Discrete Power Functions on a Hexagonal Lattice I: Derivation of defining equations from the symmetry of the Garnier System in two variables
Nalini Joshi, Kenji Kajiwara, Tetsu Masuda, Nobutaka Nakazono
aa r X i v : . [ n li n . S I] A ug DISCRETE POWER FUNCTIONS ON A HEXAGONAL LATTICE I:DERIVATION FROM ABS EQUATIONS AND GARNIER SYSTEMS
NALINI JOSHI, KENJI KAJIWARA, TETSU MASUDA, AND NOBUTAKA NAKAZONOA bstract . We derive the cross-ratio equations and similarity constraint that lead to dis-crete power functions and associated circle patterns on a hexagonal lattice from two start-ing points: the ABS equations and the Garnier systems. This provides a di ff erent origin forthe equations used in earlier studies and moreover provides a new correspondence betweenthe ABS and Garnier systems.
1. I ntroduction
Following the revival of the study of circle packings by Thurston [21, 23], and relateddevelopments, we consider the explicit construction of circle patterns from a geometricviewpoint based on reflection groups. Given a planar graph, a circle pattern is generatedby inscribing each face in a circle. (See Figure 1.1 for an example.)Figure 1.1. A circle pattern on a planar graph.Recent interest is due to their relation to analytic functions, including discrete conformalmappings and discrete power functions. A limited number of explicit examples are knownin the literature, and they turn out to be related to integrable systems. In this paper, weshow for the first time how to relate the Garnier system and the Adler-Bobenko-Suris(ABS) lattice equations to a discrete power function on a hexagonal lattice proposed byAgafonov, Bobenko et al. [4–6].The solutions of these equations in the complex plane give rise to kite-shaped quadrilat-erals (see Figure 1.2), which can be associated with intersecting circles. Here, the param-eters of equation (1.1) define the angle at which the circles intersect and the solutions givethe combinatorial construction of circle patterns shown in Figure 1.3.
Mathematics Subject Classification.
Key words and phrases. discrete power function; Garnier system; a ffi ne Weyl group; projective reduction;ABS equation; τ function . Figure 1.2. The shaded region arising from two intersecting circles is a kite-shaped quadri-lateral.In our geometric approach, the dynamics of the primary equations, (1.1) and (1.2), aredescribed by the actions of an a ffi ne Weyl group. This geometric perspective was used inour previous study of the discrete power function on a square lattice associated with thePainlev´e VI equation (P VI ) [14]. In that study, the starting point was the cross-ratio equationand similarity constraint, both well-known from the theory of P VI [18], whose symmetrygroup is given by the extended a ffi ne Weyl group of type D (1)4 denoted by e W ( D (1)4 ). Weshowed that the lattice structure associated with the discrete power function is given by e W (3 A (1)1 ), which is a subgroup of e W ( D (1)4 ) and of e W ( B (1)3 ). The latter group is associatedwith the cubic lattice and starting with the ABS equations consistently placed on a 3-cube,we showed how the discrete power function arose from certain reflection hyperplanes.In this paper, we extend this perspective to the 4-cube with the symmetry group e W ( B (1)4 ),which includes e W (4 A (1)1 ) as a subgroup. We derive a system of di ff erence equations con-sisting of three cross-ratio equations:( f l , l , l − f l + , l , l )( f l + , l + , l − f l , l + , l )( f l + , l , l − f l + , l + , l )( f l , l + , l − f l , l , l ) = x , (1.1a)( f l , l , l − f l , l + , l )( f l , l + , l + − f l , l , l + )( f l , l + , l − f l , l + , l + )( f l , l , l + − f l , l , l ) = x , (1.1b)( f l , l , l − f l , l , l + )( f l + , l , l + − f l + , l , l )( f l , l , l + − f l + , l , l + )( f l + , l , l − f l , l , l ) = x , (1.1c)and a similarity constraint: α f l , l , l = ( l − α ) ( f l + , l , l − f l , l , l )( f l , l , l − f l − , l , l ) f l + , l , l − f l − , l , l + ( l − α ) ( f l , l + , l − f l , l , l )( f l , l , l − f l , l − , l ) f l , l + , l − f l , l − , l + ( l − α ) ( f l , l , l + − f l , l , l )( f l , l , l − f l , l , l − ) f l , l , l + − f l , l , l − , (1.2)from the birational representation of e W (4 A (1)1 ). Here, l , l , l are integers, and x i , i = , , α j , j = , ,
3, and α are complex parameters satisfying x x x =
1. A solution f l , l , l ofEquations (1.1) and (1.2), which satisfies the initial conditions f , , = , f , , = e c ( a + a ) √− , f , , − = e ca √− , with the identification x j = e a j √− ( a j > , a + a + a = π ), j = , ,
3, and α = c / < c < α = α = α =
0, is associated with a hexagonal circle pattern [4, Definition1]. We refer to such solutions as discrete power functions. See Figure 1.3 for an examplewith a i = π/ i = , ,
3, and c = / Figure 1.3. A hexagonal circle pattern.Surprisingly, we show that the same system of di ff erence equations arises from theGarnier system in two variables (see [16, 22] for details of the Garnier system). This leadsto an unexpected connection between the ABS and Garnier systems. Our main result isexpressed by the following theorem. Note that the term B¨acklund transformations refers toactions of the respective symmetry groups. Theorem 1.1.
The system of equations (1.1) and (1.2) can be derived from (i)
B¨acklund transformations of the system (2.1) of ABS equations; (ii)
B¨acklund transformations of the Garnier system (3.1) , with Hamiltonian given by (3.2) . The proofs of (i) and (ii) in Theorem 1.1 are given in Section 2.2 and Section 3.4 re-spectively.It is known that the Garnier system in two variables admits special solutions expressedby Appel’s hypergeometric functions. Consequently, we expect to deduce an explicit for-mula of the discrete power function in terms of the hypergeometric τ -functions of theGarnier system. This will be reported in detail in a separate paper.1.1. Notation.
We refer the reader to the standard notation for reflection groups, Coxetergroups and associated lattices in [12]; see also [14].Where direct sums of the same type of groups occur, we use an integer coe ffi cient as anabbreviation, e.g., A ⊕ A ⊕ A ⊕ A = : 4 A . For conciseness, we replace ⊕ by + , e.g., D ⊕ A ⊕ A = D + A .1.2. Plan of the paper.
The plan of this paper is as follows. In Section 2 we introducethe system of equations (2.1) which has a property called multi-dimensional consistencyon a cubic lattice. Moreover, we give a nontrivial symmetry of the system (2.4) and (2.5),which is a special case of the system (2.1), and then we construct the equations (1.1) and(1.2) from this symmetry. In Section 3 we derive the equations (1.1) and (1.2) from thesymmetries of the Garnier system in two variables. The relationship between the ABSequations under consideration and the Garnier system is revealed in Section 4. Concludingremarks are then given in Section 5.
NALINI JOSHI, KENJI KAJIWARA, TETSU MASUDA, AND NOBUTAKA NAKAZONO
2. D erivation from
ABS equations
Adler, Bobenko and Suris [1,2] and Boll [7–9] classified quad-equations that are placedon faces of cubes in such a way that, given initial values on 4 vertices of the cube, the so-lutions of the equations are uniquely defined at each remaining vertex. Such equations aresaid to be consistent-around-a-cube (CAC) [1, 17]. Integrable partial di ff erence equationsarise by iterating such consistent quad-equations on neighbouring cubes that tile the lattice Z . These are collectively called the ABS equations.Such equations also extend to multiple dimensions by extending to hypercubes. Inthis setting, the associated equations are said to be multi-dimensionally consistent, if theequations on every sub-cube of each n -dimensional hypercube have the CAC property.Extension to n -dimensional hypercubes that are space-filling polytopes in the lattice Z n leads to a system of ABS equations in n -variables.In this section, we consider the system of ABS equations (2.1), which is an extensionof the system discussed in [14] to four dimensions. Moreover, we consider nontrivialsymmetries of the special case of the system (2.1) and then construct the system (1.1) and(1.2), which define the discrete power function on a hexagonal lattice, from the symmetrieswe find.2.1. System of ABS equations (Q1, H1).
Consider the 4-dimensional cubic lattice con-structed by a space-filling of 4-dimensional hypercube (see Figure 2.1). We place thefollowing equations on each respective face of a sub-3-cube of each 4-cube:( z l , l , l , l + z l + , l , l , l )( z l , l + , l , l + z l + , l + , l , l )( z l , l , l , l + z l , l + , l , l )( z l + , l + , l , l + z l + , l , l , l ) = µ (1) l µ (2) l , (2.1a)( z l , l , l , l + z l , l + , l , l )( z l , l , l + , l + z l , l + , l + , l )( z l , l , l , l + z l , l , l + , l )( z l , l + , l + , l + z l , l + , l , l ) = µ (2) l µ (3) l , (2.1b)( z l , l , l , l + z l , l , l + , l )( z l + , l , l , l + z l + , l , l + , l )( z l , l , l , l + z l + , l , l , l )( z l + , l , l + , l + z l , l , l + , l ) = µ (3) l µ (1) l , (2.1c) z l , l , l , l + z l + , l , l , l ! ( z l , l , l , l + + z l + , l , l , l + ) = − µ (1) l µ (0) l , (2.1d) z l , l , l , l + z l , l + , l , l ! ( z l , l , l , l + + z l , l + , l , l + ) = − µ (2) l µ (0) l , (2.1e) z l , l , l , l + z l , l , l + , l ! ( z l , l , l , l + + z l , l , l + , l + ) = − µ (3) l µ (0) l , (2.1f)where { . . . , µ ( i ) − , µ ( i )0 , µ ( i )1 , . . . } i = , , , are parameters. We can verify that the system of equa-tions (2.1) is multidimensionally consistent in the 4-cubic lattice. Note that each equation(2.1a)-(2.1c) and each equation (2.1d)-(2.1f) are called Q1 and H1 respectively in the ABSlist [1, 2, 7–9].In this paper we consider a special case of the system (2.1) obtained as follows. Let µ (1) l = x , µ (2) l = , µ (3) l = x , µ (0) l = ( α + l )( α + l + , (2.2)and set x = x x . (2.3)Then, the system (2.1) can be rewritten as the following:( z l , l , l , l + z l + , l , l , l )( z l , l + , l , l + z l + , l + , l , l )( z l , l , l , l + z l , l + , l , l )( z l + , l + , l , l + z l + , l , l , l ) = x , (2.4a) Figure 2.1. A 4-dimensional hypercube. Note that u = z l , l , l , l and the subscript i means + l i -direction, e.g. u = z l + , l , l , l , u = z l + , l , l , l + and u = z l + , l + , l + , l + .( z l , l , l , l + z l , l + , l , l )( z l , l , l + , l + z l , l + , l + , l )( z l , l , l , l + z l , l , l + , l )( z l , l + , l + , l + z l , l + , l , l ) = x , (2.4b)( z l , l , l , l + z l , l , l + , l )( z l + , l , l , l + z l + , l , l + , l )( z l , l , l , l + z l + , l , l , l )( z l + , l , l + , l + z l , l , l + , l ) = x , (2.4c) z l , l , l , l + z l + , l , l , l ! ( z l , l , l , l + + z l + , l , l , l + ) = − ( α + l )( α + l + x , (2.5a) z l , l , l , l + z l , l + , l , l ! ( z l , l , l , l + + z l , l + , l , l + ) = − ( α + l )( α + l + , (2.5b) z l , l , l , l + z l , l , l + , l ! ( z l , l , l , l + + z l , l , l + , l + ) = − x ( α + l )( α + l + . (2.5c)2.2. Symmetry of the system of ABS equations.
In this subsection, we show that thesystem of equations (2.4) and (2.5) admits the symmetry isomorphic to the extended a ffi neWeyl group of type 4 A (1)1 . In our previous work [14], we clarified that the symmetry of thesystem of equations (2.4a), (2.5a) and (2.5b) can be described by the group of type 3 A (1)1 ,which inspires the idea in this subsection. Moreover, we give the proof of (i) in Theorem1.1.Let us consider the parameters α ki ( k = , . . . , , i = , x j ( j = , ,
3) satisfyingthe following conditions: α k = − α k , k = , . . . , , x x x = , (2.6)and define z -variables z i ( i = − , . . . ,
3) satisfying the following conditions: α + α + α − α − α z + α + α + α + α − α z = α ( z − z )( z − + z ) z − z − + α ( z − z )( z − + z ) z − z − , (2.7a) α + α − α − α − α z + α + α − α + α − α z − = α ( z − z − )( z − + z ) z − z − + α ( z − z − )( z − + z ) z − z − . (2.7b)Equations (2.7) and (2.13) are required to guarantee a consistent set of symmetry relationsbelow. Note that the number of essential parameters is 7 and that of essential z -variables is5. We now define the transformations s ki ( k = , . . . , , i = , π k ( k = , . . . , σ , and σ as follows. The definitions are motivated by our aim to find symmetries of the ABSequations (2.4) and (2.5) as explained in the statement of Lemma 2.2.Their actions on the parameters α ki ( k = , . . . , , i = , x j ( j = , ,
3) are givenby s k : ( α k , α k ) ( − α k , − α k ) , s k : ( α k , α k ) (2 − α k , − α k ) , (2.8a) π k : ( α k , α k , α , α ) ( α k , α k , α , α ) , (2.8b) σ : ( α , α , α , α , x , x , x ) ( α , α , α , α , x − , x − , x − ) , (2.8c) σ : ( α , α , α , α , x , x , x ) ( α , α , α , α , x − , x − , x − ) , (2.8d)for k = , . . . ,
3. Moreover, actions on the z -variables are given by s k : ( z − k , z k ) ( z k , z − k ) , k = , , , (2.9a) s : ( z − , z − , z − , z , z , z , z ) ( z − − , z − − , z − − , z − , z − , z − , z − ) , (2.9b) σ : ( z − , z − , z , z ) ( z − , z − , z , z ) , (2.9c) σ : ( z − , z − , z , z ) ( z − , z − , z , z ) , (2.9d) π : z − x z ( z + z − ) − z − ( z + z ) z + z − x ( z + z − ) , z − x z − ( z + z ) − z ( z + z − ) z + z − − x ( z + z ) , z − α π ( z )( π ( z − ) + z ) π ( z − ) − π ( z ) + α π ( z )( π ( z − ) + z ) π ( z − ) − π ( z ) + ( − α + α + α − α − α ) z α π ( z − ) + z π ( z − ) − π ( z ) + α π ( z − ) + z π ( z − ) − π ( z ) − ( − α + α + α + α − α ) , z z , z z , z x z ( z + z ) − z ( z + z ) z + z − x ( z + z ) , z x z ( z + z ) − z ( z + z ) z + z − x ( z + z ) , (2.9e) π : z − x z − ( z + z ) − z ( z + z − ) z + z − − x ( z + z ) , z − x z ( z + z − ) − z − ( z + z ) z + z − x ( z + z − ) , z − α π ( z )( π ( z − ) + z ) π ( z − ) − π ( z ) + α π ( z )( π ( z − ) + z ) π ( z − ) − π ( z ) + ( α − α + α − α − α ) z α π ( z − ) + z π ( z − ) − π ( z ) + α π ( z − ) + z π ( z − ) − π ( z ) − ( α − α + α + α − α ) , z z , z π ( z ) , z z , z x z ( z + z ) − z ( z + z ) z + z − x ( z + z ) , (2.9f) π : z − α π ( z )( π ( z − ) + z ) π ( z − ) − π ( z ) + α π ( z )( π ( z − ) + z ) π ( z − ) − π ( z ) + ( α + α − α − α − α ) z α π ( z − ) + z π ( z − ) − π ( z ) + α π ( z − ) + z π ( z − ) − π ( z ) − ( α + α − α + α − α ) , z − x z ( z + z − ) − z − ( z + z ) z + z − x ( z + z − ) , z − x z − ( z + z ) − z ( z + z − ) z + z − − x ( z + z ) , z z , z π ( z ) , z π ( z ) , z z , (2.9g) π : z −
7→ − α α x (cid:16) π ( z − ) − s π ( z − ) (cid:17) + α π ( z − ) − s π ( z − ) + ( α + x z − s π ( z − ) , z −
7→ − α α x (cid:16) π ( z − ) − s π ( z − ) (cid:17) + α + z − s π ( z − ) + α x π ( z − ) − s π ( z − ) , z −
7→ − α α + x (cid:16) z − s π ( z − ) (cid:17) + α π ( z − ) − s π ( z − ) + α x π ( z − ) − s π ( z − ) , z
7→ − α α x ( z − z − ) + α z − z − + α x z − z − , z
7→ − α α − x (cid:16) π ( z − ) − z (cid:17) + α π ( z ) − π ( z − ) + α x π ( z ) − π ( z − ) , z
7→ − α α x (cid:16) π ( z ) − π ( z − ) (cid:17) + α − π ( z − ) − z + α x π ( z ) − π ( z − ) , z
7→ − α α x (cid:16) π ( z ) − π ( z − ) (cid:17) + α π ( z ) − π ( z − ) + ( α − x π ( z − ) − z . (2.9h)Note that we follow the convention that the parameters and z -variables not explicitly in-cluded in the actions listed in Equations (2.8) and (2.9) are the ones that remain unchangedunder the action of the corresponding transformation. That is, the transformation acts asan identity on those parameters or variables. The action of the remaining transformations s k , k = , ,
3, and s on the z -variables are defined by s = π s π , s = π s π , s = π s π , s = π s π . (2.10)These transformations collectively form the extended a ffi ne Weyl group of type 4 A (1)1 : e W (4 A (1)1 ) = W (4 A (1)1 ) ⋊ h π , π , π , π , σ , σ i , (2.11)where W (4 A (1)1 ) is the a ffi ne Weyl group of type 4 A (1)1 : W (4 A (1)1 ) = h s , s i × h s , s i × h s , s i × h s , s i . (2.12)Indeed, e W (4 A (1)1 ) satisfies the following fundamental relations:( s ki ) = , k = , . . . , , i = , , (2.13a)( s k s k ) ∞ = , k = , . . . , , (2.13b)( s ki s lj ) = , k , l ∈ { , . . . , } , k , l , i , j ∈ { , } , (2.13c)( σ ) = ( σ ) = ( σ σ ) = , (2.13d)( π k ) = , k = , . . . , , (2.13e)( π k π l ) = , k , l ∈ { , . . . , } , k , l , (2.13f) π k s k { , } = s k { , } π k , k = , . . . , , (2.13g) π k s li = s li π k , k , l ∈ { , . . . , } , k , l , i ∈ { , } , (2.13h) σ s { , , , } i = s { , , , } i σ , i ∈ { , } , (2.13i) σ s { , , , } i = s { , , , } i σ , i ∈ { , } , (2.13j) σ π { , , , } = π { , , , } σ , σ π { , , , } = π { , , , } σ . (2.13k)Note that for the z -variables the extended parts are modified to become σ π ( z k ) = x π σ ( z k ) , σ π ( z k ) = x x π σ ( z k ) , (2.14)where k = − , . . . ,
3. Here, the relation ( ww ′ ) ∞ = w and w ′ meansthat there is no positive integer N such that ( ww ′ ) N = NALINI JOSHI, KENJI KAJIWARA, TETSU MASUDA, AND NOBUTAKA NAKAZONO
Figure 2.2. Coxeter diagram for W (4 A (1)1 ) and its automorphisms. Remark 2.1.
The transformations s ki ( k = , . . . , , i = , are reflections with respect tothe A (1)1 type root parameters α ki . The remaining transformations are automorphisms ofthe Coxeter diagram (see Figure 2.2). The translations in e W (4 A (1)1 ) are given by ρ = π s , ρ = π s , ρ = π s , ρ = s π , (2.15)whose actions on the parameters are given by ρ k : ( α k , α k , α , α ) ( α k + , α k − , α , α ) , k = , . . . , . (2.16)Let z l , l , l , l = ρ l ρ l ρ l ρ l ( z ) . (2.17)Then, we can verify that the z -variables can be expressed by z − = z , , − , , z − = z , − , , , z − = z − , , , , z = z , , , , (2.18a) z = z , , , , z = z , , , , z = z , , , . (2.18b) Lemma 2.2.
The extended a ffi ne Weyl group of type A (1)1 describes the symmetry of theABS equations (2.4) and (2.5) .Proof. From the action of ρ i , i = , , ,
3, we obtain( z + z )( z + ρ ( z ))( z + z )( ρ ( z ) + z ) = x , z + z ! (cid:16) ρ ( z ) + ρ ( z ) (cid:17) = − α ( α + x , (2.19a)( z + z )( z + ρ ( z ))( z + z )( ρ ( z ) + z ) = x , z + z ! (cid:16) ρ ( z ) + ρ ( z ) (cid:17) = − α ( α + , (2.19b)( z + z )( z + ρ ( z ))( z + z )( ρ ( z ) + z ) = x , z + z ! (cid:16) ρ ( z ) + ρ ( z ) (cid:17) = − x α ( α + , (2.19c)which give Equations (2.4) and (2.5). Moreover, the transformations s ki ( k = , . . . , , i = ,
1) and π k ( k = , . . . ,
3) are reflections of the 4-dimensional cubic lattice given by thesystem of equations (2.4) and (2.5) with the following hyperplanes: s : l = , π : l = , s : l = , s : l = , π : l = , s : l = , s : l = , π : l = , s : l = , s : l = , π : l = − , s : l = − , (2.20)and the transformations σ and σ change the axes of the 4-dimensional cubic lattice asthe following σ : l ↔ l , σ : l ↔ l . (2.21)Therefore, we have completed the proof. (cid:3) We are now ready to prove (i) in Theorem 1.1, that is, Equations (1.1) and (1.2) can bederived from the birational action of e W (4 A (1)1 ). Indeed, letting f l , l , l = ( − l + l + l z l , l , l , = ( − l + l + l ρ l ρ l ρ l ( z ) , (2.22)then, Equations (1.1) follow from Equations (2.4). Moreover, from the condition (2.7), weobtain α z = − α ( z + z )( z + z − ) z − z − − α ( z + z )( z + z − ) z − z − − α ( z + z )( z + z − ) z − z − , (2.23)which gives Equation (1.2).3. D erivation from the G arnier system in two variables In this section, we derive Equations (1.1) and (1.2) from the bilinear relations for τ -variables of the Garnier system in two independent variables. The first two subsectionsare devoted to formulating the symmetries of the Garnier system. Then we introduce τ -variables through the Hamiltonians of the Garnier system and derive bilinear relationsamong the τ -variables. On the basis of these results, we derive Equations (1.1) and (1.2).3.1. Garnier system in two variables and its symmetry.
The Garnier system in twovariables [10, 13, 16] is equivalent to the Hamiltonian system of partial di ff erential equa-tions ∂ q j ∂ t i = ∂ H i ∂ p j , ∂ p j ∂ t i = − ∂ H i ∂ q j , ( i , j = , , (3.1)with the Hamiltonians t i ( t i − H i = q i ( q p + q p + α )( q p + q p + α + κ ∞ ) + t i p i ( q i p i − θ i ) − t j ( t i − t i − t j ( q j p j − θ j ) q i p j − t i ( t i − t i − t j ( q i p i − θ i ) q j p i − t i ( t j − t j − t i q j p j ( q i p i − θ i ) − t i ( t j − t j − t i q i p i ( q j p j − θ j ) − ( t i + q i p i − θ i ) q i p i + ( κ t i + κ − q i p i , ( i , j ) = (1 , , (2 , , (3.2)where α = −
12 ( θ + θ + κ + κ + κ ∞ − σ : θ ↔ θ , t ↔ t , q ↔ q , p ↔ p ,σ : θ ↔ κ , t t − t t − , t t t − , q t − t t ( t − q , q t − t ( g t − , p t − t − t ( t p − t p ) , p (1 − t ) p ,σ : κ ↔ κ , t i t i , q i q i t i , p i t i p i ,σ : κ ↔ κ ∞ , t i t i t i − , q i q i g − , p i ( g − p i − α − q p − q p ) , (3.3)where g t = q t + q t , g = q + q . (3.4) Note that these transformations act on the parameters θ = ( θ , θ , κ , κ , κ ∞ ) by transpos-ing adjacent pairs, and generate the group isomorphic to the symmetric group S . Thefollowing transformations r i : θ i
7→ − θ i , p i p i − θ i q i ( i = , , r : κ
7→ − κ , p i p i − κ t i ( g t − , r : κ
7→ − κ , p i p i − κ g − , r : κ ∞
7→ − κ ∞ , (3.5)are also the B¨acklund transformations for the Garnier system. These satisfy the fundamen-tal relations( r i ) = i = , . . . , , r i r j = r j r i ( i , j ) , σ r i σ − = r σ ( i ) ( σ ∈ S ) . (3.6)Further there exists the transformation [24] r : θ i
7→ − θ i ( i = , , κ
7→ − κ + , κ
7→ − κ + , κ ∞
7→ − κ ∞ , q i t i p i ( q i p i − θ i )( q p + q p + α )( q p + q p + α + κ ∞ ) , q i p i
7→ − q i p i . (3.7)We have the fundamental relations( r ) = , r i r = r r i ( i , , . (3.8)and r σ = σ r , r σ ab = σ ab r ( a , b , , , (3.9)where σ ab = σ ba ∈ S is a transposition for mutually distinct indices a , b ∈ { , . . . , } . Remark 3.1.
The group generated by r i (i = , . . . , ), σ ab (a , b ∈ { , , . . . , } ), andr constitute a symmetry group of the Garnier system. This group includes the a ffi neWeyl group of type B (1)5 [22]. In this sense, the above group can be regarded as a certainextension of the a ffi ne Weyl group of type B (1)5 , but we note that the full symmetry group ofthe Garnier system has not yet been completely identified. Based on r , for later convenience, we introduce the transformations r ab = r ba for otherpairs of distinct subscripts by r σ ( a ) σ ( b ) = σ r ab σ − ( σ ∈ S ) , (3.10)where a , b ∈ { , . . . , } , a , b . It is easy to see that we have( r ab ) = , r i r ab = r ab r i ( i , a , b ) . (3.11)The action of r ab on the parameters and the dependent variables is given by r ab : θ a
7→ − θ a + , θ b
7→ − θ b + , θ c
7→ − θ c ( c , a , b ) , q i Q ( ab ) i , p i P ( ab ) i , (3.12)where ( θ , θ , θ ) = ( κ , κ , κ ∞ ), and Q ( ab ) i , P ( ab ) i ( i = ,
2) are certain rational functions inthe canonical variables q i , p i ( i = , T ab = r a r b r c r d r e r ab , T ab = r a r c r d r e r ab r b , (3.13)where a , b , c , d , e ∈ { , . . . , } are mutually distinct. These transformations commute witheach other, and act on the parameters by T ab : θ a θ a + , θ b θ b + , θ c θ c ( c , a , b ) , T ab : θ a θ a + , θ b θ b − , θ c θ c ( c , a , b ) . (3.14) Transformations of the Hamiltonians.
One can introduce the τ -variables throughthe Hamiltonians [13] in a similar manner to the Painlev´e equations (see, for example,[19, 20]). For this purpose, we here describe the actions of r i ( i = , . . . ,
5) and r on theHamiltonians.Following [22], we first introduce the modified Hamiltonians h (3) i ( i = ,
2) by addingthe correction terms which do not depend on p i , q i to the Hamiltonian H (3.2) as h (3)1 = H − θ θ t − − t − t ! + θ ( κ − t − t − ! − θ κ t − + θ t − t − + t − t ! − θ t + t − − t − t ! + ( κ − t − t − − t − t ! − κ t + t − + t − t ! − κ ∞ t − t − + t − t ! , (3.15)and h (3)2 = σ ( h (3)1 ). Then we have r i ( h (3) j ) = h (3) j ( i , , r ( h (3) j ) = h (3) j − κ q j t j ( g t − + κ t − t − − t − t ! . (3.16)We see that the 1-form ω = X j = h (3) j dt j , (3.17)is invariant with respect to the action of σ ab ( a , b , σ ab ( ω ) = ω ( a , b , . (3.18)These observations suggest further modified Hamiltonians obtained by applying σ ∈ S . We introduce the 1-form ω k , k = , . . . , , and the Hamiltonians h ( k ) j , j = , , k = , . . . , , by σ ( ω k ) = ω σ ( k ) ( σ ∈ S ) , (3.19)and ω k = X j = h ( k ) j dt j , (3.20)respectively. For a technical reason, we also define an auxiliary 1-form ω by ω = ω + r ( ω ) . (3.21)Then it follows that r i ( ω k ) = ω k , ( i , k ) , r ab ( ω a ) = ω b , r ab ( ω b ) = ω a . (3.22)It is possible to verify the following formulas among the above Hamiltonians by directcomputation: r ( h (1)1 ) + h (1)1 − r ( h (5)1 ) − h (5)1 − ∂∂ t log q = t − t − t ! , r ( h (1)2 ) + h (1)2 − r ( h (5)2 ) − h (5)2 − ∂∂ t log q = − t − t + t − ! , r ( h (3) j ) + h (3) j − r ( h (5) j ) − h (5) j − ∂∂ t j log q t + q t − ! = − t j − − t j ! , r ( h (4) j ) + h (4) j − r ( h (5) j ) − h (5) j − ∂∂ t j log( q + q − = −
13 1 t j − , (3.23) h (1)1 + r ( h (1)1 ) − h (4)1 − h (3)1 − ∂∂ t log( q p − θ ) = − t − t − − t − t ! , h (1)2 + r ( h (1)2 ) − h (4)2 − h (3)2 − ∂∂ t log( q p − θ ) = − t + t − + t − t ! , (3.24) r ( h (5) j ) + r r ( h (5) j ) − h (4) j − h (3) j − ∂∂ t j log( q p + q p + α ) = − t j − t j − + t j − t k ! , h (5) j + r ( h (5) j ) − h (4) j − h (3) j − ∂∂ t j log( q p + q p + α + κ ∞ ) = − t j − t j − + t j − t k ! . (3.25)From Equations (3.23), (3.24) and (3.25) we can deduce the following relations amongthe 1-forms ω k ( k = , . . . , d log q = r ( ω ) + ω − r ( ω ) − ω + dt t + dt t − − dt − dt t − t , d log q = r ( ω ) + ω − r ( ω ) − ω + dt t − + dt t − dt − dt t − t , d log q t + q t − ! = r ( ω ) + ω − r ( ω ) − ω + X j = t j − − t j ! dt j , d log( q + q − = r ( ω ) + ω − r ( ω ) − ω + X j = dt j t j − , (3.26)and d log( q p − θ ) = ω + r ( ω ) − ω − ω + t − t − ! dt + t + t − ! dt − dt − dt t − t , d log( q p − θ ) = ω + r ( ω ) − ω − ω + t + t − ! dt + t − t − ! dt − dt − dt t − t . (3.27)Furthermore, we have d log( q p + q p + α ) = ω + r r ( ω ) − ω − ω + X j = t j − t j − ! + dt − dt t − t , d log( q p + q p + α + κ ∞ ) = ω + r ( ω ) − ω − ω + X j = t j − t j − ! + dt − dt t − t . (3.28) τ -variables and bilinear relations. Here we introduce the τ -variables for the Gar-nier system and formulate the B¨acklund transformations for them; see also [22]. For thispurpose, we consider the action of the subgroup G = h r i , r ab i , i = , . . . , , a , b ∈ { , . . . , } . (3.29)Note that the independent variables t i ( i = ,
2) are invariant under the action of this sub-group.It is known that the 1-forms ω k ( k = , . . . ,
5) given in Equations (3.20) and (3.21) areclosed [13], so that one can define the τ -variables τ k ( k = , . . . ,
5) and τ by ω k = d log τ k ( k = , . . . , , ω = d log τ, (3.30)up to constant multiples. Below, we focus on deriving symmetry actions on these τ -variables.From Equations (3.22), (3.26) and (3.30), one can define the action of r i ( i = , . . . , τ k and the auxiliary variable τ by r i ( τ k ) = τ k ( i , k ) , r i ( τ ) = τ ( i = , . . . , , r ( τ ) = t − / ( t − − / ( t − t ) / q ττ , r ( τ ) = t − / ( t − − / ( t − t ) / q ττ , r ( τ ) = Y j = t / j ( t j − − / q t + q t − ! ττ , r ( τ ) = Y j = ( t j − − / (cid:0) q + q − (cid:1) ττ , r ( τ ) = ττ . (3.31)The actions of r ab ( a , b ∈ { , . . . , } ) on the τ -variables can be determined from Equations(3.27) and (3.28).To this end, we define some preliminary notations. For mutually distinct indices k , a , b ∈{ , . . . , } we define ϕ ab , k = ϕ ba , k by ϕ , k = q k p k − θ k ( k = , , ϕ , = − ( q p + q p + α + κ ∞ ) , (3.32)and σ ( ϕ ab , k ) = ϕ σ ( a ) σ ( b ) ,σ ( k ) ( σ ∈ S ) . (3.33)Further we introduce ϕ a ( a = , , ,
4) by ϕ = σ σ r ( g − g − ϕ , r ( ϕ , ) , ϕ = σ σ r ( g − g − ϕ , r ( ϕ , ) ,ϕ = σ r ( g − g − ϕ , r ( ϕ , ) , ϕ = σ r ( g t − g t − ϕ , r ( ϕ , ) , (3.34)where g and g t are given in Equation (3.4). Note that these are polynomials of the canon-ical variables q k , p k ( k = , r ab on the τ -variables can be expressed as follows r ab : τ a τ b , τ b τ a , τ k t ab , k ϕ ab , k τ a τ b τ k , k , a , b ,τ t ab , ϕ ab , r ( ϕ ab , ) τ a τ b τ , a , b , ,τ t ab ϕ ab τ a τ b τ , b = . (3.35)Here t ab , k = t ba , k and t ab = t ba are certain power products of t , t , t − , t − t − t defined in terms of χ ( e , e , e , e , e ) = t e t e ( t − e ( t − e ( t − t ) e , (3.36) and given by t , = χ ( − , − , − , − , ) , t , = χ ( − , − , − , − , ) , t , = χ ( − , − , − , − , ) , (3.37) t , = χ ( , − , − , , − ) , t , = √− χ ( , − , − , − , − ) , t , = χ ( , − , − , , − ) , (3.38) t , = − χ ( − , , − , , ) , t , = − χ ( − , , − , − , − ) , t , = − χ ( − , − , − , , − ) , (3.39) t , = − χ ( − , , , − , − ) , t , = − χ ( − , , − , − , − ) , t , = √− χ ( − , , , − , − ) , (3.40) t , = − √− χ ( , − , , − , − ) , t , = − χ ( , − , − , − , − ) , t , = − χ ( − , − , , − , − ) , (3.41) t , = √− χ ( − , − , , − , ) , t , = √− χ ( − , − , − , , ) , t , = √− χ ( − , − , , , − ) , (3.42) t , = χ ( − , , , − , − ) , t , = √− χ ( − , , , − , − ) , t , = √− χ ( − , − , , − , − ) , t = χ ( − , − , , , − ) , (3.43) t , = − χ ( , − , − , , − ) , t , = − χ ( , − , − , , − ) , t , = − χ ( − , − , − , , − ) , t = − χ ( − , − , , , − ) , (3.44) t , = χ ( − , − , − , − , ) , t , = χ ( − , − , − , − , ) , t , = χ ( − , − , − , − , − ) , t = χ ( − , − , , , − ) , (3.45) t , = − χ ( , − , − , − , ) , t , = − χ ( − , , − , − , ) , t , = − √− χ ( , , − , − , − ) , t = − χ ( − , − , , , − ) . (3.46)On the level of the τ -variables, the fundamental relations (3.11) are preserved exceptfor some modifications. First we have( r ab ) ( τ k ) = − τ k ( k , a , b ) . (3.47)The commutativity of the Schlesinger transformations T µ , T µ ′ ∈ (cid:8) T ab , T ab (cid:12)(cid:12)(cid:12) a , b ∈ { , . . . , } , a , b (cid:9) are modified by T µ T µ ′ ( τ k ) = c µ,µ ′ T µ ′ T µ ( τ k ) ( k = , . . . , , T µ T µ ′ ( τ ) = c µ,µ ′ T µ ′ T µ ( τ ) , (3.48)where c µ,µ ′ are certain constants depending on the pair ( µ, µ ′ ).One can derive the various bilinear relations from the above formulation. It is easy toverify that we have r k ( ϕ ab , k ) − ϕ ab , k = θ k . (3.49)Substituting t ab , k ϕ ab , k = τ k r ab ( τ k ) τ a τ b into the above formula, we get the bilinear relations τ k , k r ab ( τ k , k ) − τ k r ab ( τ k ) = θ k t ab , k τ a τ b (3.50)for mutually distinct indices a , b , k ∈ { , . . . , } , where τ k , k = r k ( τ k ). From the formulas(3.31) we get the bilinear relation Y j = t − / j ( t j − / τ τ , = t − / ( t − / ( t − t ) − / τ τ , + t − / ( t − / ( t − t ) − / τ τ , − τ τ , . (3.51) Derivation of the equations.
Here we derive the cross-ratio equations (1.1) and thesimilarity constraint (1.2) from the bilinear relations for the τ -variables of the Garniersystem. The crucial point is how to choose the appropriate dependent variable. Suggestedby our preceding works [3, 11], here we choose the variable f in terms of the τ -variablesof the Garnier system by f = τ , τ . (3.52)Note that the action of the Schlesinger transformations T ab , T ab ( a , b ∈ { , . . . , } ) on f commute with each other, since the variable f is defined by a ratio of τ -variables. Proposition 3.2.
The variable f satisfies the cross-ratio equationsf − T − ( f ) T − ( f ) − T − ( f ) T − ( f ) − T − ( f ) T − ( f ) − f = − t − t , f − T − ( f ) T − ( f ) − T − ( f ) T − ( f ) − T − ( f ) T − ( f ) − f = − t , f − T − ( f ) T − ( f ) − T − ( f ) T − ( f ) − T − ( f ) T − ( f ) − f = − t . (3.53) Proof.
The bilinear relations (3.50) with ( a , b , k ) = (1 , ,
5) and (2 , ,
5) yield f − T − ( f ) = − κ ∞ t − / t − / ( t − − / ( t − / ( t − t ) − / τ τ τ r ( τ , ) , (3.54) f − T − ( f ) = − κ ∞ t − / t − / ( t − / ( t − − / ( t − t ) − / τ τ τ r ( τ , ) , (3.55)from which we have f − T − ( f ) f − T − ( f ) = ( t − − / ( t − / τ r ( τ , ) τ r ( τ , ) . (3.56)Applying the transformation T − r to the relation (3.54), we get T − ( f ) − T − ( f ) = − κ ∞ t − / t − / ( t − − / ( t − / ( t − t ) − / r ( τ , ) τ r ( τ , ) r r r ( τ ) . (3.57)Similarly we obtain T − ( f ) − T − ( f ) = − κ ∞ t − / t − / ( t − / ( t − − / ( t − t ) − / r ( τ , ) τ r ( τ , ) r r r ( τ ) (3.58)by applying T − r to Equation (3.55). Due to the relation r ( τ , ) r r r ( τ ) = r ( τ , ) r r r ( τ ) , (3.59)we have T − ( f ) − T − ( f ) T − ( f ) − T − ( f ) = ( t − − / ( t − / τ r ( τ , ) τ r ( τ , ) . (3.60)The formulas (3.56) and (3.60) yield the first equation of Proposition 3.2.Similar computations give the other equations of Proposition 3.2. Although they arerepetition of the similar process, we show the derivations in detail for readers’ convenience.First we show the third equation in Equation (3.53). The bilinear relation (3.50) with( a , b , k ) = (3 , ,
5) gives f − T − ( f ) = √− κ ∞ t − / t − / ( t − / ( t − / ( t − t ) − / τ τ τ r ( τ , ) . (3.61) Then we have f − T − ( f ) f − T − ( f ) = √− t − − / τ r ( τ , ) τ r ( τ , ) . (3.62)from Equations (3.54) and (3.61). Applying the transformation T − r to the relation (3.54),we get T − ( f ) − T − ( f ) = − κ ∞ t − / t − / ( t − − / ( t − / ( t − t ) − / r ( τ , ) τ r ( τ , ) r r r ( τ ) . (3.63)Similarly we obtain T − ( f ) − T − ( f ) = √− κ ∞ t − / t − / ( t − / ( t − / ( t − t ) − / r ( τ , ) τ r ( τ , ) r r r ( τ ) (3.64)by applying T − r to Equation (3.61). Noticing the relation r ( τ , ) r r r ( τ ) = r ( τ , ) r r r ( τ ) , (3.65)we have T − ( f ) − T − ( f ) T − ( f ) − T − ( f ) = √− t − − / τ r ( τ , ) τ r ( τ , ) . (3.66)Equations (3.62) and (3.66) yield the third equation of Equation (3.53).Finally we show the second equation of Equation (3.53). We have from Equations (3.55)and (3.61) f − T − ( f ) f − T − ( f ) = √− t − − / τ r ( τ , ) τ r ( τ , ) . (3.67)Applying the transformation T − r to Equation (3.55), we get T − ( f ) − T − ( f ) = − κ ∞ t − / t − / ( t − / ( t − − / ( t − t ) − / r ( τ , ) τ r ( τ , ) r r r ( τ ) . (3.68)Similarly we obtain T − ( f ) − T − ( f ) = √− κ ∞ t − / t − / ( t − / ( t − / ( t − t ) − / r ( τ , ) τ r ( τ , ) r r r ( τ ) (3.69)by applying T − r to Equation (3.61). Then the relation r ( τ , ) r r r ( τ ) = r ( τ , ) r r r ( τ ) , (3.70)gives T − ( f ) − T − ( f ) T − ( f ) − T − ( f ) = √− t − − / τ r ( τ , ) τ r ( τ , ) . (3.71)Equations (3.67) and (3.71) yield the second equation of Equation (3.53), which completesthe proof of Proposition 3.2. (cid:3) Proposition 3.3.
The variable f satisfies the similarity constraint κ ∞ f = θ ( T ( f ) − f )( f − T − ( f )) T ( f ) − T − ( f ) + θ ( T ( f ) − f )( f − T − ( f )) T ( f ) − T − ( f ) + κ ( T ( f ) − f )( f − T − ( f )) T ( f ) − T − ( f ) . (3.72) Proof.
We express each term of the right hand side of Equation (3.72) in terms of the τ -variables. To this end, we start with the relations f − T − ( f ) = − κ ∞ t − / t − / ( t − − / ( t − / ( t − t ) − / τ τ τ r ( τ , ) , f − T ( f ) = − κ ∞ t − / t − / ( t − − / ( t − / ( t − t ) − / τ , τ τ r r ( τ , ) , (3.73)the first equation of which is Equation (3.54), and the second equation can be obtained byapplying r to the first. Then we get1 T ( f ) − f + f − T − ( f ) = − κ ∞ t / t / ( t − / ( t − − / ( t − t ) / × τ τ τ τ , (cid:16) τ , r ( τ , ) − τ r r ( τ , ) (cid:17) . (3.74)The bilinear relation (3.50) with ( a , b , k ) = (4 , , τ , r ( τ , ) − τ r r ( τ , ) = − θ t / t − / ( t − − / ( t − − / ( t − t ) / τ τ , (3.75)simplifies the right hand side of Equation (3.74) to give θ ( T ( f ) − f )( f − T − ( f )) T ( f ) − T − ( f ) = κ ∞ t − / ( t − / ( t − t ) − / τ τ , τ . (3.76)By the similar discussions one can express the other terms of the right-hand side ofEquation (3.72) by the τ -variables. For the second term we start with the relations f − T − ( f ) = − κ ∞ t − / t − / ( t − / ( t − − / ( t − t ) − / τ τ τ r ( τ , ) , f − T ( f ) = − κ ∞ t − / t − / ( t − / ( t − − / ( t − t ) − / τ , τ τ r r ( τ , ) , (3.77)where the first equation is Equation (3.55), and the second equation can be obtained byapplying r to the first. Then we get1 T ( f ) − f + f − T − ( f ) = − κ ∞ t / t / ( t − − / ( t − / ( t − t ) / × τ τ τ τ , (cid:16) τ , r ( τ , ) − τ r r ( τ , ) (cid:17) . (3.78)By using the bilinear relation (3.50) with ( a , b , k ) = (4 , , τ , r ( τ , ) − τ r r ( τ , ) = − θ t − / t / ( t − − / ( t − − / ( t − t ) / τ τ , (3.79)we obtain θ ( T ( f ) − f )( f − T − ( f )) T ( f ) − T − ( f ) = κ ∞ t − / ( t − / ( t − t ) − / τ τ , τ . (3.80)Similarly, starting with the relations f − T − ( f ) = √− κ ∞ t − / t − / ( t − / ( t − / ( t − t ) − / τ τ τ r ( τ , ) , f − T ( f ) = √− κ ∞ t − / t − / ( t − / ( t − / ( t − t ) − / τ , τ τ r r ( τ , ) , (3.81) where the first equation is Equation (3.61) and the second can be obtained by applying r to the first, we get1 T ( f ) − f + f − T − ( f ) = √− κ ∞ t / t / ( t − − / ( t − − / ( t − t ) / × τ τ τ τ , (cid:16) τ , r ( τ , ) − τ r r ( τ , ) (cid:17) . (3.82)Then the bilinear relation (3.50) with ( a , b , k ) = (4 , , τ , r ( τ , ) − τ r r ( τ , ) = − √− κ t / t / ( t − − / ( t − − / ( t − t ) − / τ τ , (3.83)gives κ ( T ( f ) − f )( f − T − ( f )) T ( f ) − T − ( f ) = − κ ∞ t − / t − / ( t − / ( t − / τ τ , τ . (3.84)Finally, Equations (3.76), (3.80) and (3.84) gives θ ( T − ( f ) − f )( f − T ( f )) T − ( f ) − T ( f ) + θ ( T − ( f ) − f )( f − T ( f )) T − ( f ) − T ( f ) + κ ( T − ( f ) − f )( f − T ( f )) T − ( f ) − T ( f ) = κ ∞ τ (cid:16) t − / ( t − / ( t − t ) − / τ τ , + t − / ( t − / ( t − t ) − / τ τ , − t − / t − / ( t − / ( t − / τ τ , (cid:17) , (3.85)which yields the similarity constraint (3.72) by using the bilinear relation (3.51) and Equa-tion (3.52). (cid:3) Derivation of equations of type H1 . In this subsection, we derive the ABS equationsof type H1 in Equation (2.5) from the bilinear relations.
Proposition 3.4.
The variable f satisfies the equations of type H1 f − T − ( f ) ( T ( f ) − T ( f )) = κ ∞ ( κ ∞ + χ ( − , − , − , , − ) , f − T − ( f ) ( T ( f ) − T ( f )) = κ ∞ ( κ ∞ + χ ( − , − , , − , − ) , f − T − ( f ) ( T ( f ) − T ( f )) = − κ ∞ ( κ ∞ + χ ( − , − , , , − ) . (3.86) where χ is given in Equation (3.36) .Proof. We start with the relations f − T − ( f ) = − κ ∞ t − / t − / ( t − − / ( t − / ( t − t ) − / τ τ τ r ( τ , ) , T ( f ) − T ( f ) = − ( κ ∞ + t − / t − / ( t − − / ( t − / ( t − t ) − / × r r ( τ , ) τ , τ r r r r ( τ , ) , (3.87)the first of which is Equation (3.54) and the second can be obtained by applying r T tothe first. Due to the relation r r ( τ , ) r r r r ( τ , ) = − r ( τ ) τ , (3.88) we have (cid:16) f − T − ( f ) (cid:17) ( T ( f ) − T ( f )) = − κ ∞ ( κ ∞ + t − / t − / ( t − − / ( t − / ( t − t ) − / f T − ( f ) , (3.89)which is equivalent to the first equation of Equation (3.86).The second and third equations can be also obtained in a similar way. Let us start withthe relations f − T − ( f ) = − κ ∞ t − / t − / ( t − / ( t − − / ( t − t ) − / τ τ τ r ( τ , ) , T ( f ) − T ( f ) = − ( κ ∞ + t − / t − / ( t − / ( t − − / ( t − t ) − / × r r ( τ , ) τ , τ r r r r ( τ , ) , (3.90)the first equation of which is Equation (3.55) and the second can be obtained by applying r T to to the first. Due to the relation r r ( τ , ) r r r r ( τ , ) = − r ( τ ) τ , (3.91)we have (cid:16) f − T − ( f ) (cid:17) ( T ( f ) − T ( f )) = − κ ∞ ( κ ∞ + t − / t − / ( t − / ( t − − / ( t − t ) − / f T − ( f ) , (3.92)which is equivalent to the second equation of Equation (3.86).Similarly we start with the relations f − T − ( f ) = √− κ ∞ t − / t − / ( t − / ( t − / ( t − t ) − / τ τ τ r ( τ , ) , T ( f ) − T ( f ) = √− κ ∞ + t − / t − / ( t − / ( t − / ( t − t ) − / × r r ( τ , ) τ , τ r r r r ( τ , ) , (3.93)the first of which is Equation (3.61) and the second can be obtained by applying r T toto the first. Due to the relation r r ( τ , ) r r r r ( τ , ) = − r ( τ ) τ , (3.94)we have (cid:16) f − T − ( f ) (cid:17) ( T ( f ) − T ( f )) = κ ∞ ( κ ∞ + t − / t − / ( t − / ( t − / ( t − t ) − / f T − ( f ) , (3.95)which is equivalent to the third equation of Equation (3.86). This completes the proof ofProposition 3.4. (cid:3)
4. R elationship between
ABS equations and G arnier system In this section we discuss the relationship between the symmetry group of the ABSequations given in Section 2 and that of the Garnier system given in Section 3. We showthat the Weyl group symmetry group of the ABS equations can be recovered from thesymmetries of the Garnier system given by G in Equation (3.29). The reader should notethat the actions s ki , π k , i = , k = , , ,
3, are defined below from actions given inSection 3 and shown after some specialization of parameters to lead to those used in Section2.
Let us define the transformations s ki ( i = , k = , , ,
3) and π k ( k = , , ,
3) by s k = r k ( k = , , , s = r ,π = r r r r , π = r r r r , π = r r r r , π = r r r r , s k = π k s k π k ( k = , , , s = π s π . (4.1)Then we have the fundamental relations( s ki ) = , s ki s lj = s lj s ki ( k , l ) , ( π k ) = , π k π l = π l π k ( k , l ) ,π k s li = s li π k ( k , l ) . (4.2)This means that for each k ∈ { , , , } , the transformations s k and s k generate the a ffi neWeyl group of type A (1)1 and π k is the automorphism of the Dynkin diagram, namely wehave h s k , s k i (cid:27) W ( A (1)1 ) ( k = , , , , h s k , s k , π k i (cid:27) e W ( A (1)1 ) ( k = , , , . (4.3)These four extended a ffi ne Weyl groups commute with each other. Let us introduce theparameters α ki ( i = , k = , , , ,
4) by α = θ , α = θ , α = κ , α = κ , α = κ ∞ ,α k = − α k ( k = , , , , α = − α . (4.4)Then the transformation s ki ( k = , , ,
3) can be regarded as the reflection with respect tothe simple root α ki ( k = , , , s k : α k
7→ − α k , α k α k + α k , s k : α k
7→ − α k , α k α k + α k . (4.5)The action of π k ( k = , , ,
3) on the parameters are given by π k : ( α k , α k , α , α ) ( α k , α k , α , α ) ( k = , , , . (4.6)The transformations ρ k ( k = , , ,
3) defined by ρ k = π k s k ( k = , , ,
3) (4.7)act on the root lattice Q ( A (1)1 ) (cid:27) M i = Z α ki ( k = , , ,
3) as the translation; ρ k : ( α k , α k ) ( α k + , α k − . (4.8)Note that we have T k T k = ( ρ k ) − ( k = , , , T T = ( ρ ) . (4.9)Let us introduce the variables f k ( k = , ± , ± , ±
3) by f = f , f ± k = ( ρ k ) ± ( f ) ( k = , , , (4.10)where f is given in Equation (3.52). Note that we have f k = T − k ( f ) , f − k = T k ( f ) ( k = , ,
3) (4.11)and ρ − ( f ) = T − ( f ) , ρ ( f ) = T ( f ) . (4.12) Lemma 4.1.
The variables f k ( k = , ± , ± , ± satisfy the relations α + α + α − α − α f − α + α + α − α + α f = α ( f − f − )( f − f ) f − f − + α ( f − f − )( f − f ) f − f − , − α + α + α − α − α f − − α + α + α − α + α f − = α ( f − f − )( f − f − ) f − f − + α ( f − f − )( f − f − ) f − f − , (4.13) α + α + α − α − α f − α + α + α − α + α f = α ( f − f − )( f − f ) f − f − + α ( f − f − )( f − f ) f − f − ,α − α + α − α − α f − α − α + α − α + α f − = α ( f − f − )( f − f − ) f − f − + α ( f − f − )( f − f − ) f − f − , (4.14) α + α + α − α − α f − α + α + α − α + α f = α ( f − f − )( f − f ) f − f − + α ( f − f − )( f − f ) f − f − ,α + α − α − α − α f − α + α − α − α + α f − = α ( f − f − )( f − f − ) f − f − + α ( f − f − )( f − f − ) f − f − . (4.15)We note that these relations are not independent. Proof.
We give a proof for the first relation of Equation (4.15) only, as other equations canbe proved in a similar manner. One can easily verify that α f − f − f − f − = − t − / t / ( t − / ( t − / ( t − t ) − / τ , r ( τ , ) τ τ (4.16)from the discussion of the previous section (see the proof of Proposition 3.3). We have alinear relation ( t − ϕ , + ϕ , − t ϕ , = , (4.17)which can be verified by using Equations (3.32), (3.33) and (3.3). Equation (4.17) impliesthe bilinear relation √− t − / τ r ( τ , ) + τ r ( τ , ) + t / τ r ( τ , ) = . (4.18)Then we compute by using Equations (3.73), (3.81) and (4.18) f − f = ( f − f ) − ( f − f ) = κ ∞ t − / t − / ( t − − / ( t − / ( t − t ) − / τ τ τ r ( τ , ) + √− κ ∞ t − / t − / ( t − / ( t − / ( t − t ) − / τ τ τ r ( τ , ) = κ ∞ t − / t − / ( t − − / ( t − / ( t − t ) − / τ τ r ( τ , ) r ( τ , ) × (cid:16) τ r ( τ , ) + √− t − / τ r ( τ , ) (cid:17) = − t / t − / ( t − − / ( t − / ( t − t ) − / α τ r ( τ , ) r ( τ , ) r ( τ , ) , (4.19)which yields α f − f − f − f − ( f − f ) = α t / ( t − / ( t − t ) − / τ , r ( τ , ) τ r ( τ , ) , (4.20)by multiplying Equation (4.16). Similarly we obtain α f − f − f − f − ( f − f ) = α t / ( t − / ( t − t ) − / τ , r ( τ , ) τ r ( τ , ) , (4.21)which is derived from α f − f − f − f − = − t / t − / ( t − / ( t − / ( t − t ) − / τ , r ( τ , ) τ τ , (4.22)and f − f = − t − / t / ( t − / ( t − − / ( t − t ) − / α τ r ( τ , ) r ( τ , ) r ( τ , ) . (4.23)Here, Equation (4.23) is derived in the same manner as Equation (4.19) by using the bilin-ear relation √− t − / τ r ( τ , ) + τ r ( τ , ) + t / τ r ( τ , ) = . (4.24)which follows from the linear relation( t − ϕ , + ϕ , − t ϕ , = . (4.25)Therefore we obtain from Equations (4.20), (4.21) and (3.81) α f − f − f − f − ( f − f ) + α f − f − f − f − ( f − f ) − α ( f − f ) = α τ r ( τ , ) (cid:16) t / ( t − / ( t − t ) − / τ , r ( τ , ) + t / ( t − / ( t − t ) − / τ , r ( τ , ) + √− α t − / t − / ( t − / ( t − / ( t − t ) − / τ τ (cid:17) , (4.26)where α = −
12 ( α + α + α − α + α ). We finally obtain the first relation of Equation(4.15) by using the bilinear relation t / ( t − / ( t − t ) − / τ , r ( τ , ) + t / ( t − / ( t − t ) − / τ , r ( τ , ) + i α t − / t − / ( t − / ( t − / ( t − t ) − / τ τ = − τ , r ( τ , ) , (4.27)which follows from the linear relation r ( ϕ , ) + r ( ϕ , ) + r ( ϕ , ) + α = . (4.28) (cid:3) Based on the above setup one can recover the symmetry of the ABS equations.
Proposition 4.2.
The actions of s ki , π k ( i = , k = , , , on the f -variables can bedescribed as follows.s ki : s k ( f k ) = f − k , s k ( f − k ) = f k ( k = , , , s ( f k ) = ( f k ) − ( k = , ± , ± , ± , (4.29) π : π ( f ) = f , π ( f ) = f , f − f − f − − π ( f − ) π ( f − ) − f f − f = − t , f − f f − π ( f ) π ( f ) − f f − f = − t , f − f − f − − π ( f − ) π ( f − ) − f − f − − f = − t − t , f − f f − π ( f ) π ( f ) − f f − f = − t − t ,α + α + α − α − α f − α + α + α − α + α π ( f − ) = α ( f − π ( f − ))( π ( f ) − π ( f − )) π ( f ) − π ( f − ) + α ( f − π ( f − ))( π ( f ) − π ( f − )) π ( f ) − π ( f − ) , (4.30) π : π ( f ) = f , π ( f ) = f , f − f − f − − π ( f − ) π ( f − ) − f f − f = − t − t , f − f f − π ( f ) π ( f ) − f f − f = − t − t , f − f f − π ( f − ) π ( f − ) − f − f − − f = − t , f − f f − π ( f ) π ( f ) − f f − f = − t ,α + α + α − α − α f − α + α + α − α + α π ( f − ) = α ( f − π ( f − ))( π ( f ) − π ( f − )) π ( f ) − π ( f − ) + α ( f − π ( f − ))( π ( f ) − π ( f − )) π ( f ) − π ( f − ) , (4.31) π : π ( f ) = f , π ( f ) = f , f − f f − π ( f − ) π ( f − ) − f − f − − f = − t , f − f f − π ( f ) π ( f ) − f f − f = − t , f − f − f − − π ( f − ) π ( f − ) − f f − f = − t , f − f f − π ( f ) π ( f ) − f f − f = − t ,α + α + α − α − α f − α + α + α − α + α π ( f − ) = α ( f − π ( f − ))( π ( f ) − π ( f − )) π ( f ) − π ( f − ) + α ( f − π ( f − ))( π ( f ) − π ( f − )) π ( f ) − π ( f − ) , (4.32) π : π ( f ) = χα α ( t − f − f − )( t − + α f − f − − α ( t − f − f − ,π ( f k ) = π k π ( f ) , π ( f − k ) = s k π ( f k ) ( k = , , , (4.33) where χ = t − / t − / ( t − / ( t − − / ( t − t ) − / . (4.34) Proof. (1) s ki (4.29): It can be verified directly by using Equations (3.31), (3.52), (4.1) and(4.10).(2) π : π ( f ), π ( f ) can be shown by the same manner as (1). π ( f ± ) and π ( f ± ) canbe determined by the cross-ratio equations in Proposition 3.2. In fact, by noticing that T ± ( f ) = π ( f ∓ ), T ± ( f ) = π ( f ∓ ), we see that the fourth and the sixth equations ofEquation (4.30) are the third and the first equations of Equation (3.53), respectively. Thethird equation is derived by applying T on the third equation of Equation (3.53). Thefifth equation is obtained by applying T on the first equation. The seventh equation canbe derived by applying T − on the second equation of Equation (4.13). We skip π , π since they can be verified in the similar manner.(3) π : π ( f ) can be determined from the bilinear relations. By subtracting the secondequation of Equation (3.73) from the first equation and using the biliner equation (3.75),we see that χκ ∞ θ f − f − = t − / ( t − t − − / ( t − t ) − / r τ τ , τ . (4.35)Similarly, we obtain χκ ∞ θ f − f − = t − / ( t − / ( t − t ) − / r τ τ , τ ,χκ ∞ κ f − f − = t − / t − / ( t − / ( t − − / r τ τ , τ . (4.36)Then the bilinear relation (3.51) implies χα α ( t − f − f − )( t − + α f − f − − α ( t − f − f − = r τ τ , τ = r ( f ) = π ( f ) . (4.37) π ( f k ) ( k = , ,
3) can be computed from f k = π k ( f ) and the commutativity of π and π k .Similarly, π ( f − k ) ( k = , ,
3) follows from f − k = s k ( f k ) and the commutativity of π and s k . (cid:3) The actions in Proposition 4.2 coincide with those for the ABS equations given in Sec-tion 2 under the correspondence x = − t − t , x = − t , x = − t (4.38)up to multiplication by the factor χ which can be easily absorbed. More precisely, the factor χ a ff ects the right hand side of the ABS equations of type H1, which can be absorbed bythe redefintion of the parameter of the equations. Remark 4.3.
For simplicity we avoid lifting the action of S introduced in Equation (3.3) to the τ -variables. However it is easy to see that the action of the transformations σ and σ on the f -variables coincides with one give in Section 2 up to multiplication by somenormalization factors.
5. C oncluding remarks
In this paper, we studied Equations (1.1) and (1.2) from a geometric viewpoint arisingfrom reflection groups. We showed that these equations can be constructed from symmetrygroups (composed of B¨acklund transformations) of the ABS and Garnier systems. Thiscreates a new bridge between these two systems. Moreover, we showed that the equations(1.1) and (1.2) admit a symmetry group that is isomorphic to the extended a ffi ne Weylgroup of type 4 A (1)1 .It is known that the Garnier system in two variables admits special solutions expressedby Appel’s hypergeometric functions. Consequently, we expect to deduce an explicit for-mula of the discrete power function in terms of the hypergeometric τ -functions of theGarnier system. This will be reported in detail in a separate paper. The two main systems from which explicit circle patterns have been constructed so farin the literature are P VI and the Garnier systems. However, the bridge with ABS equa-tions suggests the tantalizing prospect of connections between circle patterns and otherintegrable systems. The question of whether such connections exist remains open. Acknowledgment.
This research was supported by an Australian Laureate Fellowship eferences [1] V. E. Adler, A. I. Bobenko, and Y. B. Suris. Classification of integrable equations on quad-graphs. Theconsistency approach.
Comm. Math. Phys. , 233(3):513–543, 2003.[2] V. E. Adler, A. I. Bobenko, and Y. B. Suris. Discrete nonlinear hyperbolic equations: classification ofintegrable cases.
Funktsional. Anal. i Prilozhen. , 43(1):3–21, 2009.[3] H. Ando, M. Hay, K. Kajiwara, T. Masuda. An explicit formula for the discrete power function associatedwith circle patterns of schramm type.
Funkcial. Ekvac.
J. Math. Phys. , 44(8):3455–3469, 2003.[5] A. I.Bobenko and T. Ho ff mann. Hexagonal circle patterns and integrable systems: patterns with constantangles. Duke Math. Journal ff mann. Conformally symmetric circle packings. A generalization of Doyle spirals. Exp. Math.
J. Nonlinear Math. Phys. , 18(3):337–365, 2011.[8] R. Boll. Corrigendum: Classification of 3D consistent quad-equations.
J. Nonlinear Math. Phys. ,19(4):1292001, 3, 2012.[9] R. Boll. Classification and Lagrangian Structure of 3D Consistent Quad-Equations.
Doctoral Thesis, Tech-nische Universit¨at Berlin , 2012.[10] R. Garnier, Sur des ´equations di ff ´erentielles du troisi`eme ordre dont l´ınt´egrale g´en´erale est uniforme et surun classe d´equations nouvelles d´order sup´erieur dont l´ınt´egrale g´en´erale a ses points critiques fixes, Ann.Sci. ´Ecole Norm. Sup. (3)
29, 1-126, 1912 (French).[11] M. Hay, K. Kajiwara, T. Masuda. Bilinearization and special solutions to the discrete Schwarzian KdVequation.
J. Math-for-Ind.
Proc. R. Soc. A. , 473(2207):20170312, 19 pages, 2017.[15] H. Kimura. Symmetries of the Garnier system and of the associated polynomial Hamiltonian system.
Proc.Japan Acad. Ser. A Math. Sci. , 66(7):176–178, 1990.[16] H. Kimura and K. Okamoto. On the polynomial Hamiltonian structure of the Garnier systems.
J. Math.Pures Appl. (9) , 63(1):129–146, 1984.[17] F. W. Nijho ff and A. J. Walker. The discrete and continuous Painlev´e VI hierarchy and the Garnier systems. Glasg. Math. J. , 43A:109–123, 2001. Integrable systems: linear and nonlinear dynamics (Islay, 1999).[18] F. W. Nijho ff , A. Ramani, B. Grammaticos, and Y. Ohta. On discrete Painlev´e equations associated with thelattice KdV systems and the Painlev´e VI equation. Stud. Appl. Math. , 106(3):261–314, 2001.[19] M. Noumi, M. Noumi.
Painlev´e equations through symmetry , volume 223 of
Translations of MathematicalMonographs . American Mathematical Society, Providence, RI, 2004. Translated from the 2000 Japaneseoriginal by the author.[20] K. Okamoto. Studies on the Painlev´e equations. I. Sixth Painlev´e equation P VI . Ann. Mat. Pura Appl. SerieQuarta , 146(1):337–381, 1987.[21] B. Rodin and D. Sullivan. The convergence of circle mappings to the Riemann mapping.
J. Di ff erentialGeom. , 26:349–360, 1987.[22] T. Suzuki. A ffi ne Weyl group symmetry of the Garnier system. Funkcial. Ekvac. , 48(2):203–230, 2005.[23] W. P. Thurston. The finite Riemann mapping theorem. Invited Address. International Symposium in Cele-bration of the Proof of the Bieberbach Conjecture. Purdue University (1985).[24] T. Tsuda. Rational solutions of the Garnier system in terms of Schur polynomials.
Int. Math. Res. Not.IMRN , (43):2341–2358, 2003. S chool of M athematics and S tatistics F07, T he U niversity of S ydney , NSW 2006, A ustralia . E-mail address : [email protected] I nstitute of M athematics for I ndustry , K yushu U niversity , 744 M otooka , F ukuoka apan . E-mail address : [email protected] D epartment of P hysics and M athematics , A oyama G akuin U niversity , S agamihara , K anagawa apan . E-mail address : [email protected] I nstitute of E ngineering , T okyo U niversity of A griculture and T echnology , 2-24-16 N akacho K oganei ,T okyo apan . E-mail address ::