the dynamical system is integrable in the sense of Liouville. The invariant curve is theunion of the N disjoint quadratic curves each of which is on the 2-dimensional subspace of P N +1 ( C ). REFERENCES S. Fomin and A. Zelevinsky, “Cluster algebras I: Foundations,” J. Amer. Math. Soc. ,497 (2002). 34 A. Fordy and J. Marsh, “Cluster mutation-periodic quivers and associated Laurent se-quences,” J. Algebr. Comb. , 19 (2011). S. Fomin and A. Zelevinsky, “Y-systems and generalized associahedra,” Ann. Math. ,977 (2003). R. Inoue, O. Iyama, A. Kuniba, T. Nakanishi, and J. Suzuki, “Periodicities of T-systemsand Y-systems,” Nagoya Math. J. , 59 (2010). R. Inoue, O. Iyama, B. Keller, A. Kuniba, and T. Nakanishi, “Periodicities of T-systemsand Y-systems, dilogarithm identities, and cluster algebras I: Type B r ,” Publ. RIMS ,1 (2013). R. Inoue, O. Iyama, B. Keller, A. Kuniba, and T. Nakanishi, “Periodicities of T-systemsand Y-systems, dilogarithm identities, and cluster algebras II: Types C r , F , and G ,”Publ. RIMS , 43 (2013). N. Okubo, “Discrete integrable systems and cluster algebras,” RIMS Kˆokyˆuroku Bessatsu , 25 (2013). T. Mase, “The Laurent phenomenon and discrete integrable systems,” RIMS KˆokyˆurokuBessatsu , 43 (2013). N. Okubo, “Bilinear equations and q -discrete Painlev´e equations satisfied by variables andcoefficients in cluster algebras,” J. Phys. A: Math. Theor. , 355201 (2015). A. Marshakov, “Lie groups, cluster variables and integrable systems,” J. Geom. Phys. ,16 (2013). T. Mase, “Investigation into the role of the Laurent property in integrability,” J. Math.Phys. , 022703 (2016). A. Nobe, “Mutations of the cluster algebra of type A (1)1 and the periodic discrete Todalattice,” J. Phys. A: Math. Theor. , 285201 (2016). M. Bershtein, P. Gavrylenko, and A. Marshakov, “Cluster integrable systems, q -Painlev´eequations and their quantization,” J. High Energy Phys. , 77 (2018). S. Fomin and A. Zelevinsky, “Cluster algebras II: Finite type classification,” Invent. Math. , 63 (2003). S. Fomin and A. Zelevinsky, “Cluster algebras IV: Coefficients,” Compos. Math. , 112(2007). A. Nobe and J. Matsukidaira, “A family of integrable and non-integrable difference equa-tions arising from cluster algebras,” RIMS Kˆokyˆuroku Bessatsu
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