Lattice equations arising from discrete Painlevé systems. II. A (1) 4 case
aa r X i v : . [ m a t h - ph ] S e p LATTICE EQUATIONS ARISING FROM DISCRETE PAINLEV ´E SYSTEMS. II. A (1)4 CASE
NALINI JOSHI, NOBUTAKA NAKAZONO, AND YANG SHIA bstract . In this paper, we construct two lattices from the τ functions of A (1)4 -surface q -Painlev´e equations, on which quad-equations of ABS type appear. Moreover, using thereduced hypercube structure, we obtain the Lax pairs of the A (1)4 -surface q -Painlev´e equa-tions. C ontents
1. Introduction 12. Construction of the ω -lattice of type A (1)4
53. Construction of the lattice ω A + A e W (( A ⋊ A ) (1) ) 28References 301. I ntroduction Two longstanding classifications of integrable discrete systems in di ff erent dimensions,one by Adler-Bobenko-Suris (ABS) [1, 2, 6–8] and the other by Sakai [55], have beenwidely studied, but the mathematical connection between them remains incomplete. Howto reduce the ABS partial di ff erence equations to Sakai’s discrete Painlev´e equations is anatural question that has inspired many authors [12, 14, 16, 17, 39, 48, 49]. However, theseearlier approaches focused on taking periodic constraints in two dimensions that lead toequations with a restricted set of parameters, manually extending these by adding gaugetransformations in order to introduce more parameters. Another rich vein of inquiry re-duces the Lax pairs of ABS equations to provide these elusive linear problems for discretePainlev´e equations. We provide a di ff erent approach grounded in higher-dimensional ge-ometry associated naturally with full-parameter discrete Painlev´e equations [23–25]. Inthis paper, we review our approach and illustrate it for A (1)4 -surface type q -discrete Painlev´eequations, providing new Lax pairs for these equations.The geometric setting of reflection groups is essential to our approach. Within thisframework, we construct higher dimensional lattices, called ω -lattices, from discrete Painlev´eequations. These lattices also arise from integer lattices associated with ABS classificationand thereby provide a bridge between the two classifications. In an earlier series of pa-pers [23–25], we constructed ω -lattices for A (1)5 - and A (1)6 - surface q -Painlev´e equations.The A (1)4 -case is a simpler (less degenerate) surface than these earlier cases, but it is wellknown that when the surface is simpler, the corresponding symmetry groups and discretePainlev´e equations become more complex [55]. Mathematics Subject Classification.
Key words and phrases.
Discrete Painlev´e equation; ABS equation; Lax pair; τ function; a ffi ne Weyl group . Despite the increasing complexity, our approach connects discrete Painlev´e equations topartial di ff erence equations through reductions of hypercubes and polytopes. We constructtwo lattices in two ways, one through reduction of polytopes and the other by reduction ofhypercubes. Both lattices arise from the τ functions of A (1)4 -surface type q -discrete Painlev´eequation. They share fundamental variables (called ω -variables) and both give rise to ABSequations and to q -discrete Painlev´e equations. The polytope case will be investigatedfurther in future work. The hypercube lattice is referred to below as ω A + A . (More detailsare given in § § ω A + A then provides uswith reductions of the Lax pairs of ABS squations, which turn out to be new Lax pairs for q -Painlev´e equations (1.1). Our results show that these equations share one monodromyproblem. Moreover, the coe ffi cient matrices in each case are factorized into product ofmatrices that are linear in the monodromy variable x . We remark that in each case, we alsoobtain Lax pairs for the scalar form of the equations.In this paper, we construct two important lattices, where quad-equations are observed,from the τ functions of A (1)4 -surface type q -discrete Painlev´e equation. One is the ω -latticeof type A (1)4 investigated in §
2. An ω -lattice provides informations about how a system ofpartial di ff erence equations can be reduced to discrete Painlev´e equations. It provides notonly the type of equation, but also the combinatorial structure of the lattice before reduction(see [24, 25] for details). The other lattice is the ω A + A investigated in §
3. The lattice ω A + A can be obtained from an integer lattice, given by the space-filling of the hypercubeon whose faces quad-equations of ABS type are assigned, by the geometric reduction. Byusing this reduced hypercube structure, we obtain the Lax pairs of the q -Painlev´e equations(1.1). These Lax pairs di ff er from the ones in the literature [35]. Moreover, our result showthat four equations of A (1)4 -surface type share the same q -discrete monodromy problem(1.4) with di ff ering deformation equations given by (1.7). Other distinctive properties ofour Lax pairs are that their coe ffi cient matrices occur as products of matrices of degree onein the spectral parameter x and elements of the coe ffi cient matrices given by the rationalfunctions of Painlev´e variables.1.1. A (1)4 -surface q -Painlev´e equations. In this paper, we collectively call the following q -di ff erence equations as A (1)4 -surface q -Painlev´e equations since they are of A (1)4 -surfacetype in Sakai’s classification [55]: q -P V : FF = c t ( c + tG )( c + tG ) c + G , GG = c t ( c − + tF )( qc c c − + tF ) c − + F , (1.1a) q -P V ∗ : ( FG − FG − = t qc c ( c − c + G )( qc c + G ) c t + G , ( FG − FG − = c t qc ( c c − + F )( q − c − c − + F ) q − c − t + F , (1.1b) q -P III ( D (1)7 ): e GG e = t ( c + tG )( p − + tG )1 + G , (1.1c) q -P IV : ( e GG − GG e − = t p c c ( p − c − c − + G )( p c c + G ) p − c − t + G , (1.1d)where t , c , c , c , q , p ∈ C ∗ and F = F ( t ) , G = G ( t ) , F = F ( qt ) , G = G ( q − t ) , e G = G ( pt ) , G e = G ( p − t ) . (1.2) ATTICE EQUATIONS ARISING FROM DISCRETE PAINLEV´E SYSTEMS. II 3
We note that q -P V (1.1a), q -P V ∗ (1.1b), q -P III ( D (1)7 ) (1.1c) and q -P IV (1.1d) are known asa q -discrete analogue of the Painlev´e V equation [55], that of the Painlev´e V equation[59], that of the Painlev´e III equation of D (1)7 -surface type [38] and that of the Painlev´e IVequation [54], respectively. Remark 1.1.
It is known that q- P V (1.1a) and q- P V ∗ (1.1b) can be reduced to q- P III ( D (1)7 )(1.1c) [38] and q- P IV (1.1d) [37] by the projective reductions:c = p − , c = , q = p , F = G e , (1.3a) c c = p − , q = p , F = G e , (1.3b) respectively. In this sense, q- P III ( D (1)7 ) and q- P IV are sometimes called as the scalar formsof q- P V and q- P V ∗ , respectively. Main results.
In this section, we outline two main results of this paper.Firstly, in § Theorem 1.2.
The lattice ω A + A has a reduced hypercube structure. The lattice ω A + A is a 3-dimensional integer lattice on which ABS equations (2.39)–(2.40) and q -Painlev´e equations (1.1) appear. This lattice is constructed from the τ func-tions of A (1)4 -surface q -Painlev´e equations (see § ω A + A can be also obtained from the 4-dimensional hypercube lattice on whose faces ABSequations are assigned. This reduced hypercube structure turn out to be essential in theconstruction of Lax pairs for discrete Painlev´e equations [23].Our second main result, Theorem 1.3, concerns the Lax pairs of the q -Painlev´e equa-tions (1.1). Equations (1.1) share one spectral linear problem, which takes the factorizedform Φ ( px ) = ∗ x ∗∗ ! . ∗ x ∗∗ ∗ x ! . ∗ x ∗∗ ∗ x ! . ∗ x ∗∗ ∗ x ! . Φ ( x ) = A . Φ ( x ) . (1.4)Here, the 2 × A = A ( x ) is given by (4.28) whose elements are expressed by thenon-zero complex parameters b i , i = , . . . ,
3, and p and unknown functions f (1) i , i = , , f (1) i satisfy the following relation: b b b + pb b f (1)1 − p b b / f (3)1 + p b b b / f (1)1 f (2)1 f (3)1 = . (1.5)We introduce the deformation operators T , T , R and R whose actions on the parame-ters b i , i = , . . . ,
3, and p are given by T : ( b , b , b , b , p ) ( pb , pb , b , b , p ) , (1.6a) T : ( b , b , b , b , p ) ( b , b , p b , b , p ) , (1.6b) R : ( b , b , b , b , p ) ( b , pb , b , b − , p ) , (1.6c) R : ( b , b , b , b , p ) ( b , b , pb , b − , p ) , (1.6d)while those on the spectral parameter x and the wave function Φ = Φ ( x ) are given by T ( x ) = T ( x ) = R ( x ) = R ( x ) = x , (1.7a) T ( Φ ) = ∗ x ∗∗ ∗ x ! . ∗ x ∗∗ ∗ x ! . Φ ( x ) = B T . Φ , (1.7b) T ( Φ ) = ∗ x ∗∗ ! . ∗ x ∗∗ ! . Φ ( x ) = B T . Φ , (1.7c) R ( Φ ) = ∗ x ∗∗ ∗ x ! . Φ ( x ) = B R . Φ , (1.7d) R ( Φ ) = ∗ x ∗∗ ! . Φ ( x ) = B R . Φ , (1.7e) NALINI JOSHI, NOBUTAKA NAKAZONO, AND YANG SHI where the 2 × B T = B T ( x ), B T = B T ( x ), B R = B R ( x ) and B R = B R ( x )are given by (4.29). Equations (1.6) and (1.7) provide us with the deformation of thespectral problem. Theorem 1.3.
The compatibility conditions of the linear equation (1.4) with the operatorsT , T , R and R : T ( A ) . B T = B T ( px ) . A , T ( A ) . B T = B T ( px ) . A , (1.8a) R ( A ) . B R = B R ( px ) . A , R ( A ) . B R = B R ( px ) . A , (1.8b) are equivalent to T ( f (3)1 ) f (3)1 = b ( b + pb f (1)1 )( − b b b / + pb f (1)1 ) p b b ( pb b + b f (1)1 ) , T − ( f (1)1 ) f (1)1 = pb ( b − + pb f (3)1 )( − p − b b b − / + pb f (3)1 ) b b − ( b b − + pb f (3)1 ) , (1.9a) T ( f (1)1 ) f (2)1 − b b f (1)1 f (2)1 − b b = b b ( pb + b b f (2)1 )(1 + pb b f (2)1 ) pb b / ( − b b + b b / f (2)1 ) , f (1)1 f (2)1 − b b f (1)1 T − ( f (2)1 ) − b b = b b ( pb b + b f (1)1 )( b + pb f (1)1 ) p b b ( − b b b / + pb f (1)1 ) , (1.9b) R ( f (3)1 ) = f (1)1 , R ( f (1)1 ) f (3)1 = b ( b + pb f (1)1 )( − b b b / + pb f (1)1 ) p b b ( pb b + b f (1)1 ) , (1.9c) R ( f (1)1 ) = f (2)1 , R ( f (2)1 ) f (2)1 − b b f (1)1 f (2)1 − b b = b b ( pb + b b f (2)1 )(1 + pb b f (2)1 ) pb b / ( − b b + b b / f (2)1 ) , (1.9d) respectively. This theorem is proven in § q -Painlev´eequations (1.1). Remark 1.4.
Equations (1.9a) and (1.9b) are equivalent to q- P V (1.1a) and q- P V ∗ (1.1b) by the following correspondences: ¯ = T , t = b , c = − b b b / pb , c = b p , c = pb b b , q = p , F = f (3)1 , G = f (1)1 , (1.10a)¯ = T , t = p / b , c = − b b , c = b p / b / , c = p / b b / , q = p , F = − b b f (1)1 , G = − b b f (2)1 , (1.10b) respectively. Moreover, letting b = p / b , b = , (1.11) and setting ˜ = R , t = b , c = − b p / , G = f (1)1 , (1.12) ATTICE EQUATIONS ARISING FROM DISCRETE PAINLEV´E SYSTEMS. II 5 we obtain q- P III ( D (1)7 ) (1.1c) from the action (1.9c) . Similarly, by lettingb = , (1.13) and setting ˜ = R , t = p / b , c = − b b , c = b p / , G = − b b f (2)1 , (1.14) the action (1.9d) gives q- P IV (1.1d) . Background.
In the 1900s, in order to find new class of special functions, Painlev´eand Gambier classified all di ff erential equations in the form of y ′′ = F ( y ′ , y , t ), where y = y ( t ), ′ = d / dt and F is a rational function, by imposing the condition that the solutionsshould admit only poles as movable singular points. As a result, they showed that theresulting equations can be reduced to one of the six equations, which are now called thePainlev´e I through VI equations, unless it can be integrated algebraically, or transformedinto a simpler equations such as a linear equation or the di ff erential equation of ellipticfunctions. Moreover, it is known that Painlev´e equations can be classified into eight typesby the geometrical classification of space of initial conditions [42, 43, 55]. From the viewpoint of this classification, P III can be divided into P
III ( D (1)6 ), P III ( D (1)7 ) and P III ( D (1)8 ) by thevalues of parameters.Discrete Painlev´e equations are nonlinear ordinary di ff erence equations of second order,which include discrete analogues of the Painlev´e equations. The geometric classificationof discrete Painlev´e equations, based on types of rational surfaces connected to a ffi ne Weylgroups, is well known [55]. Together with the Painlev´e equations, they are now regardedas one of the most important classes of equations in the theory of integrable systems (see,e.g., [13, 30]).It is well known that the τ functions, which gives rise to various bilinear equations,play a crucial role in the theory of integrable systems [34]. The same is true in the theoryof continuous and discrete Painlev´e equations [19–21, 40, 44–47]. A representation of thea ffi ne Weyl groups can be lifted to the level of the τ functions [25–27, 32, 33, 60, 62].Discrete Painlev´e equations are called integrable because they arise as compatibilityconditions of associated linear problems called Lax pairs. The search for and construc-tion of Lax pairs of discrete Painlev´e equations has been a very active research area.Noteworthy approaches include extensions of Birkho ff ’s study of linear q -di ff erence equa-tions [22,56,57], periodic-type reductions from ABS equations or the discrete KP / UC hier-archy [15,16,23,31,48,50,51,53,61], extensions of Schlesinger transformations [4,10,11],search for linearizable curves in the space of initial values [65, 66], Pad´e approximation orinterpolation [18, 36, 41] and the theory of orthogonal polynomials [3, 5, 9, 52, 63, 64].1.4.
Plan of the paper.
This paper is organized as follows: in §
2, we introduce the τ functions of A (1)4 -surface q -Painlev´e equations, which have the extended a ffi ne Weyl groupsymmetry e W ( A (1)4 ), and show that the q -Painlev´e equations (1.1) can be derived from abirational representation of e W ( A (1)4 ). Moreover, we construct the ω -lattice of type A (1)4 andthen derive various quad-equations of ABS type, as relations on the ω -lattice. In §
3, weconstruct the lattice ω A + A and show its properties. In §
4, we give the proofs of Theorems1.2 and 1.3 by using the geometric reduction from the integer lattice Z with the integrableP ∆ Es to the lattice ω A + A . Some concluding remarks are given in § onstruction of the ω - lattice of type A (1)4 In this section, we define τ functions by using the transformation group e W ( A (1)4 ). Then,we derive the q -Painlev´e equations (1.1) and construct the ω -lattice of type A (1)4 from the τ functions. NALINI JOSHI, NOBUTAKA NAKAZONO, AND YANG SHI
For convenience, throughout this paper we use the following notation for compositionsof arbitrary mappings w i , i = , . . . , n : w · · · w n : = w ◦ · · · ◦ w n . (2.1)2.1. τ functions. In this section, we define the τ functions by using the transformationgroup e W ( A (1)4 ) = h s , s , s , s , s , σ, ι i , which forms the extended a ffi ne Weyl group oftype A (1)4 (see Appendix A).Below, we describe the actions of e W ( A (1)4 ) on the five parameters a , . . . , a ∈ C ∗ and onthe ten variables τ ( j ) i , i = , j = , . . . ,
5, which satisfy the following three relations: τ (1)2 = a a ( a τ (3)1 τ (5)1 + a τ (4)1 τ (3)2 ) a a τ (5)2 , (2.2a) τ (2)2 = a a ( a τ (1)1 τ (4)1 + a τ (5)1 τ (4)2 ) a a τ (1)2 , (2.2b) τ (4)2 = a a ( a τ (1)1 τ (3)1 + a τ (2)1 τ (1)2 ) a a τ (3)2 . (2.2c) Remark 2.1.
Below we use the index j to denote an element of Z / Z with a slightlydi ff erent enumeration for transformations s , . . . , s , parameters a , . . . , a and variables τ (1) i , . . . , τ (5) i ( i = , . To avoid confusion, we point out, for example, that j = for s j anda j would imply j = for τ ( j ) i . Lemma 2.2.
The action of e W ( A (1)4 ) on the parameters are given bys i ( a j ) = a j a i − a ij , σ ( a i ) = a i + , ι ( a i ) = a − i − , (2.3) where i , j ∈ Z / Z and ( a i j ) i , j = = − − − − − − − − − − (2.4) is the Cartan matrix of type A (1)4 , while their actions on the variables are given bys j ( τ ( j )1 ) = τ ( j + , s j ( τ ( j + ) = a j + a j + ( a j a j + τ ( j + τ ( j + + a j + τ ( j + τ ( j + ) a j + τ ( j )1 , (2.5a) s j ( τ ( j + ) = τ ( j )1 , s j ( τ ( j )2 ) = a j + ( a j + τ ( j + τ ( j + + a j a j + τ ( j + τ ( j + ) a j a j + a j + τ ( j )1 , (2.5b) σ ( τ ( j )1 ) = τ ( j + , σ ( τ ( j )2 ) = τ ( j + , ι ( τ ( j )1 ) = τ (5 − j )1 , ι ( τ ( j )2 ) = τ (3 − j )2 , (2.5c) where j ∈ Z / Z . In general, for a function F = F ( a i , τ ( k ) j ) , we let an element w ∈ e W ( A (1)4 ) act as w . F = F ( w . a i , w .τ ( k ) j ) , that is, w acts on the arguments from the left. The proof of Lemma 2.2 is given in Appendix A.
Remark 2.3.
The action of e W ( A (1)4 ) in Lemma 2.2 was first obtained by Tsuda in [60]without the details of the proof. The notations in this paper are related to those in [60] by ATTICE EQUATIONS ARISING FROM DISCRETE PAINLEV´E SYSTEMS. II 7 the following correspondence: ( s , s , s , s , s , σ, ι ) → ( s , s , s , s , s , π , ι ) , (2.6a)( a , a , a , a , a , q ) → ( a , a , a , a , a , q ) , (2.6b)( τ (1)1 , τ (2)1 , τ (3)1 , τ (4)1 , τ (5)1 ) → ( τ , τ , τ , τ , τ ) , (2.6c)( τ (1)2 , τ (2)2 , τ (3)2 , τ (4)2 , τ (5)2 ) → ( π ( τ ) , π ( τ ) , τ , π ( τ ) , τ ) . (2.6d) We also note that in [60] each element w ∈ e W ( A (1)4 ) acts on the arguments from the right,whereas in the present paper it acts from the left. To iterate each variable τ ( j ) i , we need the following transformations: T = σ s s s s , T = σ s s s s , T = σ s s s s , (2.7a) T = σ s s s s , T = σ s s s s , (2.7b)which are translations on the root lattice ˆ Q ( A (1)4 ) (A.8) (see Appendix A). Note that T i , i = , . . . ,
4, commute with each other and T T T T T = . (2.8)Their actions on the parameters are given by T i ( a i ) = qa i , T i ( a i + ) = q − a i + , i ∈ Z / Z , (2.9)where q = a a a a a is invariant under the actions of T i . We define τ functions by τ l , l , l , l , l = T l T l T l T l T l ( τ (3)2 ) , (2.10)where l i ∈ Z . We note that τ (1)1 = τ , , , , , τ (2)1 = τ , , , , , τ (3)1 = τ , , , , , τ (4)1 = τ , , , , , (2.11a) τ (5)1 = τ , , , , , τ (1)2 = τ , , , , , τ (2)2 = τ , , , , , τ (3)2 = τ , , , , , (2.11b) τ (4)2 = τ , , , , , τ (5)2 = τ , , , , . (2.11c)2.2. Discrete Painlev´e equations.
In this section, we define the f -variables by rationalfunctions of the τ -variables. Then, we demonstrate that elements of infinite order of e W ( A (1)4 ) give various q -Painlev´e equations.Let us define the ten f -variables by f ( j )1 = τ ( j + τ ( j )2 τ ( j )1 τ ( j + , f ( j )2 = s j + ( f ( j )1 ) = a j a j + ( a j + a j + + a j f ( j + ) a j + f ( j + , (2.12)where j ∈ Z / Z . From the definition above and the relations (2.2), the following relationshold: a j + a j + f ( j )1 f ( j + = a j a j + ( a j + + a j f ( j + ) , (2.13)where j ∈ Z / Z . The relations above look like five equations, but the relations representonly three. Therefore, there are only two essential f -variables. The action of e W ( A (1)4 ) onthese variables f ( j ) i is given by the lemma below, which follows from the actions (2.5). Lemma 2.4.
The action of e W ( A (1)4 ) on variables f ( j ) i is given bys j ( f ( j + ) = f ( j + , s j ( f ( j )1 ) = a j + ( a j + + a j a j + f ( j + ) a j a j + a j + f ( j + , s j ( f ( j + ) = f ( j + , (2.14a) s j ( f ( j + ) = a j a j + a j + ( a j + + a j a j + f ( j + + a j a j + a j + f ( j + ) a j + f ( j + f ( j + , (2.14b) s j ( f ( j + ) = a j a j + a j + f ( j + f ( j )1 f ( j + a j + ( a j + + a j a j + f ( j + ) , (2.14c) NALINI JOSHI, NOBUTAKA NAKAZONO, AND YANG SHI s j ( f ( j )2 ) = a j a j + a j + + a j + a j + f ( j + + a j a j + a j + f ( j + a j a j + a j + f ( j + f ( j + , σ ( f ( j )1 ) = f ( j + , (2.14d) σ ( f ( j )2 ) = f ( j + , ι ( f ( j )1 ) = f (3 − j )1 , ι ( f ( j )2 ) = a − j ( a − j + a − j a − j f (5 − j )1 ) a − j a − j a − j f (2 − j )1 , (2.14e) where j ∈ Z / Z . It is well known that the translation part of e W ( A (1)4 ) give discrete Painlev´e equations [55].For examples, from the translations T i , i = , . . . ,
4, we obtain q -P V (1.1a) and from thetranslations T i T j , 0 ≤ i < j ≤
4, we obtain q -P V ∗ (1.1b). Indeed, the action of T : T : ( a , a , a , a , a ) ( qa , q − a , a , a , a ) , (2.15a) T ( f (3)1 ) f (3)1 = a a a a ( a + a a f (1)1 )( a + a f (1)1 ) a a + a f (1)1 , (2.15b) T − ( f (1)1 ) f (1)1 = a a a ( a a + a f (3)1 )( a + a f (3)1 ) a + a a f (3)1 , (2.15c)leads to q -P V (1.1a) by the correspondences (1.10a) and b = a / q / , b = a / a / , b = − q / a / a / , (2.16a) b = a a a q / , p = q / , (2.16b)or, equivalently, a = p b , a = − b b / pb b , a = − pb b b b / , a = − b b b b / , (2.16c) a = − b b / b b , q = p . (2.16d)Moreover, the action of T = T T : T : ( a , a , a , a , a ) ( a , qa , q − a , qa , q − a ) , (2.17a) T ( f (1)1 ) f (2)1 − a a a a f (1)1 f (2)1 − a a a a = a a a a ( a + a a f (2)1 )( a + a f (2)1 ) a a + a f (2)1 , (2.17b) f (1)1 f (2)1 − a a a a f (1)1 T − ( f (2)1 ) − a a a a = a a a a a ( a + a f (1)1 )( a a + a f (1)1 ) a + a a f (1)1 , (2.17c)gives q -P V ∗ (1.1b) by the correspondences (1.10b) and (2.16).It is also known that discrete Painlev´e equations can be obtained from elements ofinfinite order of e W ( A (1)4 ) which are not necessarily translations of e W ( A (1)4 ) [29, 58]. Wehere show that how q -P III ( D (1)7 ) (1.1c) and q -P IV (1.1d) can be derived from the actions of e W ( A (1)4 ). Let R = σ s s , R = σ s s s , (2.18)where R = T and R = T . Actions of these transformations in the parameter spaceare not translational motion: R : ( a , a , a , a , a ) ( a a a , q − a a a , a , a a , a ) , (2.19a) R : ( a , a , a , a , a ) ( a , a a a , a − , qa − , q − a a a ) , (2.19b) ATTICE EQUATIONS ARISING FROM DISCRETE PAINLEV´E SYSTEMS. II 9 but under the special values of the parameters these actions become translational motion.Indeed, by imposing a = a , a = a a , (2.20)which implies q / = a a = a a a = a a = a a a , (2.21)the action of R becomes R : ( a , a , a , a , a ) ( q / a , q − / a , a , a , a ) . (2.22)Similarly, under the condition of the parameters q / = a a = a a a , (2.23)the action of R becomes R : ( a , a , a , a , a ) ( a , q / a , q − / a , q / a , q − / a ) . (2.24)Therefore, the action of R : R ( f (3)1 ) = f (1)1 , R ( f (1)1 ) f (3)1 = a a a a ( a + a a f (1)1 )( a + a f (1)1 ) a a + a f (1)1 , (2.25)with the condition (2.20), gives q -P III ( D (1)7 ) (1.1c) by the correspondences (1.12) and (2.16).Moreover, the action of R : R ( f (1)1 ) = f (2)1 , (2.26a) R ( f (2)1 ) f (2)1 − a a a a f (2)1 f (1)1 − a a a a = a a a a ( a a f (2)1 + a )( a f (2)1 + a ) a f (2)1 + a a , (2.26b)with the condition (2.23), gives q -P IV (1.1d) by the correspondences (1.14) and (2.16).2.3. ω -lattice. In this section, we define the ω -variables by the ratios of the τ -variablesand then construct the ω -lattice of type A (1)4 .Let us define the fifteen ω -variables by ω ( j )1 = τ ( j )1 τ ( j + , ω ( j )2 = τ ( j )2 τ ( j + , ω ( j )3 = τ ( j − τ ( j − , j ∈ Z / Z , (2.27)which satisfy f ( j )1 = ω ( j )2 ω ( j )1 , f ( j )2 = a j a j + a j + ω ( j + ( a j + a j + ω ( j + + a j ω ( j + ) ω ( j + ω ( j + , j ∈ Z / Z . (2.28)From the definition above and the relations (2.2), they satisfy the following nine relations: ω (5)2 = ω (1)1 ω (5)1 ω (1)3 , ω (5)3 = ω (4)1 ω (5)1 ω (4)2 , ω (2)2 = ω (2)1 ω (3)1 ω (3)3 , ω (3)2 = ω (3)1 ω (4)1 ω (4)3 , (2.29a) ω (5)1 = ω (1)1 ω (2)1 ω (3)1 ω (4)1 , ω (2)1 = ω (1)2 ω (2)3 ω (1)1 , (2.29b) ω (3)1 = a ω (3)3 ( a a ω (1)1 ω (4)2 − a a ω (4)1 ω (1)3 ) a a ω (4)1 ω (1)3 , (2.29c) ω (4)1 = a ω (4)3 ( a a ω (1)2 − a a ω (1)3 ) a a ω (1)3 , ω (4)2 = a a ω (4)3 ( a ω (1)1 + a ω (1)2 ) a a ω (1)1 . (2.29d)By inspection, we see that there are six essential ω -variables. The action of e W ( A (1)4 ) onthe ω -variables is given by the lemma below, which follows from the action (2.5) and thedefinition (2.27). Lemma 2.5.
The action of e W ( A (1)4 ) on the fifteen ω -variables is given bys j ( ω ( j + ) = ω ( j )3 , s j ( ω ( j )1 ) = ω ( j + , s j ( ω ( j )3 ) = ω ( j + , s j ( ω ( j + ) = ω ( j )1 , (2.30a) s j ( ω ( j + ) = a j + a j + ω ( j + ω ( j + ( a j a j + ω ( j + + a ω ( j + ) a j + ω ( j + ω ( j + , (2.30b) s j ( ω ( j )2 ) = a j + ω ( j + ( a a ω ( j + + a ω ( j + ) a j a j + a j + ω ( j + , (2.30c) s j ( ω ( j + ) = a j a j + a j + ω ( j + ω ( j + ω ( j + a j + ω ( j + ( a j + ω ( j + + a j a j + ω ( j + ) , (2.30d) s j ( ω ( j + ) = a j + ω ( j + ω ( j )1 a j + a j + ( a j a j + ω ( j + + a j + ω ( j + ) , (2.30e) σ ( ω ( j )1 ) = ω ( j + , σ ( ω ( j )2 ) = ω ( j + , σ ( ω ( j )3 ) = ω ( j + , (2.30f) ι ( ω ( j )1 ) = ω (4 − j )1 , ι ( ω ( j )2 ) = ω (4 − j )3 , ι ( ω ( j )3 ) = ω (4 − j )2 , (2.30g) where j ∈ Z / Z . We define ω -functions by ω ( j ) l , l , l , l , l = T l T l T l T l T l ( ω ( j )3 ) , (2.31)where j = , . . . , l , . . . , l ∈ Z . We note that ω (1)1 = ω (1)1 , , , , , ω (1)2 = ω (1)1 , , , , , ω (1)3 = ω (1)0 , , , , , (2.32a) ω (2)1 = ω (2)0 , , , , , ω (2)2 = ω (2)0 , , , , , ω (2)3 = ω (2)0 , , , , , (2.32b) ω (3)1 = ω (3)0 , , , , , ω (3)2 = ω (3)0 , , , , , ω (3)3 = ω (3)0 , , , , , (2.32c) ω (4)1 = ω (4)0 , , , , , ω (4)2 = ω (4)1 , , , , , ω (4)3 = ω (4)0 , , , , , (2.32d) ω (5)1 = ω (5)0 , , , , , ω (5)2 = ω (5)0 , , , , , ω (5)3 = ω (5)0 , , , , . (2.32e)Now we are in a position to construct the ω -lattice of type A (1)4 . Let us consider thefollowing lattice (see Figure 1): X i = l i v i ⊂ Z , (2.33)whose vertices v i , i = , . . . ,
4, are defined by v = ( − , − , − , − , , v = (4 , − , − , − , − , v = ( − , , − , − , − , (2.34a) v = ( − , − , , − , − , v = ( − , − , − , , − , (2.34b)and satisfy v + v + v + v + v = . For simplicity, we here use the following notation: v k ... k n = n X i = v k i , k i ∈ { , . . . , } . (2.35) ATTICE EQUATIONS ARISING FROM DISCRETE PAINLEV´E SYSTEMS. II 11
Let us assign the τ functions τ l , l , l , l , l and the ω -functions ω ( j ) l , l , l , l , l to the vertices andthe edges of the lattice (2.33) by the following correspondence: τ l , l , l , l , l ↔ l + v , (2.36a) ω (1) l , l , l , l , l ↔ edge( l + v ; 1) , (2.36b) ω (2) l , l , l , l , l ↔ edge( l ; 2) , (2.36c) ω (3) l , l , l , l , l ↔ edge( l + v ; 3) , (2.36d) ω (4) l , l , l , l , l ↔ edge( l + v ; 4) , (2.36e) ω (5) l , l , l , l , l ↔ edge( l + v ; 0) , (2.36f)where l = P i = l i v i . Here, edge( A ; i ) is a edge connecting a vertex A to a vertex ( A + v i ).We refer to the lattice (2.33) with the ω -functions ω ( j ) l , l , l , l , l as ω -lattice of type A (1)4 . Wenote that the configurations of the τ -variables on the ω -lattice are given by( τ (1)1 , τ (2)1 , τ (3)1 , τ (4)1 , τ (5)1 ) ↔ ( , v , v , v , v ) , (2.37a)( τ (1)2 , τ (2)2 , τ (3)2 , τ (4)2 , τ (5)2 ) ↔ ( v , v , v , v , v ) , (2.37b)while those of the ω -variables are given by( ω (1)1 , ω (1)2 , ω (1)3 ) ↔ (cid:16) edge( ; 1) , edge( v ; 1) , edge( v ; 1) (cid:17) , (2.38a)( ω (2)1 , ω (2)2 , ω (2)3 ) ↔ (cid:16) edge( v ; 2) , edge( v ; 2) , edge( ; 2) (cid:17) , (2.38b)( ω (3)1 , ω (3)2 , ω (3)3 ) ↔ (cid:16) edge( v ; 3) , edge( v ; 3) , edge( v ; 3) (cid:17) , (2.38c)( ω (4)1 , ω (4)2 , ω (4)3 ) ↔ (cid:16) edge( v ; 4) , edge( v ; 4) , edge( v ; 4) (cid:17) , (2.38d)( ω (5)1 , ω (5)2 , ω (5)3 ) ↔ (cid:16) edge( v ; 0) , edge( v ; 0) , edge( v ; 0) (cid:17) . (2.38e)On the ω -lattice various quad-equations of ABS-type can be derived, e.g. T T ( ω (1)3 ) ω (1)3 = a a a (1 − a ) a a T ( ω (1)3 ) + (1 − a ) T ( ω (1)3 ) a a T ( ω (1)3 ) − T ( ω (1)3 ) , (2.39a) T T ( ω (1)3 ) ω (1)3 = a a a a a (1 − a a ) T ( ω (1)3 ) + (1 − a ) T ( ω (1)3 ) a a a T ( ω (1)3 ) − T ( ω (1)3 ) , (2.39b) T T ( ω (1)3 ) ω (1)3 = a a a a a a a (1 − a a a ) T ( ω (1)3 ) + (1 − a ) T ( ω (1)3 ) a a a a T ( ω (1)3 ) − T ( ω (1)3 ) , (2.39c) T T ( ω (1)3 ) ω (1)3 = a a a (1 − a a ) T ( ω (1)3 ) − (1 − a ) T ( ω (1)3 ) a T ( ω (1)3 ) − T ( ω (1)3 ) , (2.39d) T T ( ω (1)3 ) ω (1)3 = a a a a a (1 − a a a ) T ( ω (1)3 ) − (1 − a ) T ( ω (1)3 ) a a T ( ω (1)3 ) − T ( ω (1)3 ) , (2.39e) T T ( ω (1)3 ) ω (1)3 = a a a a a (1 − a a a ) T ( ω (1)3 ) − (1 − a a ) T ( ω (1)3 ) a T ( ω (1)3 ) − T ( ω (1)3 ) , (2.39f) Figure 1. The lattice (2.33) around the origin, which is a 2-dimensional projection of theVoronoi cell of type A . Refer to (2.34) and (2.35) for v . The directions from to v i , i = , . . . ,
4, correspond to the T i -directions, i = , . . . ,
4, respectively. ω (1)1 ω (1)3 − a a a a ω (3)2 ω (3)3 = − a a , (2.40a) ω (1)1 ω (1)3 − a a a T ( ω (2)2 ) ω (2)1 = − a a a , (2.40b) ω (1)1 ω (1)3 − a a a ω (4)1 T − T − ( ω (4)1 ) = − a a a . (2.40c)Note that Equations (2.39) are relations between the ω -function ω (1) l , l , l , l , l , but Equations(2.40) are the relations between ω (1) l , l , l , l , l and ω (3) l , l , l , l , l , ω (1) l , l , l , l , l and ω (2) l , l , l , l , l and ω (1) l , l , l , l , l and ω (4) l , l , l , l , l , respectively. Each equation of Equations (2.39) and that of Equa-tions (2.40) are of H
3- and D ω -lattice of type A (1)4 will be discussed in a forthcoming paper (N. Joshi, N.Nakazono and Y. Shi, in preparation).3. C onstruction of the lattice ω A + A In this section, we consider the extended a ffi ne Weyl group e W (( A ⋊ A ) (1) ) given by thefollowing six generators: w = s , w = s s s , w = s s s , r = ι, r = σι s s , π = σ ι s . (3.1) ATTICE EQUATIONS ARISING FROM DISCRETE PAINLEV´E SYSTEMS. II 13
The details of e W (( A ⋊ A ) (1) ) is discussed in Appendix B. Using this group, we constructanother important lattice ω A + A . Moreover, we show that the q -Painlev´e equations (1.1)can be derived also as the relations on the lattice ω A + A .3.1. A ffi ne Weyl group e W (( A ⋊ A ) (1) ) . In this section, we consider the birational actionof e W (( A ⋊ A ) (1) ) on the parameters b i , i = , , ,
3, and p defined by (2.16) and on theparticular ω -variables ω ( j ) i , i = , , j = ,
3, given by (2.27). We note that from therelations (2.29), these six ω -variables satisfy the following two relations: ω (1)2 = b b / ω (1)3 ( p b b / ω (3)2 − b b ω (3)1 ) b ω (3)1 , (3.2a) ω (3)3 = p b b b ω (3)2 ω (1)3 ω (1)1 − pb b b b / ω (1)3 . (3.2b)Therefore, essential ω -variables used here are four. The action of e W (( A ⋊ A ) (1) ) on theparameters is given by w : ( b , b , b , b , p ) ( b − p − , p − b − b , p − b − b , b , p ) , (3.3a) w : ( b , b , b , b , p ) ( b b − , b − , b − b , b , p ) , (3.3b) w : ( b , b , b , b , p ) ( b , b , b , b , p ) , (3.3c) r : ( b , b , b , b , p ) ( b − , b − b , p − b − b , b − , p − ) , (3.3d) r : ( b , b , b , b , p ) ( pb , pb , p − b , b − , p − ) , (3.3e) π : ( b , b , b , b , p ) ( pb b − , b − , p − b − b , b , p − ) , (3.3f)while that on the six ω -variables is given by w ( ω (1)3 ) = p ω (3)1 ω (3)3 ( pb ω (1)2 + b ω (1)1 ) b ω (3)2 ( pb b ω (3)1 + ω (3)3 ) , (3.4a) w ( ω (3)2 ) = p b ω (3)2 ( pb b ω (1)1 + b ω (1)2 ) pb ω (1)2 + b ω (1)1 , (3.4b) w ( ω (1)1 ) = ω (1)1 ( pb b ω (3)2 + ω (3)1 ) b ( pb b ω (3)2 + b ω (3)1 ) , (3.4c) w ( ω (3)3 ) = pb ω (1)3 ω (3)3 ( pb b ω (3)2 + b ω (3)1 ) b ( − pb b b / ω (1)3 ω (3)3 + pb b ω (3)1 ω (1)3 + b ω (1)1 ω (3)3 ) , (3.4d) w ( ω (1)2 ) = b b ω (1)3 ( p b b b ω (1)1 ω (3)2 + pb ω (3)1 ω (1)2 − b b b / ω (1)1 ω (3)1 ) b ω (3)1 ( pb b ω (1)3 + ω (1)1 ) , (3.4e) w ( ω (3)1 ) = b ω (3)1 ( pb b ω (1)3 + ω (1)1 ) b ( pb b ω (1)3 + ω (1)1 ) , r ( ω (1) i ) = ω (3) − i + , r ( ω (3) i ) = ω (1) − i + , (3.4f) r ( ω (1)1 ) = ω (3)2 , r ( ω (1)2 ) = ω (3)1 , r ( ω (1)3 ) = − pb b ω (1)1 b ω (3)1 ( b b b / ω (1)1 − pb ω (1)2 ) , (3.4g) r ( ω (3)1 ) = ω (1)2 , r ( ω (3)2 ) = ω (1)1 , (3.4h) r ( ω (3)3 ) = − p b b ω (3)2 pb b b / ω (1)1 ω (3)2 + b b b / ω (1)1 ω (3)1 − pb ω (1)2 ω (3)1 , (3.4i) π ( ω (1) i ) = ω (1) − i + , π ( ω (3)1 ) = ω (3)2 , π ( ω (3)2 ) = ω (3)1 , (3.4j) π ( ω (3)3 ) = − pb b ω (1)1 b ω (3)1 ( b b b / ω (1)1 − pb ω (1)2 ) , (3.4k)where i ∈ Z / Z , which follow from (2.3), (2.16), (2.30) and (3.1).Let ρ = π r w w , ρ = π r w w , ρ = π r w w , ρ = π r r r . (3.5)Note here that the transformations ρ i , i = , . . . ,
4, are translations on the root system Q (( A + A ) (1) ) (B.1) (see Appendix B for details). The translations ρ i , i = , . . . ,
4, com-mute with each other and ρ ρ ρ ρ = . (3.6)Their actions on the parameters are given by ρ : ( b , b , b , b ) ( pb , b , b , b − ) , (3.7a) ρ : ( b , b , b , b ) ( b , pb , b , b − ) , (3.7b) ρ : ( b , b , b , b ) ( p − b , p − b , p − b , b − ) , (3.7c) ρ : ( b , b , b , b ) ( b , b , pb , b − ) , (3.7d)where p is invariant under their actions.3.2. Lattice ω A + A . In this section, we define the ω -functions associated with the trans-lations on the root system Q (( A + A ) (1) ) and then construct the lattice ω A + A .We define ω -functions by using the translations ρ i , i = , . . . ,
4, as follows: ω l , l , l , l = ρ l ρ l ρ l ρ l ( ω (1)3 ) , (3.8)where l i ∈ Z . We note that ω (1)1 = ω , , , , ω (1)2 = ω , , , , ω (1)3 = ω , , , , (3.9a) ω (3)1 = ω , , , , ω (3)2 = ω , , , , ω (3)3 = ω , , , . (3.9b)Let us assign the ω -functions ω l , l , l , l to the vertices of the lattice X i = l i v i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l , . . . , l ∈ Z (3.10)by the following correspondence: ω l , l , l , l ↔ l v + l v + l v + l v . (3.11)Here, v i , i = , . . . ,
4, are defined by v = (1 , , , v = ( − , − , , v = (1 , − , − , v = ( − , , − , (3.12)and satisfy v + v + v + v = . We here refer to the lattice (3.10) with the ω -functions ω l , l , l , l as lattice ω A + A . We note that the configurations of the ω -variables on the lattice ω A + A are given by( ω (1)1 , ω (1)2 , ω (1)3 , ω (3)1 , ω (3)2 , ω (3)3 ) ↔ ( v + v , v − v , , v , − v , − v ) . (3.13)See the example given in Figure 2 to see the quadrilateral associated with ω (1)1 , ω (1)3 , ω (3)2 and ω (3)3 .The 14 vertices around l ∈ ω A + A : { l ± v i , l + v i + v j | i , j = , . . . , , i , j } , (3.14)collectively forms the rhombic dodecahedron (see Figure 3). Letting ¯ V ( l ) be the rhombicdodecahedron with the center l ∈ ω A + A :¯ V ( l ) = { l } ∪ V ( l ) , (3.15) ATTICE EQUATIONS ARISING FROM DISCRETE PAINLEV´E SYSTEMS. II 15
Figure 2. A quadrilateral associated with the ω -variables ω (1)1 , ω (1)3 , ω (3)2 and ω (3)3 .Figure 3. The rhombic dodecahedron around . Refer to (3.12) for v . The directions from to v i , i = , . . . ,
4, correspond to the ρ i -directions, i = , . . . ,
4, respectively.then the following holds: ω A + A = [ l ∈ ω A + A ¯ V ( l ) . (3.16)Henceforth, let us consider the quad-equations appearing on the lattice ω A + A . Lemma 3.1.
The following quad-equations hold on the lattice ω A + A : ω l + , l + , l , l ω l , l , l , l = p l − l ) + b λ l + l + l + l ω l + , l , l , l − p l − l ) b b − ω l , l + , l , l p l − l ) b b − ω l , l + , l , l − ω l + , l , l , l , (3.17a) ω l , l + , l + , l ω l , l , l , l = p − l + l ) + λ l + l + l + l b ω l , l + , l , l − p l − l ) b ω l , l , l + , l p l − l ) b ω l , l , l + , l − ω l , l + , l , l , (3.17b) ω l + , l , l + , l ω l , l , l , l = p − l + l ) + λ l + l + l + l b ω l + , l , l , l − p l − l ) b ω l , l , l + , l p l − l ) b ω l , l , l + , l − ω l + , l , l , l , (3.17c) ω l , l , l , l ω l + , l , l , l + = p − l + l + l − b λ l + l + l + l b ω l + , l , l , l ω l , l , l , l + + p − l + l + l + l − b b λ l + l + l + l / b , (3.17d) ω l , l , l , l ω l , l + , l , l + = p l − l + l − b λ l + l + l + l b ω l , l + , l , l ω l , l , l , l + + p l − l + l + l − b b λ l + l + l + l / b , (3.17e) ω l , l , l , l ω l , l , l + , l + = (cid:16) p l + l − l − b b λ l + l + l + l (cid:17) ω l , l , l + , l ω l , l , l , l + + p l + l − l + l − b b b λ l + l + l + l / , (3.17f) where λ l = b − l . (3.18) Note that each equation of Equations (3.17a) – (3.17c) and that of Equations (3.17d) – (3.17f) are of H - and D -types in the ABS classification [1, 2, 6–8], respectively.Proof. Recalling the definitions of ρ i given in (3.5) and the relations (3.2), we have theactions shown below: ρ ( ω (1)3 ) = b ω (3)1 ( pb b ω (1)3 + ω (1)1 ) b ( pb b ω (1)3 + ω (1)1 ) , (3.19a) ρ ( ω (3)1 ) = ω (1)1 ( pb b ω (3)2 + ω (3)1 ) b ( pb b ω (3)2 + b ω (3)1 ) , (3.19b) ρ − ( ω (3)1 ) = p b ω (1)3 ( pb ω (3)3 + b b ω (3)1 ) pb b ω (3)1 + ω (3)3 , (3.19c) ρ − ( ω (3)1 ) = ω (1)2 = b b / ω (1)3 ( p b b / ω (3)2 − b b ω (3)1 ) b ω (3)1 , (3.19d) ρ ( ω (3)1 ) = b ω (1)1 ω (3)3 b b / ( b b / ω (3)1 − b b ω (3)3 ) . (3.19e)This leads to ω (1)1 ω (1)3 = pb b ω (3)1 − b b − ρ ( ω (1)3 ) b b − ρ ( ω (1)3 ) − ω (3)1 , (3.20a) ω (3)2 ω (3)1 = b − pb ω (1)1 − b ρ ( ω (3)1 ) b ρ ( ω (3)1 ) − ω (1)1 , (3.20b) ω (3)1 ω (3)3 = b − pb ρ − ( ω (3)1 ) − p b ω (1)3 p b ω (1)3 − ρ − ( ω (3)1 ) , (3.20c) ω (3)2 ω (3)1 = b b − pb ρ − ( ω (3)1 ) ω (1)3 + b b b − / p b , (3.20d) ω (3)1 ω (3)3 = b b − b ω (1)1 ρ ( ω (3)1 ) + b b b − / b , (3.20e)which in turn lead immediately to Equations (3.17a)–(3.17e). Moreover, we get Equation(3.17f) from the relation (3.2b) or, equivalently, ω (1)1 ω (1)3 = ( pb b b ) ω (3)2 ω (3)3 + pb b b b / . (3.21)Therefore we have completed the proof. (cid:3) Lemma 3.2.
The quad-equations (3.17) are fundamental relations on the lattice ω A + A .Proof. In this proof we will show that any ω -function ω l , l , l , l can be calculated by thequad-equations (3.17) with four initial values: ω (1)1 , ω (1)3 , ω (3)1 and ω (3)2 (or, ω , , , , ω , , , , ω , , , and ω , , , ).First, we obtain the values of all ω -functions on ¯ V ( ) from the initial values by thefollowing steps. ATTICE EQUATIONS ARISING FROM DISCRETE PAINLEV´E SYSTEMS. II 17
Step 1:
By using Equations (3.17a) (0 , , , , (3.17b) (1 , , , and (3.17f) (1 , , , , the func-tions on v , v + v and − v can be calculated, respectively. Step 2:
By using Equations (3.17c) (0 , , , , (3.17c) (0 , , , , (3.17d) (0 , , , , (3.17e) (1 , , , and (3.17e) (1 , , , , the functions on v , v + v , v + v , v + v and − v can be cal-culated, respectively. Step 3:
By using Equations (3.17a) (0 , , , , (3.17d) (0 , , , and (3.17f) (0 , , , , the func-tions on v , v + v and − v can be calculated, respectively.Note that the subscripts of the equation numbers ( l , l , l , l ) denote the values of the pa-rameters l i , i = , . . . ,
4, in the equations.Next, we consider ¯ V ( v ). From the determined ω -functions on¯ V ( ) ∩ ¯ V ( v ) = { , v , v + v i , − v i | i = , , } , (3.22)we can obtain the values of the ω -functions on¯ V ( v ) − ¯ V ( ) = { v − v i , v , v + v i | i = , , } , (3.23)by the following steps. Step 1:
By using Equations (3.17a) (0 , − , , , (3.17c) (0 , , − , and (3.17d) (0 , , , − , thefunctions on v − v , v − v and v − v can be calculated, respectively. Step 2:
By using Equations (3.17c) (1 , , , , (3.17a) (1 , , , and (3.17a) (1 , , , , the func-tions on 2 v + v , 2 v + v and 2 v + v can be calculated, respectively. Step 3:
By using Equation (3.17b) (2 , , , , the function on 2 v can be calculated.In a similar manner, we can calculate all ω -functions on ¯ V ( l + v i ), i = , . . . ,
4, fromthose on ¯ V ( l ) for any l ∈ ω A + A . Therefore we have completed the proof. (cid:3) For later convenience, we here make the mention of R briefly. Its action on the param-eters b i and p is given by R : ( b , b , b , b , p ) ( b , pb , b , b − , p ) , (3.24)while that on the restricted ω -functions, which are on the following sublattice: X i = l i v i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l = l , l i ∈ Z ∪ X i = l i v i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l = l + , l i ∈ Z ⊂ ω A + A , (3.25)is given by R : ω l , l , l , l ω l + , l , l , l if l = l ,ω l , l + , l , l if l = l + . (3.26)3.3. Discrete Painlev´e equations.
In this section we consider the particular f -variables f ( j )1 , j = , ,
3, given by (2.12), which can be expressed by the ratios of the ω -functions ω l , l , l , l as follows: f (1)1 = ω (1)2 ω (1)1 = ω , , , ω , , , , f (2)1 = ω (2)2 ω (2)1 = ω (3)1 ω (3)3 = ω , , , ω , , , , f (3)1 = ω (3)2 ω (3)1 = ω , , , ω , , , . (3.27)These f -variables satisfy the relation (1.5), which follows from the relations (2.13). Theaction of e W (( A ⋊ A ) (1) ) on the three f -variables is given by w ( f (3)1 ) = p b f (3)1 ( pb b + b f (1)1 ) b + pb f (1)1 , w ( f (1)1 ) = b f (1)1 ( b + pb b f (3)1 )1 + pb b f (3)1 , (3.28a) w ( f (2)1 ) = b pb b / f (3)1 b r ( f (3)1 )(1 + pb b f (3)1 ) pb b / ( b + pb b f (3)1 ) − b , (3.28b) w ( f (1)1 ) = pb f (3)1 (cid:16) pb + b π ( f (2)1 ) (cid:17) pb b + r ( f (3)1 ) , w ( f (2)1 ) = b f (2)1 (cid:16) pb b + r ( f (3)1 ) (cid:17) b (cid:16) pb b + r ( f (3)1 ) (cid:17) , (3.28c) w ( f (3)1 ) = b f (3)1 (cid:16) pb b + r ( f (3)1 ) (cid:17) b (cid:16) pb b + r ( f (3)1 ) (cid:17) , r ( f (1)1 ) = f (2)1 , r ( f (2)1 ) = f (1)1 , (3.28d) r ( f (3)1 ) = b b / ( − b b + p b b / f (3)1 ) b f (1)1 , r ( f (1)1 ) = f (3)1 , r ( f (3)1 ) = f (1)1 , (3.28e) r ( f (2)1 ) = − b b / p b b f (1)1 f (3)1 − b b b / p b b f (1)1 + b p b f (3)1 , (3.28f) π ( f (1)1 ) = r ( f (3)1 ) , π ( f (2)1 ) = b ( − b b b / + pb f (1)1 ) pb b f (3)1 . (3.28g)Note that r ( f (3)1 ) = ω (1)1 ω (1)3 . (3.29)Moreover, the time evolutions of the q -Painlev´e equations shown in § e W (( A ⋊ A ) (1) ) as follows: T = ρ ρ , T = ρ , R = π r w , R = ρ , (3.30)where ρ i are defined by (3.5). Therefore, the birational actions of T , T , R and R are given by (1.6) and (1.9). As mentioned in Remark 1.4, these actions give q -Painlev´eequations (1.1). 4. P roofs of T heorems and ff erence equations: u ( l + ǫ + ǫ ) u ( l ) = − α l u ( l + ǫ ) − β l u ( l + ǫ ) α l u ( l + ǫ ) − β l u ( l + ǫ ) , (4.1a) u ( l + ǫ + ǫ ) u ( l ) = − β l u ( l + ǫ ) − γ l u ( l + ǫ ) β l u ( l + ǫ ) − γ l u ( l + ǫ ) , (4.1b) u ( l + ǫ + ǫ ) u ( l ) = − γ l u ( l + ǫ ) − α l u ( l + ǫ ) γ l u ( l + ǫ ) − α l u ( l + ǫ ) , (4.1c) u ( l + ǫ + ǫ ) u ( l ) + u ( l + ǫ ) u ( l + ǫ ) = − α l K l , (4.1d) u ( l + ǫ + ǫ ) u ( l ) + u ( l + ǫ ) u ( l + ǫ ) = − β l K l , (4.1e) u ( l + ǫ + ǫ ) u ( l ) + u ( l + ǫ ) u ( l + ǫ ) = − γ l K l , (4.1f)where l = P i = l i ǫ i ∈ Z and { ǫ , . . . , ǫ } is a standard basis for R . Here, u ( l ) is a functionfrom Z to C and { α l } l ∈ Z , { β l } l ∈ Z , { γ l } l ∈ Z and { K l } l ∈ Z are complex parameters. This system isobtained by assigning the quad-equations of ABS type to the faces of each 4-dimensionalhypercube (4-cube) (see [23] and references therein). The Lax equations for System (4.1)are given by the following [23]: Ψ l + , l , l , l = δ (1) µα l − u ( l + ǫ )1 u ( l ) − µα l u ( l + ǫ ) u ( l ) . Ψ l , l , l , l , (4.2a) ATTICE EQUATIONS ARISING FROM DISCRETE PAINLEV´E SYSTEMS. II 19 P ∆ E Lax pair(4.1a) (4.2a), (4.2b)(4.1b) (4.2b), (4.2c)(4.1c) (4.2a), (4.2c)(4.1d) (4.2a), (4.2d)(4.1e) (4.2b), (4.2d)(4.1f) (4.2c), (4.2d) P ∆ E Lax pair(4.5a) (4.18a), (4.18b)(4.5b) (4.18b), (4.18c)(4.5c) (4.18a), (4.18c)(4.5d) (4.18a), (4.18d)(4.5e) (4.18b), (4.18d)(4.5f) (4.18c), (4.18d) P ∆ E Lax pair(3.17a) (4.19a), (4.19b)(3.17b) (4.19b), (4.19c)(3.17c) (4.19a), (4.19c)(3.17d) (4.19a), (4.19d)(3.17e) (4.19b), (4.19d)(3.17f) (4.19c), (4.19d)Table 1. The correspondences between P ∆ Es and Lax pairs. Ψ l , l + , l , l = δ (2) µβ l − u ( l + ǫ )1 u ( l ) − µβ l u ( l + ǫ ) u ( l ) . Ψ l , l , l , l , (4.2b) Ψ l , l , l + , l = δ (3) µγ l − u ( l + ǫ )1 u ( l ) − µγ l u ( l + ǫ ) u ( l ) . Ψ l , l , l , l , (4.2c) Ψ l , l , l , l + = δ (4) − µ K l − u ( l + ǫ )1 u ( l ) 0 . Ψ l , l , l , l , (4.2d)where δ ( i ) , i = , . . . ,
4, are arbitrary constants and µ is a spectral parameter. The pairs ofEquations (4.2) give the Lax pairs of P ∆ Es (4.1) (see Table 1).4.1.
Proof of Theorem 1.2.
In this section, we show that the lattice ω A + A can be obtainedfrom the integer lattice Z with the P ∆ Es (4.1) by a geometric reduction.Let u ( l ) = λ l + l + l + l ( l + l + l − l ) / U ( l ) , (4.3)where l = P i = l i ǫ i ∈ Z . Here, λ is a non-zero complex parameter and λ l = λ if l = n , λ if l = n + . (4.4)Then, System (4.1) can be rewritten as the following: U ( l + ǫ + ǫ ) U ( l ) = − λ l + l + l + l α l U ( l + ǫ ) − β l U ( l + ǫ ) α l U ( l + ǫ ) − β l U ( l + ǫ ) , (4.5a) U ( l + ǫ + ǫ ) U ( l ) = − λ l + l + l + l β l U ( l + ǫ ) − γ l U ( l + ǫ ) β l U ( l + ǫ ) − γ l U ( l + ǫ ) , (4.5b) U ( l + ǫ + ǫ ) U ( l ) = − λ l + l + l + l γ l U ( l + ǫ ) − α l U ( l + ǫ ) γ l U ( l + ǫ ) − α l U ( l + ǫ ) , (4.5c) U ( l ) U ( l + ǫ + ǫ ) + λ l + l + l + l U ( l + ǫ ) U ( l + ǫ ) = − α l K l λ l + l + l + l / , (4.5d) U ( l ) U ( l + ǫ + ǫ ) + λ l + l + l + l U ( l + ǫ ) U ( l + ǫ ) = − β l K l λ l + l + l + l / , (4.5e) U ( l ) U ( l + ǫ + ǫ ) + λ l + l + l + l U ( l + ǫ ) U ( l + ǫ ) = − γ l K l λ l + l + l + l / . (4.5f) Moreover, by imposing the following (1 , , , U ( l ) = U ( l + ǫ + ǫ + ǫ + ǫ ) , (4.6)for l ∈ Z , with the following condition of the parameters: α l = p − l α , β l = p − l β , γ l = p − l γ , K l = p l K , (4.7)where p is a non-zero complex parameter, System (4.5) becomes the system of q -di ff erenceequations (in this case the shift parameter is given by p ).We define the transformations ˆ ρ i , i = , . . . ,
4, by the following actions:ˆ ρ : ( U ( l ) , α , β , γ , K , λ , p ) ( U ( l + ǫ ) , p − α , β , γ , K , λ − , p ) , (4.8a)ˆ ρ : ( U ( l ) , α , β , γ , K , λ , p ) ( U ( l + ǫ ) , α , p − β , γ , K , λ − , p ) , (4.8b)ˆ ρ : ( U ( l ) , α , β , γ , K , λ , p ) ( U ( l + ǫ ) , α , β , p − γ , K , λ − , p ) , (4.8c)ˆ ρ : ( U ( l ) , α , β , γ , K , λ , p ) ( U ( l + ǫ ) , α , β , γ , pK , λ − , p ) , (4.8d)which imply that ˆ ρ i is a shift operator of ǫ i -direction on Z . In addition, we also introducea transformation ˆ R as follows. Its action on the parameters is defined byˆ R : ( α , β , γ , K , λ , p ) ( β , p − α , γ , K , λ − , p ) , (4.9)while that on the function U ( l ) is defined byˆ R ( U ( l )) = U ( l + ǫ ) if l ∈ ∇ (1) , U ( l + ǫ ) if l ∈ ∇ (2) , (4.10)where ∇ (1) = X i = l i ǫ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l i ∈ Z , l = l , ∇ (2) = X i = l i ǫ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l i ∈ Z , l = l + . (4.11)These imply that ˆ R is a zigzag-shift operator on the sublattice R = ∇ (1) ∪ ∇ (2) ⊂ Z , (4.12)that is, ˆ R ( l ) = ˆ ρ ( l ) if l ∈ ∇ (1) , ˆ ρ ( l ) if l ∈ ∇ (2) , ˆ R − ( l ) = ˆ ρ − ( l ) if l ∈ ∇ (1) , ˆ ρ − ( l ) if l ∈ ∇ (2) . (4.13)In general, for a function F = F ( U ( l ) , α , β , γ , K , λ , p ), we let a transformation w ∈h ˆ ρ , . . . , ˆ ρ , ˆ R i act as w ( F ) = F (cid:16) w ( U ( l )) , w ( α ) , w ( β ) , w ( γ ) , w ( K ) , w ( λ ) , w ( p ) (cid:17) , (4.14)that is, the transformation w act on the arguments from the left.Finally, letting ω l , l , l , l = H l , l , l , l U ( l ) , b = γ α , b = γ β , b = γ K , (4.15)where l = P i = l i ǫ i , we obtain the fundamental relations on the lattice ω A + A (3.17) fromSystem (4.5). Here, the gauge factor H l , l , l , l is defined by H l , l , l , l = i (log α l β l γ l K l ) / log p e( (log p α l β l γ l − ) + α l β l − ) ) /
16 log p × γ l α l / β l / / , (4.16)where i = √−
1. Furthermore, the actions of transformations ˆ ρ i , i = , . . . ,
4, and ˆ R correspond to those of ρ i , i = , . . . ,
4, and R which are elements of e W (( A ⋊ A ) (1) ), ATTICE EQUATIONS ARISING FROM DISCRETE PAINLEV´E SYSTEMS. II 21
Figure 4. A (1 , , , Z → Z / Z ( ǫ + ǫ + ǫ + ǫ ) (cid:27) ω A + A . The reduction from Z with System (4.1) to the lattice ω A + A is referred to as geometricreduction [24] and then the lattice ω A + A is said to have the reduced hypercube structure.Therefore, we have completed the proof of Theorem 1.2.4.2. Proof of Theorem 1.3.
In this section, we construct the Lax pairs of the q -Painlev´eequations (1.1) from the Lax equations (4.2) by using the reduction given in § Ψ l , l , l , l = i − l − l − l + l U ( l ) −
00 i λ l + l + l + l ( l + l + l − l ) / ! .φ l , l , l , l , (4.17)the Lax equations (4.2) can be rewritten as the following: φ l + , l , l , l = δ (1) i µα l U ( l + ǫ ) U ( l ) 1 λ l + l + l + l / λ l + l + l + l / − i µα l U ( l ) U ( l + ǫ ) .φ l , l , l , l , (4.18a) φ l , l + , l , l = δ (2) i µβ l U ( l + ǫ ) U ( l ) 1 λ l + l + l + l / λ l + l + l + l / − i µβ l U ( l ) U ( l + ǫ ) .φ l , l , l , l , (4.18b) φ l , l , l + , l = δ (3) i µγ l U ( l + ǫ ) U ( l ) 1 λ l + l + l + l / λ l + l + l + l / − i µγ l U ( l ) U ( l + ǫ ) .φ l , l , l , l , (4.18c) φ l , l , l , l + = δ (4) − i µ K l U ( l + ǫ ) U ( l ) λ l + l + l + l λ l + l + l + l .φ l , l , l , l . (4.18d)These give the Lax pairs of P ∆ Es (4.5) (see Table 1). Moreover, by the reduction (4.6)with (4.7) and the replacement (4.15), the Lax equations (4.18) can be rewritten as the following: φ l + , l , l , l = δ (1) − p − l + l + l − b b ω l + , l , l , l ω l , l , l , l x λ l + l + l + l / λ l + l + l + l / − p l − l − l + b b ω l , l , l , l ω l + , l , l , l x .φ l , l , l , l , (4.19a) φ l , l + , l , l = δ (2) − p l − l + l − b b ω l , l + , l , l ω l , l , l , l x λ l + l + l + l / λ l + l + l + l / − p − l + l − l + b b ω l , l , l , l ω l , l + , l , l x .φ l , l , l , l , (4.19b) φ l , l , l + , l = δ (3) − p l + l − l − b b ω l , l , l + , l ω l , l , l , l x λ l + l + l + l / λ l + l + l + l / − p − l − l + l + b b ω l , l , l , l ω l , l , l + , l x .φ l , l , l , l , (4.19c) φ l , l , l , l + = δ (4) p l b ω l , l , l , l + ω l , l , l , l x λ l + l + l + l λ l + l + l + l .φ l , l , l , l , (4.19d)where x = µγ . (4.20)These give the Lax pairs of P ∆ Es (3.17) (see Table 1).Now we are in a position to construct the Lax pairs of the q -Painlev´e equations. We firstlift the action of h ˆ ρ , . . . , ˆ ρ , ˆ R i up to the Lax equations (4.19) byˆ ρ : ( δ (1) , δ (2) , δ (3) , δ (4) , µ, φ l , l , l , l ) ( δ (1) , δ (2) , δ (3) , δ (4) , µ, φ l + , l , l , l ) , (4.21a)ˆ ρ : ( δ (1) , δ (2) , δ (3) , δ (4) , µ, φ l , l , l , l ) ( δ (1) , δ (2) , δ (3) , δ (4) , µ, φ l , l + , l , l ) , (4.21b)ˆ ρ : ( δ (1) , δ (2) , δ (3) , δ (4) , µ, φ l , l , l , l ) ( δ (1) , δ (2) , δ (3) , δ (4) , µ, φ l , l , l + , l ) , (4.21c)ˆ ρ : ( δ (1) , δ (2) , δ (3) , δ (4) , µ, φ l , l , l , l ) ( δ (1) , δ (2) , δ (3) , δ (4) , µ, φ l , l , l , l + ) , (4.21d)ˆ R : ( δ (1) , δ (2) , δ (3) , δ (4) , µ, φ l , l , l , l ) ( δ (2) , δ (1) , δ (3) , δ (4) , µ, φ Rl , l , l , l ) , (4.21e)where φ Rl , l , l , l = φ l + , l , l , l if l = l ,φ l , l + , l , l if l = l + . (4.22)By letting φ , , , − = ω , , , − ω , , , ! . Φ , (4.23)the action of h ˆ ρ . . . , ˆ ρ , ˆ R i on Φ is given byˆ ρ ( Φ ) = δ − b b p x b b b ( p b b / f (3)1 − b b ) f (3)1 b / − pb b f (3)1 b / ( p b b / f (3)1 − b b ) x . Φ , (4.24a) ATTICE EQUATIONS ARISING FROM DISCRETE PAINLEV´E SYSTEMS. II 23 ˆ ρ ( Φ ) = δ − b pb x b b b (cid:16) p b b / ˆ ρ − ( f (1)1 ) − b b (cid:17) ˆ ρ − ( f (1)1 ) b / − pb b ˆ ρ − ( f (1)1 ) b / (cid:16) p b b / ˆ ρ − ( f (1)1 ) − b b (cid:17) x . Φ , (4.24b)ˆ ρ ( Φ ) = δ − b b p x b / ˆ ρ (cid:16) f (2)1 f (3)1 (cid:17) b b (cid:16) b b ˆ ρ (cid:16) f (2)1 f (3)1 (cid:17) + b b / (cid:17) p b / − b b ˆ ρ (cid:16) f (2)1 f (3)1 (cid:17) + b b / pb ˆ ρ (cid:16) f (2)1 f (3)1 (cid:17) x . Φ , (4.24c)ˆ ρ ( Φ ) = δ b p x b pb f (3)1 (cid:16) pb ˆ ρ ( f (3)1 ) − b b b / (cid:17) b b . Φ , (4.24d)ˆ R ( Φ ) = ˆ ρ ( Φ ) , (4.24e)where f ( j )1 , j = , ,
3, are given by (3.27) and satisfy the relation (1.5). Next, let us defineˆ T SP = ˆ ρ ˆ ρ ˆ ρ ˆ ρ , ˆ T = ˆ ρ ˆ ρ , ˆ T = ˆ ρ , ˆ R = ˆ ρ . (4.25) Remark 4.1.
Under the actions on the f -variables f ( j )1 , j = , , , and the parameters b i ,i = , . . . , , and p, the transformations ˆ T , ˆ T , ˆ R and ˆ R are respectively equivalent tothe transformations T , T , R and R , which are elements of e W (( A ⋊ A ) (1) ) , and thespectral operator ˆ T SP can be regarded as an identity mapping. The actions of ˆ T SP , ˆ T , ˆ T , ˆ R and ˆ R on the spectral parameter x are given byˆ T SP ( x ) = px , ˆ T ( x ) = ˆ T ( x ) = ˆ R ( x ) = ˆ R ( x ) = x , (4.26)while those on the wave function Φ are given by the following:ˆ T SP ( Φ ) = δ δ δ δ A . Φ , ˆ T ( Φ ) = δ δ B T . Φ , ˆ T ( Φ ) = δ B T . Φ , (4.27a)ˆ R ( Φ ) = δ B R . Φ , ˆ R ( Φ ) = δ B R . Φ , (4.27b)where A = b p x b − b / ( p b b / f (3)1 − b b )( b b b / + pb b b / f (3)1 − pb f (1)1 ) p b b f (1)1 f (3)1 . − pb b x − p b b b / f (3)1 b b b / + pb b b / f (3)1 − pb f (1)1 b ( pb f (1)1 − b b b / ) pb b / f (3)1 pb b ( pb f (1)1 − b b b / ) b b b / + pb b b / f (3)1 − pb b f (1)1 x . − b b x pb b / b ( pb f (1)1 − b b b / ) b / f (1)1 pb b b f (1)1 b b b / − pb f (1)1 x . − b pb x b b b ( p b b / f (3)1 − b b ) f (3)1 b / − pb b f (3)1 b / ( p b b / f (3)1 − b b ) x , (4.28) B T = − b b x pb b / b ( pb f (1)1 − b b b / ) b / f (1)1 pb b b f (1)1 b b b / − pb f (1)1 x . − b pb x b b b ( p b b / f (3)1 − b b ) f (3)1 b / − pb b f (3)1 b / ( p b b / f (3)1 − b b ) x , (4.29a) B T = b x b pb b / f (1)1 ( b b / f (2)1 − b b ) (cid:16) pb b / T ( f (3)1 ) − pb b (cid:17) b ( p b b / f (3)1 − b b ) 0 . b p x b pb f (3)1 b b pb f (1)1 ( b b − b b / f (2)1 ) b b − p b b / f (3)1 − b b / , (4.29b) B R = − b pb x b b b ( p b b / f (3)1 − b b ) f (3)1 b / − pb b f (3)1 b / ( p b b / f (3)1 − b b ) x , (4.29c) B R = b p x b pb f (3)1 (cid:16) pb R ( f (3)1 ) − b b b / (cid:17) b b . (4.29d)Therefore, we finally obtain Theorem 1.3 by the following correspondence:ˆ T = T , ˆ T = T , ˆ R = R , ˆ R = R , (4.30a) δ = δ = δ = δ = . (4.30b)5. C oncluding remarks In this paper, we constructed the ω -lattice of type A (1)4 . The ω -lattice provides the infor-mations about how a system of partial di ff erence equations can be reduced to A (1)4 -surface q -Painlev´e equations. We will show how to use this information in forthcoming paper (N.Joshi, N. Nakazono and Y. Shi, in preparation). We also constructed another importantlattice ω A + A and showed that it has the reduced hypercube structure. Moreover, by usingthis structure, we constructed the Lax pairs of the q -Painlev´e equations (1.1). The distin-guishing feature of the Lax pairs given in this paper as compared with those in the otherworks, e.g. [22, 35], is that their coe ffi cient matrices can be factorized into the products ofmatrices which are of degree one in the spectral parameter x . This property enables us to ATTICE EQUATIONS ARISING FROM DISCRETE PAINLEV´E SYSTEMS. II 25 construct the Lax pairs of symmetric discrete Painlev´e equations, e.g. q -P III ( D (1)7 ) (1.1c)and q -P IV (1.1d), which can be obtained by projective reductions [28, 29]. Acknowledgment.
This research was supported by an Australian Laureate Fellowship ppendix
A. P roof of L emma e W ( A (1)4 ) with its linear action andshow it forms the extended a ffi ne Weyl group of type A (1)4 . Moreover, we lift its action tothe birational action on the parameters and the τ -variables.First, we define the transformation group e W ( A (1)4 ) = h s , s , s , s , s , σ, ι i . Let ( f , g )be inhomogeneous coordinate of P × P . We consider the following eight base points of P × P : p : ( f , g ) = ( − a − a , , p : ( f , g ) = ( − a − a − a , , (A.1a) p : ( f , g ) = ( − a − a a , ∞ ) , p : ( f , g ) = (0 , − a a − ) , (A.1b) p : ( f , g ) = (0 , − a a − a ) , p : ( f , g ) = ( ∞ , − a a − a − ) , (A.1c) p ∞ : ( f , g ) = ( ∞ , ∞ ) , p : ( f , g ; f / g ) = ( ∞ , ∞ ; − a − a a ) , (A.1d)where a i , i = , . . . ,
4, are non-zero complex parameters. Let ǫ : X → P × P denoteblow up of P × P at the points (A.1). The linear equivalence classes of the total transformof the coordinate lines f = constant and g = constant are denoted by h and h , respectively.The Picard group of X , denoted by Pic( X ), is given byPic( X ) = Z h M Z h M i = Z e i , (A.2)where e i = ǫ − ( p i ), i = , . . . ,
8, ( p = p ∞ ) are exceptional divisors. The intersection form( | ) is defined by ( h i | h j ) = − δ i j , ( h i | e j ) = , ( e i | e j ) = − δ i j . (A.3)The anti-canonical divisor of X , denoted by − K X , is uniquely decomposed into the primedivisors: − K X = h + h − X i = e i = X i = d i = : δ, (A.4)where d = h − e − e , d = e − e , d = h − e − e , (A.5a) d = h − e − e , d = h − e − e . (A.5b)The corresponding Cartan matrix( d i j ) i , j = = − − − − − − − − − − , d i j = d i | d j )( d j | d j ) , (A.6)and Dynkin diagram (see Figure 5) are of type A (1)4 . Thus, we can set the root lattice as Q ( A (1)4 ) = M i = Z d i , (A.7) Figure 5. Dynkin diagram of type A (1)4 .and identify the surface X as being type A (1)4 in Sakai’s classification [55]. Moreover, weobtain the following root lattice orthogonal to Q ( A (1)4 ):ˆ Q ( A (1)4 ) = M i = Z α i , (A.8)where α = h + h − e − e − e − e , α = e − e , α = h − e − e , (A.9a) α = h − e − e , α = e − e , (A.9b)and δ = α + α + α + α + α , (A.10)by searching for elements of Pic( X ) that are orthogonal to all divisors d i , i = , . . . ,
4. Theroot lattice ˆ Q ( A (1)4 ) is also of A (1)4 -type.Let us consider the Cremona isometries for this setting. A Cremona isometry is definedby an automorphism of Pic( X ) which preserves (i): the intersection form on Pic( X ); (ii): the canonical divisor K X ; (iii): e ff ectiveness of each e ff ective divisor of Pic( X ).The reflections s i for simple roots α i , i = , . . . ,
4, defined by the following right actions: v . s i = v − v | α i )( α i | α i ) α i , (A.11)for all v ∈ Pic( X ) and the automorphisms of the Dynkin diagram:( d , d , d , d , d ; α , α , α , α , α ) .σ = ( d , d , d , d , d ; α , α , α , α , α ) , (A.12a)( d , d , d , d , d ; α , α , α , α , α ) .ι = ( d , d , d , d , d ; α , α , α , α , α ) , (A.12b)defined by the following right actions: (cid:16) h , h , e , . . . , e (cid:17) .σ = (cid:16) h , h , e , . . . , e (cid:17) . − − − − − − −
10 0 0 0 0 0 0 0 1 00 0 0 0 0 0 1 0 0 00 0 0 1 0 0 0 0 0 0 − − , (A.13a) ATTICE EQUATIONS ARISING FROM DISCRETE PAINLEV´E SYSTEMS. II 27 (cid:16) h , h , e , . . . , e (cid:17) .ι = (cid:16) h , h , e , . . . , e (cid:17) . , (A.13b)are Cremona isometries and collectively form extended a ffi ne Weyl group of type A (1)4 .Namely, we can easily verify that the following fundamental relations hold: s i = , ( s i s i ± ) = , ( s i s j ) = , j , i ± , (A.14a) σ = , σ s i = s i + σ, ι = , ι s i = s − i ι, σι = ισ − , (A.14b)where i , j ∈ Z / Z . Note here that the transformations T i , i = , . . . ,
4, defined by (2.7) aretranslations on ˆ Q ( A (1)4 ): α i . T i = α i − δ, α i + . T i = α i + + δ, (A.15)where i ∈ Z / Z .Next, we lift the action of e W ( A (1)4 ) to the birational action. We first define the variables f u , f d , g u and g d by f = f u f d , g = g u g d , (A.16)and their polynomial F Λ by F Λ = F Λ ( f u , f d , g u , g d ) , (A.17)where Λ = mh + nh − P i = µ i e i , which corresponds to a curve of bi-degree ( m , n ) on P × P passing through base points p i with multiplicity µ i . For example, F h + h − e − e − e = γ ( a a a f u g d + a a f d g u + a a a f d g d ) , (A.18)where γ is an arbitrary non-zero complex parameter. We next define a mapping τ by thefollowing definition. Definition A.1.
We define a mapping τ on the setM = n e i . w | w ∈ e W ( A (1)4 ) , i = , . . . , o (A.19) by the following: (i): if under the blowing down map an exceptional line e i collapses to a base pointp j , put τ ( e i ) = τ ( e j ); (A.20) (ii): if Λ = mh + nh − P i = µ i e i ∈ { d , d , d , d } , thenF Λ ( f u , f d , g u , g d ) F Λ (1 , , , = τ ( e ) µ · · · τ ( e ) µ , (A.21) which givef u = τ ( e ) τ ( e ) , f d = τ ( e ) τ ( e ) , g u = τ ( e ) τ ( e ) , g d = τ ( e ) τ ( e ); (A.22) (iii): for Λ = mh + nh − P i = µ i e i ∈ M, τ ( Λ ) is defined by τ ( Λ ) = F Λ ( f u , f d , g u , g d ) τ ( e ) µ · · · τ ( e ) µ ; (A.23) Figure 6. Relations between the τ -variables. (iv): w ∈ e W ( A (1)4 ) act on τ ( Λ ) asw .τ ( Λ ) = τ ( Λ . w − ) , (A.24) where Λ ∈ M. Finally, Lemma 2.2 follows from the setting τ (1)1 = τ ( e ) , τ (2)1 = τ ( e ) , τ (3)1 = τ ( e ) , τ (4)1 = τ ( e ) , τ (5)1 = τ ( e ) = τ ( e ) , (A.25a) τ (1)2 = τ ( e .σ ) = a a ( a τ (3)1 τ (5)1 + a τ (4)1 τ (3)2 ) a a τ (5)2 , (A.25b) τ (2)2 = τ ( e .σ ) = a a ( a τ (1)1 τ (4)1 + a τ (5)1 τ (4)2 ) a a τ (1)2 , τ (3)2 = τ ( e ) , (A.25c) τ (4)2 = τ ( e .σ ) = a a ( a τ (1)1 τ (3)1 + a τ (2)1 τ (1)2 ) a a τ (3)2 , τ (5)2 = τ ( e ) , (A.25d)and the normalization of the polynomials F Λ to be designed to hold the fundamental re-lations (A.14). We note that the action of e W ( A (1)4 ) on the τ -variables are directly obtainedfrom the definition of the mapping τ . For example, s ( τ (5)2 ) = τ ( e . s ) = γ ′ ( a a τ (3)1 τ (5)1 + a τ (4)1 τ (3)2 ) τ (2)1 , (A.26)where γ ′ is an arbitrary non-zero complex parameter. Moreover, Figure 6 shows simplerelations between the τ -variables.A ppendix B. T he linear action of e W (( A ⋊ A ) (1) )In this section, we give explanations of the transformation group e W (( A ⋊ A ) (1) ) and itstranslation part h ρ , ρ , ρ , ρ i with their linear actions on the root systems.We here consider the following submodule of the root lattice ˆ Q ( A (1)4 ) (A.8): Q (( A + A ) (1) ) = Z β M Z β M Z β M Z γ M Z γ , (B.1)where the simple roots β i , i = , ,
2, and γ i , i = ,
1, are defined by β = α , β = α + α , β = α + α , (B.2a) γ = α − α + α − α , γ = α − α + α + α , (B.2b)and satisfy δ = β + β + β = γ + γ . (B.3) ATTICE EQUATIONS ARISING FROM DISCRETE PAINLEV´E SYSTEMS. II 29
The root lattices Q ( A (1)2 ) = L i = Z β i and Q ( A (1)1 ) = L i = Z γ i are of A (1)2 - and A (1)1 -types,respectively: ( b i j ) i , j = = − − − − − − , ( c i j ) i , j = = − − ! , (B.4)where b i j = β i | β j )( β j | β j ) , c i j = γ i | γ j )( γ j | γ j ) . (B.5)Let us discuss Cremona transformations for Q (( A + A ) (1) ). The transformations w i , i = , , r i , i = ,
1, and π , defined by (3.1), act on Q (( A + A ) (1) ) as the following:( β , β , β , γ , γ ) . w = ( − β , β + β , β + β , γ , γ ) , (B.6a)( β , β , β , γ , γ ) . w = ( β + β , − β , β + β , γ , γ ) , (B.6b)( β , β , β , γ , γ ) . w = ( β + β , β + β , − β , γ , γ ) , (B.6c)( β , β , β , γ , γ ) . r = ( β , β , β , − γ , γ + γ ) , (B.6d)( β , β , β , γ , γ ) . r = ( β , β , β , γ + γ , − γ ) , (B.6e)( β , β , β , γ , γ ) .π = ( β , β , β , γ , γ ) . (B.6f)The transformations w i , i = , ,
2, correspond to the reflections for the simple roots β i , i = , ,
2, respectively, that is, they satisfy v . w i = v − v | β i )( β i | β i ) β i , i = , , , (B.7)for all v ∈ Pic( X ). Moreover, the transformation π corresponds to the automorphism of theDynkin diagram:( d , d , d , d , d ; β , β , β ; γ , γ ) .π = ( d , d , d , d , d ; β , β , β ; γ , γ ) . (B.8)Note that there are no Cremona transformations correspond to the reflections for the simpleroots γ i , i = ,
1, since 2( h | γ i )( γ i | γ i ) γ i = − γ i ∈ / Pic( X ) . (B.9)From the fundamental relations (A.14), we can verified that the group of transformations h w , w , w , r , r , π i satisfy the following relations: w i = ( w i w i ± ) = , r = r = ( r r ) ∞ = , π = , (B.10a) r w i = w − i r , r w i = w − i + r , π w i = w − i π, π r = r π, (B.10b)where i ∈ Z / Z . We note that the relation ( ww ′ ) ∞ = w and w ′ meansthat there is no positive integer N such that ( ww ′ ) N =
1. Therefore, transformation group h w , w , w , r , r , π i forms the extended a ffi ne Weyl group of type ( A ⋊ A ) (1) , denotedby e W (( A ⋊ A ) (1) ). Here, W ( A (1)2 ) = h w , w , w i and W ( A (1)1 ) = h r , r i form a ffi ne Weylgroups of types A (1)2 and A (1)1 , respectively. Moreover, W (( A ⋊ A ) (1) ) = h w , w , w , r , r i is the semi direct product of W ( A (1)2 ) and W ( A (1)1 ).The transformations ρ i , i = , . . . ,
4, defined by (3.5) are translations on Q (( A + A ) (1) )since they act on Q (( A + A ) (1) ) as the following:( β , β , β , γ , γ ) .ρ = ( β , β , β , γ , γ ) + ( − , , , , − δ, (B.11a)( β , β , β , γ , γ ) .ρ = ( β , β , β , γ , γ ) + (0 , , − , , − δ, (B.11b)( β , β , β , γ , γ ) .ρ = ( β , β , β , γ , γ ) + (1 , − , , , − δ, (B.11c)( β , β , β , γ , γ ) .ρ = ( β , β , β , γ , γ ) + (0 , , , − , δ. (B.11d) Note that ρ i , i = , . . . ,
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