Reduction of quad-equations consistent around a cuboctahedron I: additive case
aa r X i v : . [ n li n . S I] J un REDUCTION OF QUAD-EQUATIONS CONSISTENT AROUND ACUBOCTAHEDRON I: ADDITIVE CASE
NALINI JOSHI AND NOBUTAKA NAKAZONOA bstract . In this paper, we consider a reduction of a new system of partial di ff erenceequations, which was obtained in our previous paper [10] and shown to be consistentaround a cuboctahedron. We show that this system reduces to A (1) ∗ -type discrete Painlev´eequations by considering a periodic reduction of a three-dimensional lattice constructedfrom overlapping cuboctahedra.
1. I ntroduction
In this paper, we consider a system of partial di ff erence equations governing a function u = u ( l ) taking values on the vertices of a face-centered cubic lattice Ω , given by Ω = l = 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 + l ∈ Z , (1.1)where { ǫ , ǫ , ǫ } is a standard basis of R . The system consists of 6 equations: u ik u ik = ( α i j + γ i ) u jk − ( α i j + γ j − γ k ) u jk ( α i j − γ j + γ k ) u jk − ( α i j − γ i ) u jk , u jk u jk = ( α i j + γ i ) u ik − ( α i j − γ j + γ k ) u ik ( α i j + γ j − γ k ) u ik − ( α i j − γ i ) u ik , (1.2)where ( i , j , k ) = (1 , , , (2 , , , (3 , , i and j denote l → l + ǫ i and l → l − ǫ j respectively and the coe ffi cients are given by α i j = α i ( l i ) − α j ( l j ) , α i ( k ) = α i (0) + k , i , j ∈ { , , } , k ∈ Z , (1.3a) γ = − c + ( − l + l δ , γ = − c + ( − l + l δ , γ = − c + ( − l + l δ , (1.3b)with α i (0), i = , , c , and δ j , j = , ,
3, being complex parameters. Figure 1.1 shows aunit cell in Ω . Figure 1.1. A unit cell of the Ω lattice. Mathematics Subject Classification.
Key words and phrases.
Consistency around a cuboctahedron; Consistency around an octahedron; quad-equation; Consistency around a cube; ABS equation; Discrete Painlev´e equation .
Our study is motivated by two considerations. Firstly, the system (1.2) satisfies the con-sistency around a cuboctahedron (CACO) property [10], which is a generalization of thefamous consistency around a cube (CAC) property [17]. (See Appendix A for a summaryof the details of the CACO property and § ff erence equations and ordinarydi ff erence equations known as the discrete Painlev´e equations.In this paper, we show that the system (1.2) reduces to discrete Painlev´e equations withinitial value space characterised as A (1) ∗ in the sense of Sakai [22]. The latter equationshave two forms in the literature given respectively by Tsuda [23] and Ramani et al. [21]and are explicitly given by: (cid:16) Y + X (cid:17)(cid:16) X + Y (cid:17) = (cid:16) ( X + c ) − c (cid:17) (cid:16) ( X − c ) − c (cid:17) ( X + t ) − c , (cid:16) X + Y ) (cid:16) X + Y (cid:17) = (cid:16) ( Y − c ) − c (cid:17) (cid:16) ( Y + c ) − c (cid:17)(cid:16) Y + t + (cid:17) − c , (1.4a) (cid:16) X + X (cid:17)(cid:16) X + X (cid:17) = (cid:16) X − c (cid:17) (cid:16) X − c (cid:17) ( X + t ) − c . (1.4b)Here, t ∈ C is an independent variable, c i , i = . . . ,
5, are complex parameters and X , Y are dependent variables: X = X ( t ) , Y = Y ( t ) , X = X ( t + , X = X ( t − , Y = Y ( t − . (1.5)We note that discrete Painlev´e equations admit special solutions when parameters take spe-cial values. For example, Equation (1.4a) has the special solution given by the generalizedhypergeometric series F when 4 c + √ c + √ c = Ω . To be explicit, consider a vertex l ∈ Ω , given by l ǫ + l ǫ + l ǫ .Define the plane H k ⊂ Ω given by l = k . We project the vertices of H to the adjacenthorizontal plane H by taking ( l , l , ( l − , l − , H forms Z . We can define such a projection from every plane H k to H by the following: ( l , l , k ) ( l − k , l − k , . We call the result of this operation a (1 , , -periodic reduction . Theorem 1.1.
The A (1) ∗ -type discrete Painlev´e equations (1.4) can be obtained from thesystem of P ∆ Es (1.2) via the (1 , , -periodic reduction. Notation and Definitions.
Throughout the paper, we use terminology to describepolynomials and quad-equations that is common in the literature. Readers who are unfa-miliar with this notation may wish to consult [1, 8, 10]. We use Q = Q ( x , y , z , w ) to denotea multivariable polynomial over C . Under certain conditions, i.e., that Q be a ffi ne linearand irreducible, we will refer to the equation Q = quad-equation or sometimes, forsuccinctness, refer to the polynomial Q as a quad-equation. We remind the reader that thecondition of irreducibility implies that Q ( x , y , z , w ) = Background.
Integrable systems are widely applicable models of science, occurringin fluid dynamics, particle physics and optics. The prototypical example is the famousKorteweg-de Vries (KdV) equation whose solitary wave-like solutions interact elasticallylike particles, leading to the invention of the term soliton . It is then natural to ask whatdiscrete versions of such equations are also integrable. This question turns out to be relatedto consistency conditions for polynomials associated to faces of cubes as we explain below.
Integrable discrete systems were discovered [15,16,18,20] from mappings that turn outto be consistent on multi-dimensional cubes. (We note that there are additional systemsthat do not fall into this class, see e.g., [8, Chapter 3].) These are quad-equations in thesense in § et al. classified quad-equations satisfyingthe consistency around a cube (CAC) property, which lead integrable P ∆ Es. We refer tosuch P ∆ Es as ABS equations. It turns out that ABS equations contain many well knownintegrable P ∆ Es [9, 14–16].Reductions of integrable PDEs lead to Painlev´e equations, which first arose in the searchfor new transcendental functions in the early 1900’s [5, 6, 19]. Again a natural question isto ask whether discrete versions exist with analogous properties. This question led tothe discovery of second-order di ff erence equations called the discrete Painlev´e equations[7, 13, 20]).It is now well-known that discrete Painlev´e equations have initial value spaces withgeometric structures that can be identified with root systems and a ffi ne Weyl groups [22].Sakai showed that there are 22 types of initial value spaces as shown in Table 1.1.Table 1.1. Types of spaces of initial values.Discrete type Type of space of initial valuesElliptic A (1)0 Multiplicative A (1) ∗ , A (1)1 , A (1)2 , A (1)3 , . . . , A (1)8 , A (1) ′ Additive A (1) ∗∗ , A (1) ∗ , A (1) ∗ , D (1)4 , . . . , D (1)8 , E (1)6 , E (1)7 , E (1)8 Outline of the paper.
This paper is organized as follows. In §
2, we show the ex-tended a ffi ne Weyl group of type E (1)6 and its subgroup which forms that of type A (1)2 .Moreover, from those birational actions we obtain the discrete Painlev´e equations (1.4)and the P ∆ Es (2.16), which are periodically reduced equations of the system (1.2). In § § § erivation of the discrete integrable systems from an extended affine W eyl groupof type E (1)6 In this section, we derive the partial / ordinary discrete integrable systems from the bira-tional actions of an extended a ffi ne Weyl group of type E (1)6 , denoted by e W ( E (1)6 ). Note thatdetails of e W ( E (1)6 ) are given in Appendix B.2.1. Extended a ffi ne Weyl group of type A (1)2 . Let a i , i = , . . . ,
6, be parameters satisfy-ing the condition a + a + a + a + a + a + a = , (2.1)and τ ( i ) j , i = , , j = , , ,
3, be variables. Moreover, we define the transformations s i , i = , . . . , ι j , j = , ,
3, by isomorphisms from the field of rational functions K ( { τ ( i ) j } ),where K = C ( { a i } ), to itself. These transformations collectively form the extended a ffi neWeyl group of type E (1)6 , denoted by e W ( E (1)6 ): e W ( E (1)6 ) = h s , . . . , s i ⋊ h ι , ι , ι i . (2.2)See Appendix B for more details.Let us define the transformations w i , i = , ,
2, and π by w = s s s s , w = s s s s , w = s s s s , π = ι ι . (2.3) NALINI JOSHI AND NOBUTAKA NAKAZONO
They collectively form the extended a ffi ne Weyl group of type A (1)2 : e W ( A (1)2 ) = h w , w , w i ⋊ h π i . (2.4)Indeed, the following fundamental relations hold:( w i w j ) a ij = , i , j ∈ { , , } , π = , π w { , , } = w { , , } π, (2.5)where ( a i j ) i , j = = . (2.6)Introduce the parameters and variables that go well with e W ( A (1)2 ) as follows. Let b = a + a + a , b = a + a + a , b = a + a + a , (2.7a) c = a + a + a + a , d = a + a − a − a , d = a − a + a − a , (2.7b) d = a − a − a + a , (2.7c)where b + b + b =
1, and y = τ (1)1 τ (1)0 , y = τ (3)3 τ (3)2 , y = τ (2)1 τ (2)0 , y = τ (2)3 τ (2)2 , y = τ (1)3 τ (1)2 , y = τ (3)1 τ (3)0 . (2.8)Then, the action of e W ( A (1)2 ) on the parameters b , b , b , c , d , d , d are given by w i ( b j ) = − b i if i = j , b j + b i if i , j , w : ( d , d ) ( d , d ) , (2.9a) w : ( d , d ) ( d , d ) , w : ( d , d ) ( − d , − d ) , (2.9b) π : ( b , b , b , d , d , d ) ( b , b , b , − d , − d , d ) , (2.9c)where i , j ∈ Z / (3 Z ), while those on the y -variables y i , i = , . . . ,
6, are given by w : y , y y , y y , ( b − c + d ) y − ( b − d + d ) y ( b + d − d ) y − ( b + c − d ) y y y , ( b − c + d ) y − ( b − d + d ) y ( b + d − d ) y − ( b + c − d ) y y , (2.10a) w : y , y y , y ( b − c + d ) y − ( b − d + d ) y ( b + d − d ) y − ( b + c − d ) y y , y y , ( b − c + d ) y − ( b − d + d ) y ( b + d − d ) y − ( b + c − d ) y y , (2.10b) w : y , y y , y ( b − c − d ) y − ( b − d − d ) y ( b + d + d ) y − ( b + c + d ) y y , y ( b − d − d ) y − ( b − c − d ) y ( b + c + d ) y − ( b + d + d ) y y , y , (2.10c) π : ( y , y , y , y , y , y ) ( y , y , y , y , y , y ) . (2.10d) Remark 2.1.
We follow the convention that the parameters and y-variables not explicitlyincluded in the actions listed in Equations (2.9) and (2.10) are the ones that remain un-changed under the action of the corresponding transformation. That is, the transformationacts as an identity on those parameters or variables.
For later convenience, we here define the translations in e W ( A (1)2 ) by T = w w π , T = w w π , T = w w π , (2.11)whose actions on the parameters b , b , b , c , d , d , d are given by T :( b , b , d , d ) ( b − , b + , − d , − d ) , (2.12a) T :( b , b , d , d ) ( b − , b + , − d , − d ) , (2.12b) T :( b , b , d , d ) ( b − , b + , − d , − d ) . (2.12c)Note that T T T = T i T j = T j T i , where i , j = , ,
3, hold.2.2.
Derivation of the partial di ff erence equations from e W ( A (1)2 ) . In this subsection, wederive the P ∆ Es (2.16) from the birational action of e W ( A (1)2 ).Let u l , l , l = T l T l T l ( y ) . (2.13)Note that u , , = y , u , , = y , u , , = y , u , , = y , u , , = y , u , , = y . (2.14)We assign the variable u l , l , l on the vertices ( l , l , l ) of the triangle lattice Z / (1 , ,
1) : = n ( l , l , l ) ∈ Z (cid:12)(cid:12)(cid:12) l + l + l = o . (2.15)Then, we obtain the following lemma. Lemma 2.2.
On the triangle lattice there are three fundamental relations (essentially two):u i u i = (cid:16) b ( i ) l i , l j − c + ( − l i + l j d i j (cid:17) u j − (cid:16) b ( i ) l i , l j − ( − l j + l k d jk + ( − l i + l k d ik (cid:17) u j (cid:16) b ( i ) l i , l j + ( − l j + l k d jk − ( − l i + l k d ik (cid:17) u j − (cid:16) b ( i ) l i , l j + c − ( − l i + l j d i j (cid:17) u j , (2.16) where ( i , j , k ) = (1 , , , (2 , , , (3 , , andb (1) l , l = b + l − l , b (2) l , l = b + l − l − , b (0) l , l = b + l − l . (2.17) Here, u = u l , l , l and the subscript i (or, i ) for a function u = u l , l , l means + shift (or, − shift) in the l i -direction.Proof. Equations (2.16) with ( i , j , k ) = (1 , , , (2 , , , (3 , ,
2) are respectively obtainedfrom the following actions: T ( y ) y = ( b − c + d ) y − ( b + d − d ) y ( b − d + d ) y − ( b + c − d ) y , (2.18a) T ( y ) y = ( b − c − d ) y − ( b − d − d ) y ( b + d + d ) y − ( b + c + d ) y , (2.18b) T ( y ) y = ( b − c + d ) y − ( b − d + d ) y ( b + d − d ) y − ( b + c − d ) y . (2.18c)Moreover, we can easily verify that using Equations (2.16) we can express any u l , l , l onthe lattice by the six initial variables y i , i = , . . . ,
6, and one of the equations (2.16) can beobtained from the other two equations. Therefore, we have completed the proof. (cid:3)
Remark 2.3.
Because of the following relations:w ( u l , l , l ) = u l , l , l , w ( u l , l , l ) = u l , l , l , w ( u l , l , l ) = u l + , l + , l , (2.19a) π ( u l , l , l ) = u l + , l + , l , (2.19b) which follow fromw T { , , } = T { , , } w , w T { , , } = T { , , } w , w T { , , } = T { , , } w , (2.20a) π T { , , } = T { , , } π, w ( u , , ) = u , , , w ( u , , ) = u , , , (2.20b) w ( u , , ) = u , , , π ( u , , ) = u , , , (2.20c) NALINI JOSHI AND NOBUTAKA NAKAZONO the transformation group e W ( A (1)2 ) can be also regarded as the symmetry of the trianglelattice (see Figure 2.1). Figure 2.1. Triangle lattice. On the vertices the variables u l , l , l are assigned, and on thequadrilaterals there exist quad-equations (2.16), e.g. Equations (2.18a), (2.18b) and (2.18c)are colored in red, blue and green, respectively.2.3. Derivation of the A (1) ∗ -type discrete Painlev´e equations from e W ( A (1)2 ) . In this sub-section, we derive the A (1) ∗ -type discrete Painlev´e equations (1.4) from the birational actionof e W ( A (1)2 ).Let f = ( c − d + d − d ) y y − y ) + b + c − d + d − d , (2.21a) g = ( c − d + d − d ) y y − y ) − b + c − d + d − d . (2.21b)Then, the action of e W ( A (1)2 ) on the variables f and g are given by w ( f ) = f − b , w ( g ) = g + b , π ( g ) = f − b + b , (2.22a)4( c − d + d − d )4 f − b − b − c − d + d + d w ( g ) + b + b + c + d − d − d ! = ( b − d + d )(4 g + b − c + d − d + d )4 f − b + c − d + d − d − ( b − c + d )(4 g + b + c − d + d − d )4 f − b − c + d − d + d , (2.22b)4( c − d + d − d )4 g + b + b + c − d − d + d w ( f ) − b + b + c − d − d + d ! = ( b + d − d )(4 f − b + c − d + d − d )4 g + b − c + d − d + d − ( b − c + d )(4 f − b − c + d − d + d )4 g + b + c − d + d − d , (2.22c) π ( f ) = − (4 f − b + c − d + d − d )(4 g + b + c − d + d − d )16( f + g ) + b + c − d + d − d . (2.22d)Using the transformation T whose action on the parameter space { b , b , b , c , d , d , d } is translational as T : ( b , b ) ( b − , b +
2) shows, we obtain the discrete Painlev´eequation (1.4a) with the following correspondence: X = f , Y = g , X = T ( f ) , Y = T − ( g ) , t = b + b − , (2.23a) c = ( b + c + d ) , c = ( b − c − d ) , c = d + d , (2.23b) c = ( c + d − d − d ) , c = ( c − d − d + d ) . (2.23c)We can also obtain the discrete Painlev´e equations from non-translation on the parame-ter space as follows [12]. The action of T on the parameter space: T : ( b , b , d , d ) ( b − , b + , − d , − d ) , is not translational, but when the parameters take the special values d = d =
0, itbecomes translational motion on the parameter sub-space { b , b , b , c , d } : T : ( b , b ) ( b − , b + T gives the discretePainlev´e equation (1.4b) with the following correspondence: X = f , X = T (2 f ) , X = T − (2 f ) , t = b + b − , (2.24a) c = ( b + c + d ) , c = ( b − c − d ) , c = ( c − d ) . (2.24b)3. P roof of T heorem ∆ Es (1.2) to the system of P ∆ Es (2.16).The following lemma holds.
Lemma 3.1.
By imposing the (1 , , -periodic condition: u ( l + ǫ + ǫ + ǫ ) = u ( l ) for l ∈ Ω , the system (1.2) can be reduced to the following system of P ∆ Es:u i u i = (cid:16) α i j − c + ( − l i + l j δ i (cid:17) u j − (cid:16) α i j − ( − l j + l k δ j + ( − l i + l k δ k (cid:17) u j (cid:16) α i j + ( − l j + l k δ j − ( − l i + l k δ k (cid:17) u j − (cid:16) α i j + c − ( − l i + l j δ i (cid:17) u j , (3.1) where ( i , j , k ) = (1 , , , (2 , , , (3 , , , u = u ( l ) and l = P i = l i ǫ i ∈ Z / ( ǫ + ǫ + ǫ ) .Proof. Applying the (1 , , i , j , k ) = (1 , , , ,
1) and (3 , ,
2) from Equations (1.2) with ( i , j , k ) = (1 , , , ,
1) and (3 , , (cid:3) Remark 3.2.(i):
The number of essential equations in the system (3.1) is two. (ii):
By the (1 , , -reduction, each cuboctahedron is reduced to a hexagram (see Fig-ure 3.1) , which causes the reduction from the face-centred cubic lattice Ω to thetriangle lattice Z / ( ǫ + ǫ + ǫ ) . Lemma 3.3.
The reduced system (3.1) is equivalent to equations the system (2.16) .Proof.
The statement follows from the following correspondences: b (1) l , l = α , b (2) l , l = α , b (0) l , l = α , d = δ , d = δ , d = δ , (3.2a) u l , l , l = u ( l ǫ + l ǫ + l ǫ ) . (3.2b) (cid:3) NALINI JOSHI AND NOBUTAKA NAKAZONO
Figure 3.1. The (1 , , Remark 3.4.
Lemma 3.3 means that the reduced system (3.1) can be obtained from thetheory of the τ -function associated with A (1) ∗ -type discrete Painlev´e equations. We are now ready to prove Theorem 1.1. The (1 , , A (1) ∗ -type discrete Painlev´e equations (1.4) given in § § oncluding remarks In this paper, we considered a reduction of a system of P ∆ Es, which is unusual in thesense that it has the CACO property but not the widely studied CAC property. We showedhow the system (1.2) can be reduced to the A (1) ∗ -type discrete Painlev´e equations (1.4)using the a ffi ne Weyl group associated with the discrete Painlev´e equations.In a forthcoming paper (N. Joshi and N. Nakazono), we will show how another systemof P ∆ Es, which also has the CACO property, can be reduced to the A (1)2 -type discretePainlev´e equations (see Table 1.1 for the distinction between A (1)2 and A (1) ∗ ). Acknowledgment.
N. Nakazono would like to thank Profs M. Noumi, Y. Ohta and Y. Ya-mada for inspiring and fruitful discussions. This research was supported by an AustralianLaureate Fellowship ppendix
A. C onsistency around a cuboctahedron property
In this appendix, we recall the definition of consistency around a cuboctahedron . Todefine it, we also introduce an additional important property called consistency around anoctahedron . We refer the reader to [10] for detailed information about these properties.A.1.
Consistency around an octahedron property.
In this subsection, we give a defini-tion of a consistency around an octahedron.Let u i , i = , . . . ,
6, be variables and consider the octahedron shown in Figure A.1. Theplanes that pass through the vertices { u , u , u , u } , { u , u , u , u } and { u , u , u , u } give3 quadrilaterals that lie in the interior of the octahedron and we assign the quad-equations Q i , i = , ,
3, to the quadrilaterals as the following: Q ( u , u , u , u ) = , Q ( u , u , u , u ) = , Q ( u , u , u , u ) = . (A.1)The consistency around an octahedron property is defined by the following. Figure A.1. An octahedron labelled with vertices u i , i = , . . . , Definition A.1 (CAO property [10]) . The octahedron with quad-equations { Q , Q , Q } issaid to have a consistency around an octahedron (CAO) property , if each quad-equation canbe obtained from the other two equations. An octahedron is said to be a CAO octahedron ,if it has the CAO property.A.2.
Consistency around a cuboctahedron property.
In this subsection, we give a def-inition of a consistency around a cuboctahedron.We consider the cuboctahedron centered around the origin whose twelve vertices aregiven by V = n ± ǫ i ± ǫ j (cid:12)(cid:12)(cid:12) i , j ∈ Z , ≤ i < j ≤ o , where { ǫ , ǫ , ǫ } form the standard basisof R . We assign the variables u ( l ) to the vertices l ∈ V and impose the following relations: Q ( u , u , v , v ) = , Q ( v , v , u , u ) = , Q ( u , u , v , v ) = , (A.2a) Q ( v , v , u , u ) = , Q ( u , u , v , v ) = , Q ( v , v , u , u ) = , (A.2b) Q ( u , u , u , u ) = , Q ( u , u , u , u ) = , Q ( u , u , u , u ) = , (A.2c)where Q i , i = , . . . ,
9, are quad-equations and u = u ( ǫ + ǫ ) , u = u ( − ǫ − ǫ ) , u = u ( ǫ + ǫ ) , u = u ( − ǫ − ǫ ) , (A.3a) u = u ( ǫ + ǫ ) , u = u ( − ǫ − ǫ ) , v = u ( ǫ − ǫ ) , v = u ( ǫ − ǫ ) , (A.3b) v = u ( ǫ − ǫ ) , v = u ( − ǫ + ǫ ) , v = u ( − ǫ + ǫ ) , v = u ( − ǫ + ǫ ) . (A.3c)Note that quad-equations Q i , i = , . . . ,
6, are assigned to the faces of the cuboctahedron(see Figure A.2a). Moreover, u i , i = , . . . ,
6, collectively form the vertices of an octahe-dron and quad-equations Q i , i = , ,
9, are assigned to the quadrilaterals that appear assections passing through four vertices of the octahedron (see Figure A.2b). (a) A cuboctahedron labelled with vertices u i and v j , i , j = , . . . ,
6. (b) An octahedron labelled with vertices u i , i = , . . . , Figure A.2. A cuboctahedron and an interior octahedron.
We are now in a position to give the following definitions.
Definition A.2 (CACO property [10]) . The cuboctahedron with quad-equations { Q , . . . , Q } is said to have a consistency around a cuboctahedron (CACO) property , if the followingproperties hold. (i): The octahedron with quad-equations { Q , Q , Q } has the CAO property. (ii): Assume that u , . . . , u are given so as to satisfy Q i = i = , ,
9, and, in addition, v k is given, for some k ∈ { , . . . , } . Then, quad-equations Q i , i = . . . ,
6, determinethe variables v j , j ∈ { , . . . , }\{ k } , uniquely.A cuboctahedron is said to be a CACO cuboctahedron , if it has the CACO property.
Definition A.3 (Square property [10]) . The CACO cuboctahedron with quad-equations { Q , . . . , Q } is said to have a square property , if there exist polynomials K i = K i ( x , y , z , w ), i = , ,
3, where deg x K i = deg w K i = ≤ deg y K i , deg z K i , satisfying K ( v , u , u , v ) = , K ( v , u , u , v ) = , K ( v , u , u , v ) = . (A.4)Then, each equation K i = square equation .A.3. CACO property of P ∆ Es.
We now explain how to associate quad-equations withP ∆ Es in three-dimensional space by using the system of P ∆ Es (1.2) as an example. Thisrequires us to consider overlapping cuboctahedra that lead to two-dimensional tessellationsconsisting of quadrilaterals. For each given cuboctahedron, there are twelve overlappingcuboctahedra.The twelve overlapping cuboctahedra around a given one provide six directions of tilingby quadrilaterals. For later convenience, we label directions by ǫ i ± ǫ j , 1 ≤ i < j ≤ Ω given by (1.1). Such vertices are interpretedas being iterated on each successive cuboctahedron. We here consider the system of P ∆ Es(1.2). For simplicity, we abbreviate each respective equation in Equations (1.2) as P (cid:16) u , u , u , u (cid:17) = , P (cid:16) u , u , u , u (cid:17) = , P (cid:16) u , u , u , u (cid:17) = , (A.5a) P (cid:16) u , u , u , u (cid:17) = , P (cid:16) u , u , u , u (cid:17) = , P (cid:16) u , u , u , u (cid:17) = . (A.5b)Conversely, given l ∈ Ω , we obtain the cuboctahedron centered around l . We refer to itsquad-equations as before by { Q ( l ) , . . . , Q ( l ) } . Moreover, the overlapped region gives anoctahedron centred around l + ǫ , and we label its quad-equations by { ˆ Q ( l ) , ˆ Q ( l ) , ˆ Q ( l ) } .Each such quad-equation is identified with the 6 partial di ff erence equations given inEquations (1.2) in the following way. For Q , . . . , Q , we use Q ( l ) = P (cid:16) u , u , u , u (cid:17) = , Q ( l ) = P (cid:16) u , u , u , u (cid:17) = , (A.6a) Q ( l ) = P (cid:16) u , u , u , u (cid:17) = , Q ( l ) = P (cid:16) u , u , u , u (cid:17) = , (A.6b) Q ( l ) = P (cid:16) u , u , u , u (cid:17) = , Q ( l ) = P (cid:16) u , u , u , u (cid:17) = , (A.6c) Q ( l ) = P (cid:16) u , u , u , u (cid:17) = , Q ( l ) = P (cid:16) u , u , u , u (cid:17) = , (A.6d) Q ( l ) = P (cid:16) u , u , u , u (cid:17) = , (A.6e)and for ˆ Q , ˆ Q , ˆ Q , we useˆ Q ( l ) = P (cid:16) u , u , u , u (cid:17) = , ˆ Q ( l ) = P (cid:16) u , u , u , u (cid:17) = , (A.7a)ˆ Q ( l ) = P (cid:16) u , u , u , u (cid:17) = . (A.7b)Then, the following proposition holds. Proposition A.4 ( [10]) . The system of P ∆ Es (1.2) has the CACO and square properties,that is, the following statements hold. (i): The cuboctahedra with quad-equations { Q i ( l ) } have the CACO and square properties. (ii): The square equations are consistent with the P ∆ Es (1.2) . (iii): The octahedra with quad-equations { ˆ Q i ( l ) } have the CAO property. A ppendix B. E xtended affine W eyl group of type E (1)6 and τ - variables In this appendix, we review the action of the extended a ffi ne Weyl group of type E (1)6 given in [23], which is the symmetry group of A (1) ∗ -type discrete Painlev´e equations.Let a i , i = , . . . ,
6, be parameters satisfying the condition (2.1) and τ ( i ) j , i = , , j = , , ,
3, be variables. The actions of transformations s i , i = , . . . ,
6, and ι j , j = , , s : ( a , a ) ( − a , a + a ) , s : ( a , a ) ( − a , a + a ) , (B.1a) s : ( a , a , a ) ( a + a , − a , a + a ) , (B.1b) s : ( a , a , a , a ) ( a + a , − a , a + a , a + a ) , (B.1c) s : ( a , a , a ) ( a + a , − a , a + a ) , s : ( a , a ) ( a + a , − a ) , (B.1d) s : ( a , a , a ) ( a + a , a + a , − a ) , (B.1e) ι a { , , , } a { , , , } , ι a { , , , } a { , , , } , ι a { , , , } a { , , , } , (B.1f)while those on the τ -variables τ ( i ) j , i = , , j = , , ,
3, are given by s : ( τ (3)2 , τ (3)3 ) ( τ (3)3 , τ (3)2 ) , s : ( τ (1)2 , τ (1)3 ) ( τ (1)3 , τ (1)2 ) , (B.2a) s : ( τ (1)1 , τ (1)2 ) ( τ (1)2 , τ (1)1 ) , (B.2b): ( τ (2)0 , τ (3)0 ) ( a + a ) τ (1)1 τ (2)0 − a τ (2)1 τ (1)0 a τ (1)2 , ( a + a ) τ (1)1 τ (3)0 − a τ (3)1 τ (1)0 a τ (1)2 , (B.2c) s : ( τ (1)1 , τ (2)1 , τ (3)1 , τ (1)0 , τ (2)0 , τ (3)0 ) ( τ (1)0 , τ (2)0 , τ (3)0 , τ (1)1 , τ (2)1 , τ (3)1 ) , (B.2d) s : ( τ (2)1 , τ (2)2 ) ( τ (2)2 , τ (2)1 ) , (B.2e): ( τ (1)0 , τ (3)0 ) ( a + a ) τ (2)1 τ (1)0 − a τ (1)1 τ (2)0 a τ (2)2 , ( a + a ) τ (2)1 τ (3)0 − a τ (3)1 τ (2)0 a τ (2)2 , (B.2f) s : ( τ (2)2 , τ (2)3 ) ( τ (2)3 , τ (2)2 ) , (B.2g) s : ( τ (3)1 , τ (3)2 ) ( τ (3)2 , τ (3)1 ) , (B.2h): ( τ (1)0 , τ (2)0 ) ( a + a ) τ (3)1 τ (1)0 − a τ (1)1 τ (3)0 a τ (3)2 , ( a + a ) τ (3)1 τ (2)0 − a τ (2)1 τ (3)0 a τ (3)2 , (B.2i) ι : ( τ (2) j , τ (3) j ) ( τ (3) j , τ (2) j ) , ι : ( τ (1) j , τ (3) j ) ( τ (3) j , τ (1) j ) , (B.2j) ι : ( τ (1) j , τ (2) j ) ( τ (2) j , τ (1) j ) , j = , , , . (B.2k) Remark B.1.(i):
Each transformation here defined is an isomorphism from the field of rational functionsK ( { τ ( i ) j } ) , where K = C ( { a i } ) , to itself. (ii): We follow the convention of Remark 2.1, for Equations (B.1) and (B.2) . That is, eachtransformation acts as an identity on parameters or variables not appearing in itsdefinition.
The transformations collectively form the extended a ffi ne Weyl group of type E (1)6 , de-noted by (2.2). Indeed, the following fundamental relations hold:( s i s j ) A ij = , ι = ι = ι = , ι ι = ι ι = ι ι , ι ι = ι ι = ι ι , (B.3a) ι s { , , , } = s { , , , } ι , ι s { , , , } = s { , , , } ι , ι s { , , , } = s { , , , } ι , (B.3b) where i , j ∈ { , , . . . , } and( A i j ) i , j = = . (B.4) Remark B.2.
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