A geometric approach to tau-functions of difference Painleve equations

A geometric approach to tau-functions of di ff erenc e P ainle v ´ e equations T eruhisa TSUD A F aculty of Mathematics, K yu shu Uni versity , Hakozaki, Fukuoka 812-8581, Japan. tudateru@math.kyushu-u.ac.jp April 7, 2008 Abstract W e presen t a unified desc ription of bira tional represe ntation of W eyl gr oups asso ciated with T -shape d Dynkin diagrams, by usin g a particula r configuration of points in the projec ti ve plane. A geometri c formulat ion of tau-functi ons is giv en in terms of defining polynomials of certain curves. If the Dynkin diagr am is of a ffi ne type ( E (1) 6 , E (1) 7 or E (1) 8 ), our representat ion gi ve s rise to the di ff erence Painle v ´ e equations. Introduction It is classically well-kn own that a certain birational representation of W eyl groups arises from point configurations . Let X m , n be the configuration space o f n points of t he projective space P m − 1 in general posit ion. Then, the W eyl group corresponding to t he D ynkin diagram T 2 , m , n − m (see Figure 1 ) acts birational ly on X m , n and is generated by permu tations of n po ints and the standard Cremona transformation with respect to each m points; see [4, 5, 11]. W e shall put o ur att ention to the two-dimensional case, that is, m = 3 case. Only if n = 9, the a ffi ne case occurs and the corresponding di agram reads T 2 , 3 , 6 = E (1) 8 . The lattice part of the a ffi ne W eyl group W ( E (1) 8 ) provides an interesting discrete dynamical system, called the elliptic- di ff er ence P ainlev ´ e equation [15, 17]. This ( m , n ) = (3 , 9) case was e xplored by Sakai [17] (cf. [16]) in order to clarify the geometric nature of the a ffi ne W eyl group symmetry of Painle v ´ e equa- tions; moreover he class ified all the degeneration of the nin e-points configuration in P 2 , and as a result he completed t he whole list of (second-order) discrete Painle v ´ e equations . In t his context th e top of all the di screte and continuous Painle v ´ e equatio ns is the elliptic-di ff erence one, from which e very other can be ob tained through an appropriate lim iting procedure. The di screte Painle v ´ e equa- tions are divided into three types: di ff erence, q -di ff erence or ellipt ic-di ff erence one, according to their correspondi ng rational su rface s (obtai ned by blowing up nine poin ts of P 2 ); see [17]. On the 2000 Mathematics Subject Classification 14E05 , 33E17 , 34M55 , 37K20, 37K35, 39A10 K eywor d s : birational transformatio n, di ff erence Painle v ´ e equation , tau-function, W eyl group. 1 other hand, ev en in t he two-dimensio nal case ( m = 3 ), b y consi dering certain particular configu- rations o f point sets that are not only nine points, one can enj oy W eyl groups ass ociated to more var ious Dynkin diagrams [12, 20]; see also [21] in the higher dimension al c ase. In this paper , we present a unified descripti on of bi rational representation of W eyl groups as- sociated with T -shaped Dy nkin diagrams (see Figure 1), arising from a particular configuration of points in the projecti ve plane. Our construction, in a ffi ne case: E (1) 6 , E (1) 7 and E (1) 8 , is rele vant to the di ff er ence Painlev ´ e equations; we refe r to [20] in the case of q -di ff erence ones. In Section 1, we first start from ℓ 1 + ℓ 2 + ℓ 3 points in P 2 restricted on three lines L i ( i = 1 , 2 , 3) meeting at a single point, where the ℓ i points lie on each L i . Let X be th e rational surface ob- tained from P 2 by blowing up the ℓ 1 + ℓ 2 + ℓ 3 points. W e can find naturally the root lattice of type T ℓ 1 ,ℓ 2 ,ℓ 3 included in the Picard group Pic( X ) of the surface . The corresponding W eyl group W = W ( T ℓ 1 ,ℓ 2 ,ℓ 3 ) acts linearly on Pic( X ). Next, in order t o lift this linear action W : Pic( X )  to the level of birational t ransformations on the surface X itself, we introduce a geometric formu la- tion of tau-fu nctions . Recall that a smo oth rational curve with self-intersection − 1 i s said to be an e xceptional curve (of th e first k ind); see e.g. [2]. An element of W induces a permutation am ong exceptional curves on X , as analogous to the classical subject: 27 lines on a cubic surface and a W eyl group of type E 6 . The tau-function is defined by m eans of appropriately normal ized defin- ing functions of the exceptional curves; see (1.3) and (1.4). T oday the no tion of tau-functions is univ ersal in the field of integrable system s and it i s character ized in sev eral di rections; howev er , such an algebro-geom etric i dea of t au-functions first appeared in the stud y o f the ellip tic-di ff erence Painle v ´ e equ ation by Kaj iwara et al. [9] (see als o [11, 20, 21] for su bsequent d e velopment). By imposing on the tau-function a certain compatibili ty with the linear acti on W : Pic( X )  , we finally obt ain a biratio nal representation of W acting on th e tau-function and the homogeneous coordinates of P 2 (Theorem 1.1). Section 2 con cerns the a ffi ne case ( E (1) 6 = T 3 , 3 , 3 , E (1) 7 = T 4 , 4 , 2 and E (1) 8 = T 6 , 3 , 2 ); we demonstrate how to deri ve the di ff erence P ai nlev ´ e equation for e ach. | {z } ℓ 1 | {z } ℓ 2                ℓ 3 ❝ . . . ❝ ❝ ❝ . . . ❝ ❝ . . . ❝ Figure 1: Dynkin diagram T ℓ 1 ,ℓ 2 ,ℓ 3 1 Birationa l r epresentation of W eyl gr oups and tau-functio ns W e begin wit h consid ering the ℓ 1 + ℓ 2 + ℓ 3 points in the comp lex projectiv e plane P 2 restricted on three lines L i ( i = 1 , 2 , 3) meeting at a sing le point P 0 , where we arrange ℓ i points on each L i \ P 0 . Let x = [ x 1 : x 2 : x 3 ] denot e the hom ogeneous coordinates of P 2 . By PGL 3 ( C )-action, we can normalize, without loss of generality , the three lines and the point configuration { P m i } 1 ≤ i ≤ ℓ m ; m = 1 , 2 , 3 to 2 be as follows: L i = { x j = x k } for { i , j , k } = { 1 , 2 , 3 } , P 1 i = [ b 1 i + 1 : b 1 i : b 1 i ] (1 ≤ i ≤ ℓ 1 ) , P 2 i = [ b 2 i : b 2 i + 1 : b 2 i ] (1 ≤ i ≤ ℓ 2 ) , P 3 i = [ b 3 i : b 3 i : b 3 i + 1] (1 ≤ i ≤ ℓ 3 ) , where 3 b m 1 = − 1 + a 0 + c 1 + c 2 + c 3 − 3 c m and b m i + 1 = b m i + a m i (1 ≤ i ≤ ℓ m − 1 ; m = 1 , 2 , 3) wi th c m = P ℓ m − 1 i = 1 (1 − i ℓ m ) a m i . Here a = ( a 0 , ( a m i ) 1 ≤ i ≤ ℓ m − 1; m = 1 , 2 , 3 ) ∈ C ℓ 1 + ℓ 2 + ℓ 3 − 2 play roles of free parameters. W e will s how later the m eaning of a as root variables correspond ing to the Dynki n diagram T ℓ 1 ,ℓ 2 ,ℓ 3 . Let π : X → P 2 be the blo w-up at the ℓ 1 + ℓ 2 + ℓ 3 points P m i . The Picar d gr oup of the surface X thus obtained is expressed as Pic( X ) = Z h ⊕ M i = 1 ,...,ℓ m ; m = 1 , 2 , 3 Z e m i   Z ℓ 1 + ℓ 2 + ℓ 3 + 1  . Note that in this case Pic( X ) is isomorphic to the second cohomology group H 2 ( X , Z ) bec ause X i s a rati onal surface. Here we denote by h the linear equiv alent class of π − 1 of a line and by e m i the class of exceptional curv e π − 1 ( P m i ). The intersection form ( | ) : Pic( X ) × Pic( X ) → Z is giv en by ( h | h ) = 1 , ( e m i | e n j ) = − δ i , j δ m , n , ( h | e m i ) = 0 . The ant i-canonical class − K X = 3 h − P i , m e m i can be decomposed as − K X = D 1 + D 2 + D 3 , where the classes D m = h − P ℓ m i = 1 e m i are repre sented by (the proper transform s of) the three lines L m ( m = 1 , 2 , 3). Let Q be the orth ogonal complem ent of { D 1 , D 2 , D 3 } with respect t o the in tersection form in Pic( X ). Then we see that Q is a root lattice generated by t he ( − 2)-vectors α m i j = e m i − e m j and α i jk = h − e 1 i − e 2 j − e 3 k . Moreov er we can choose a basis B = { α 0 = α 111 , α m i = α m i , i + 1 (1 ≤ i ≤ ℓ m − 1 , m = 1 , 2 , 3) } of Q . T he intersection graph (Dynki n diagram) of B is o f type T ℓ 1 ,ℓ 2 ,ℓ 3 and looks as follows: ❝ α 1 ℓ 1 − 1 . . . ❝ α 1 1 ❝ α 0 ❝ α 2 1 . . . ❝ α 2 ℓ 2 − 1 ❝ α 3 1 . . . ❝ α 3 ℓ 3 − 1 W e define the action of the simple reflection corresponding to a root α ∈ Q (i.e., α 2 = − 2) by R α ( v ) = v + ( v | α ) α for v ∈ Pic( X ). Note that the in tersection form is preserved by R α . W e also prepare the not ations s 0 = R α 0 and s m i = R α m i , for con venience. W e summ arize below the linear action of generators s 0 and s m i on the basis of Pic( X ): s 0 ( h ) = 2 h − e 1 1 − e 2 1 − e 3 1 , s 0 ( e k 1 ) = h − e i 1 − e j 1 for { i , j , k } = { 1 , 2 , 3 } , (1.1) s m i ( e m { i , i + 1 } ) = e m { i + 1 , i } . 3 It is easy to check that the W e yl group W = { R α } α ∈ Q = h s 0 , s m i (1 ≤ i ≤ ℓ m − 1 , m = 1 , 2 , 3) i acti ng on Pic( X ) indeed sati sfies the fundamental relations [7] specified by the Dynkin diagram T ℓ 1 ,ℓ 2 ,ℓ 3 . Now we shall e xtend the a bove lin ear action of W : Pic( X )  to the le vel of birational transfor- mations on the su rface X . W e first fix th e action on the root var iables a 0 and a m i (1 ≤ i ≤ ℓ m − 1 , m = 1 , 2 , 3 ) as s 0 ( a 0 ) = − a 0 , s 0 ( a m 1 ) = a 0 + a m 1 , s m i ( a m i ) = − a m i , s m i ( a m i ± 1 ) = a m i + a m i ± 1 , (1.2) where a m 0 = a 0 . W e consider a sub-lattice M = F n = 1 , 2 , 3 M n of Pic( X ), where M n def = W . { e n 1 } =        Λ = d h − X i , j µ i j e i j ∈ Pic( X )      Λ 2 = − 1 , ( Λ | D n ) = 1 , ( Λ | D m ) = 0 ( m , n ) d ≥ 0; i f d > 0, then µ i j ≥ 0 for ∀ ( i , j ).        . A divisor class Λ ∈ M ⊂ Pic( X ) is represented by an exceptional curve because Λ 2 = ( Λ | K X ) = − 1. Moreover , each class Λ = d h − P i , m µ m i e m i ∈ M corresponds to (the proper transform of) a unique plane curve C Λ passing throug h P m i with mult iplicity µ m i as its representati ve. Let F Λ = F Λ ( a ; x ) be the normaliz ed defi ning polynomial of C Λ such that the following key condition is satisfied: F Λ ( a ; 1 , 1 , 1) = 1 . (1.3) Notice that C Λ nev er pass es through the po int P 0 = [1 : 1 : 1] = L 1 ∩ L 2 ∩ L 3 by definition. Let us take for e xample Λ = h − e m i − e n j ∈ M , which is represented by the line passing through P m i and P n j ; thus we hav e F h − e m i − e n j ( a ; x ) = (1 + b m i + b n j ) x ℓ − b m i x m − b n j x n for { ℓ , m , n } = { 1 , 2 , 3 } . W e introduce new variables τ m i attached to the poi nts P m i , and pre pare a field L = K ( x ; τ ) of rational functions in in determinates x m and τ m i ( m = 1 , 2 , 3 ; 1 ≤ i ≤ ℓ m ) with coe ffi cient field K = C ( a ). By means of th e normalized defining pol ynomial, we define a fun ction τ : M → L = C ( a )( x ; τ ), called the tau-functi on , b y the formula τ ( Λ ) Y i , m τ ( e m i ) µ m i = F Λ ( a ; x ) (1.4) for Λ = d h − P i , m µ m i e m i ∈ M ( d > 0), and τ ( e m i ) = τ m i . (1.5) By impos ing the following a ssumpti on: w .τ ( Λ ) = τ ( w . Λ ) , (1.6) one can fix t he action of W eyl group W on L = C ( a )( x ; τ ) in the following manner . For a ratio nal function ϕ ( a ; x ; τ ) ∈ L , we suppose that an element w ∈ W acts as w .ϕ ( a ; x ; τ ) = ϕ ( a . w ; x . w ; τ . w ) , 4 that is, w acts on the i ndependent variables from the rig ht. W e now determine the acti on of t he generators s 0 and s m i from (1.6) as a necessary condition. In view of s 0 ( e k 1 ) = h − e i 1 − e j 1 ( { i , j , k } = { 1 , 2 , 3 } ), we hav e s 0 ( τ k 1 ) = F h − e i 1 − e j 1 ( a ; x ) τ i 1 τ j 1 . The other τ m i ’ s ( i , 1) are ke pt still by s 0 -action. Applying s 0 to F h − e i 1 − e j 1 ( a ; x ) = τ ( h − e i 1 − e j 1 ) τ i 1 τ j 1 , we ha ve (LHS) = s 0 ( F h − e i 1 − e j 1 ( a ; x )) = F h − e i 1 − e j 1 ( s 0 ( a ); s 0 ( x )) = (1 + ˜ b i 1 + ˜ b j 1 ) s 0 ( x k ) − ˜ b i 1 s 0 ( x i ) − ˜ b j 1 s 0 ( x j ) , (RHS) = s 0 ( τ ( h − e i 1 − e j 1 ) τ i 1 τ j 1 ) = τ k 1 τ ( h − e j 1 − e k 1 ) τ ( h − e i 1 − e k 1 ) = F h − e j 1 − e k 1 ( a ; x ) F h − e i 1 − e k 1 ( a ; x ) τ 1 1 τ 2 1 τ 3 1 , where { i , j , k } = { 1 , 2 , 3 } and ˜ b i 1 : = s 0 ( b i 1 ) . (1.7) Solving the above equations with respect to s 0 ( x i ), we obtain s 0 ( x i ) = (1 + ˜ b i 1 ) f j f k + ˜ b j 1 f i f k + ˜ b k 1 f i f j (1 + ˜ b 1 1 + ˜ b 2 1 + ˜ b 3 1 ) τ 1 1 τ 2 1 τ 3 1 , where f i : = F h − e j 1 − e k 1 ( a ; x ) = (1 + b j 1 + b k 1 ) x i − b j 1 x j − b k 1 x k for { i , j , k } = { 1 , 2 , 3 } . (1.8) The action of s m i is much simpler and is realized as (just a permutation of τ -v ariables) s m i ( τ m { i , i + 1 } ) = τ m { i + 1 , i } , s m i ( x ) = x . Summarizing above, w e ha ve the Theor em 1.1. (I) Define the birational transformation s s 0 and s m i by s 0 ( τ i 1 ) = f i τ j 1 τ k 1 , (1.9a) s 0 ( x i ) = (1 + ˜ b i 1 ) f j f k + ˜ b j 1 f i f k + ˜ b k 1 f i f j (1 + ˜ b 1 1 + ˜ b 2 1 + ˜ b 3 1 ) τ 1 1 τ 2 1 τ 3 1 , (1.9b) wher e { i , j , k } = { 1 , 2 , 3 } , and s m i ( τ m { i , i + 1 } ) = τ m { i + 1 , i } . (1.9c) Then (1.9) with (1.2) r ealize th e W e yl gr oup W = W ( T ℓ 1 ,ℓ 2 ,ℓ 3 ) over the field C ( a )( x ; τ ) . (II) Mor eover (1.4) and (1.6) ar e consistent. 5 Pr oof. (I) By direct computation, we can verify that (1.9 ) indeed satisfy t he fundamental r elations of W . (II) W e wi ll show that the formul a (1.4) can b e recovered inductiv ely by (1.6) and (1.9). First, if Λ = h − e i 1 − e j 1 ∈ M , then (1.4 ) follows immediately from (1.9a). Next we assum e (1.4) i s true for Λ ∈ M . It is enoug h to verify for Λ ′ = w ( Λ ) where w ∈ { s 0 , s m i } is a generator o f W . Since the action of s m i is just a p ermutation of τ -variables (see (1.9c)), we will concentrate our attent ion on the only nontrivial c ase w = s 0 . Applying s 0 to (1.4), we have s 0 ( F Λ ( a ; x )) = s 0         τ ( Λ ) Y i , j ( τ i j ) µ i j         = τ ( s 0 . Λ ) Y i , j ( τ i j ) µ i j Y i = 1 , 2 , 3 s 0 ( τ i 1 ) τ i 1 ! µ i 1 = τ ( s 0 . Λ ) Y i , j ( τ i j ) µ i j Y i = 1 , 2 , 3 f i τ 1 1 τ 2 1 τ 3 1 ! µ i 1 . Noticing s 0 . Λ = Λ + ( Λ | α 0 ) α 0 = Λ + ( d − µ 1 1 − µ 2 1 − µ 3 1 )( h − e 1 1 − e 2 1 − e 3 1 ), we can verify (1.4) for Λ ′ = s 0 . Λ immediately from the lemma belo w .  Lemma 1.2. W e have s 0 ( F Λ ( a ; x )) = F s 0 . Λ ( a ; x ) Q i = 1 , 2 , 3 ( f i ) µ i 1 ( τ 1 1 τ 2 1 τ 3 1 ) d (1.10) for Λ = d h − P i , j µ i j e i j ∈ M . Pr oof. The mul tiplicity of the curve C Λ = { F Λ = 0 } at P i 1 = { f j = f k = 0 } ( { i , j , k } = { 1 , 2 , 3 } ) is giv en by µ i 1 = ord P i 1 ( F Λ ). That is, F Λ can be expressed as a polyn omial in f i ( i = 1 , 2 , 3) of the form F Λ = X k 1 + k 2 + k 3 = d A k 1 , k 2 , k 3 ( f 1 ) k 1 ( f 2 ) k 2 ( f 3 ) k 3 , A k 1 , k 2 , k 3 ∈ C ( a ) such that A k 1 , k 2 , k 3 = 0 unless 0 ≤ k i ≤ d − µ i 1 ( i = 1 , 2 , 3). By using s 0 ( f i ) = f j f k / ( τ 1 1 τ 2 1 τ 3 1 ), we ha ve s 0 ( F Λ ) ( τ 1 1 τ 2 1 τ 3 1 ) d Q i = 1 , 2 , 3 ( f i ) µ i 1 = X k 1 + k 2 + k 3 = d A k 1 , k 2 , k 3 ( f 1 ) d − µ 1 1 − k 1 ( f 2 ) d − µ 2 1 − k 2 ( f 3 ) d − µ 3 1 − k 3 = : G ( x ) . W e let ˜ k i = d − µ i 1 − k i , ˜ d = 2 d − µ 1 1 − µ 2 1 − µ 3 1 and e A ˜ k 1 , ˜ k 2 , ˜ k 3 = A k 1 , k 2 , k 3 . The polynom ial G ( x ) can be then written as G ( x ) = X ˜ k 1 + ˜ k 2 + ˜ k 3 = ˜ d e A ˜ k 1 , ˜ k 2 , ˜ k 3 ( f 1 ) ˜ k 1 ( f 2 ) ˜ k 2 ( f 3 ) ˜ k 3 , where e A ˜ k 1 , ˜ k 2 , ˜ k 3 = 0 unless 0 ≤ ˜ k i ≤ d − µ i 1 = ˜ d − ( d − µ j 1 − µ k 1 ). Therefore, the curve { G ( x ) = 0 } represents the divisor class s 0 . Λ . Since f i | x = (1 , 1 , 1) = 1, the normalizing condition (1.3) of F Λ yields P A k 1 , k 2 , k 3 = 1. Hence G (1 , 1 , 1) = P e A ˜ k 1 , ˜ k 2 , ˜ k 3 = P A k 1 , k 2 , k 3 = 1, and so G = F s 0 . Λ as desired.  6 Remark 1.3 . By virtue of the geometric characterization (1.4) of tau-function, we can trace the resulting v alue for any i teration of the W eyl g roup acti on (Theorem 1.1) by usi ng only the d efining polynomial s of appropriate plane curves, although it is in general di ffi cult to compute c omposit ion of giv en rati onal maps. Also, note that the defining polynom ial of a curv e has a determinantal expression in volving information with respect to the multipli cities at enough points; see e.g. [11]. 2 A ffi ne case and di ff erence Painl ev ´ e equations of type E From Theorem 1.1, one can als o realize t he W eyl group W = W ( T ℓ 1 ,ℓ 2 ,ℓ 3 ) as b irational transforma- tions of inh omogeneous coordinates of P 2 . In t his context, t au-functions play roles of heights in t he sense that the orig inal inhomogeneous coordinate is reco vered as a ratio of them. In this section, we consider the a ffi ne case: E (1) 6 = T 3 , 3 , 3 , E (1) 7 = T 4 , 4 , 2 and E (1) 8 = T 6 , 3 , 2 , in which o ur realization of W eyl groups on inhomo geneous coordinates is equiv alent to that of Sakai’ s formulation [17], and therefore i t is rele vant to discrete Painle v ´ e equations. Recall that discrete Painle v ´ e equations are divided into three types: di ff erence, q -di ff erence or ell iptic-di ff erence; our setti ng corresponds to th e di ff erence one. For each case, we wi ll derive the di ff ere nce P ainlev ´ e equations in terms of appropriately chosen coordinates. 2.1 E (1) 6 case ( ℓ 1 = ℓ 2 = ℓ 3 = 3 ) The Dynkin diagram of type E (1) 6 = T 3 , 3 , 3 is ❝ 1 ❝ 2 ❝ 3 ❝ 4 ❝ 5 ❝ 6 ❝ 0 Let ( a 0 , a 1 , a 2 , a 3 , a 4 , a 5 , a 6 ) be the root var iables of E (1) 6 . The transformati on from the ori ginal var iables a 0 , a m i (used in Section 1) is giv en a s follows: a 0 : ⇒ a 3 , a 1 { 1 , 2 } : ⇒ a { 2 , 1 } , a 2 { 1 , 2 } : ⇒ a { 4 , 5 } , a 3 { 1 , 2 } : ⇒ a { 6 , 0 } . W e define the action of simple reflections s i on the root variables by s i ( a j ) = a j − a i C i j , where C i j is th e Cartan matrix of E (1) 6 . Let ι i ( i = 1 , 2 , 3) b e t he d iagram autom orphisms d efined by ι 1 ( a { 0 , 1 , 2 , 3 , 4 , 5 , 6 } ) = a { 5 , 1 , 2 , 3 , 6 , 0 , 4 } , ι 2 ( a { 0 , 1 , 2 , 3 , 4 , 5 , 6 } ) = a { 1 , 0 , 6 , 3 , 4 , 5 , 2 } and ι 3 ( a { 0 , 1 , 2 , 3 , 4 , 5 , 6 } ) = a { 0 , 5 , 4 , 3 , 2 , 1 , 6 } . Let us consider the change of the var iables ϕ : P 2 → P 1 × P 1 giv en by ϕ : [ x 1 : x 2 : x 3 ] 7→ ( u , v ) = F h − e 1 1 − e 2 1 ( a ; x ) x 2 − x 3 , F h − e 1 1 − e 2 1 ( a ; x ) x 1 − x 3 ! . This tra nsformation is obtained by blo wing up the points P 1 1 and P 2 1 , and blowing down (the proper transform of) the line { F h − e 1 1 − e 2 1 = 0 } whi ch passes through the t wo points. Note that th e sum of three l ines L 1 ∪ L 2 ∪ L 3 , which is in v ariant under the action of the W eyl group, i s generically transformed into { u = ∞} ∪ { v = ∞} ∪ { u = v } by ϕ . 7 From Theorem 1.1, w e have t he fol lowing realization of e W ( E (1) 6 ) = h s 0 , . . . , s 6 , ι 1 , ι 2 , ι 3 i on ( u , v ): s 2 ( u ) = u ( v − a 2 ) v , s 2 ( v ) = v − a 2 , s 3 ( u ) = u + a 3 , s 3 ( v ) = v + a 3 , s 4 ( u ) = u − a 4 , s 4 ( v ) = v ( u − a 4 ) u , ι 1 ( u ) = − u − a 3 , ι 1 ( v ) = v ( u + a 3 ) u − v , ι 2 ( u ) = u ( v + a 3 ) v − u , ι 2 ( v ) = − v − a 3 , ι 3 ( u ) = v , ι 3 ( v ) = u . Here the s ymbol e W stands for the extended group of W by its diagram automorphisms. The birational actio n o f a translation in e W ( E (1) 6 ) yi elds the di ff erence P ainle v ´ e equat ion. W e take an element T = ( ι 3 ι 1 s 6 s 0 s 3 s 4 s 6 s 3 s 5 s 4 ) 2 acting on the root variables as their shifts: T ( a 0 , a 1 , a 2 , a 3 , a 4 , a 5 , a 6 ) = ( a 0 , a 1 , a 2 + δ , a 3 , a 4 − δ , a 5 , a 6 ), where δ = a 0 + a 1 + 2 a 2 + 3 a 3 + 2 a 4 + a 5 + 2 a 6 corresponds to t he null root. Let u = T ( u ) and v = T − 1 ( v ). Then we ha ve the system of functional equations: ( u − v )( u − v ) = v ( v + a 3 )( v + a 3 + a 6 )( v + a 3 + a 6 + a 0 ) ( v − a 2 )( v − a 1 − a 2 ) , (2.1a) ( v − u )( v − u ) = u ( u + a 3 )( u + a 3 + a 6 )( u + a 3 + a 6 + a 0 ) ( u − a 4 )( u − a 4 − a 5 ) . (2.1b) This is called the di ff er ence P ai nlev ´ e equation of type E (1) 6 . 2.2 E (1) 7 case ( ℓ 1 = ℓ 2 = 4 , ℓ 3 = 2 ) The Dynkin diagram of type E (1) 7 = T 4 , 4 , 2 is ❝ 0 ❝ 1 ❝ 2 ❝ 3 ❝ 4 ❝ 5 ❝ 6 ❝ 7 Let ( a 0 , a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 ) be th e roo t variables o f E (1) 7 . Here we have replaced the variables a 0 , a m i (used in Section 1) as a 0 : ⇒ a 3 , a 1 { 1 , 2 , 3 } : ⇒ a { 2 , 1 , 0 } , a 2 { 1 , 2 , 3 } : ⇒ a { 4 , 5 , 6 } , a 3 1 : ⇒ a 7 . Define the actio n of simple reflections s i on the root variables by s i ( a j ) = a j − a i C i j , where C i j is the Cartan matrix of E (1) 7 . Let ι be the diagram autom orphism defined by ι ( a { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 } ) = a { 6 , 5 , 4 , 3 , 2 , 1 , 0 , 7 } . 8 W e first blo w u p th e two points P 3 1 , P 3 2 and blow down (t he proper transform of) the line L 3 = { x 1 = x 2 } . This procedure yields the birational map ϕ 1 : P 2 → P 1 × P 1 giv en by ϕ 1 : [ x 1 : x 2 : x 3 ] 7→ ( u 1 , v 1 ) = ( b 3 1 + 1)( x 1 + x 2 ) − 2 b 3 1 x 3 x 1 − x 2 , ( b 3 2 + 1)( x 1 + x 2 ) − 2 b 3 2 x 3 x 1 − x 2 ! . In addition, we apply ϕ 2 : ( u 1 , v 1 ) 7→ ( u , v ) = u 1 + a 0 + 2 a 1 + 3 a 2 − 3 a 4 − 2 a 5 − a 6 4 , v 1 + a 0 + 2 a 1 + 3 a 2 − 3 a 4 − 2 a 5 − a 6 4 ! . Note that L 1 ∪ L 2 ∪ L 3 is generically transformed by ϕ = ϕ 2 ◦ ϕ 1 into two curve s { u − v + a 7 = 0 } ∪ { u − v − a 7 = 0 } ⊂ P 1 × P 1 of bidegree (1 , 1) tangent to each other at ( ∞ , ∞ ). From Theorem 1.1, we hav e the follo wing realization of e W ( E (1) 7 ) on ( u , v ): s 2 ( u ) = u − a 2 , s 2 ( v ) = v − a 2 , s 3 ( v ) = ˜ v , s 4 ( u ) = u + a 4 , s 4 ( v ) = v + a 4 , ι ( u ) = − u , ι ( v ) = − v , where ˜ v = ˜ v ( u , v ) is a rational function determined by u − ˜ v + a 3 + a 7 u − ˜ v − a 3 − a 7 = ( u + a 3 )( u − v + a 7 ) ( u − a 3 )( u − v − a 7 ) . Let us take an element T = ( s 7 s 3 s 2 s 1 s 0 s 4 s 5 s 6 s 3 s 2 s 1 s 4 s 5 s 3 s 2 s 4 s 3 ) 2 ∈ W ( E (1) 7 ) acting on t he root variables as T ( a 0 , a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 ) = ( a 0 , a 1 , a 2 , a 3 − δ , a 4 , a 5 , a 6 , a 7 + 2 δ ), where δ = a 0 + 2 a 1 + 3 a 2 + 4 a 3 + 3 a 4 + 2 a 5 + a 6 + 2 a 7 . Let u = T ( u ) and v = T − 1 ( v ). Then we ha ve the system of functional equations: ( v − u + a 7 − δ )( v − u + a 7 ) ( v − u − a 7 + δ )( v − u − a 7 ) (2.2a) = ( u + a 3 )( u + a 3 + 2 a 4 )( u + a 3 + 2 a 4 + 2 a 5 )( u + a 3 + 2 a 4 + 2 a 5 + 2 a 6 ) ( u − a 3 )( u − a 3 − 2 a 2 )( u − a 3 − 2 a 2 − 2 a 1 )( u − a 3 − 2 a 2 − 2 a 1 − 2 a 0 ) , ( u − v − a 7 − δ )( u − v − a 7 ) ( u − v + a 7 + δ )( u − v + a 7 ) (2.2b) = ( v + a 3 + a 7 )( v + a 3 + 2 a 4 + a 7 )( v + a 3 + 2 a 4 + 2 a 5 + a 7 )( v + a 3 + 2 a 4 + 2 a 5 + 2 a 6 + a 7 ) ( v − a 3 − a 7 )( v − a 3 − 2 a 2 − a 7 )( v − a 3 − 2 a 2 − 2 a 1 − a 7 )( v − a 3 − 2 a 2 − 2 a 1 − 2 a 0 − a 7 ) . This is called the di ff er ence P ai nlev ´ e equation of type E (1) 7 . 2.3 E (1) 8 case ( ℓ 1 = 6 , ℓ 2 = 3 , ℓ 3 = 2 ) The Dynkin diagram of type E (1) 8 = T 6 , 3 , 2 is ❝ 0 ❝ 1 ❝ 2 ❝ 3 ❝ 4 ❝ 5 ❝ 6 ❝ 7 ❝ 8 9 Let ( a 0 , a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , a 8 ) be the root v ariables o f E (1) 8 . The interpretation from t he orig inal var iables a 0 , a m i (in Section 1) is gi ven by a 0 : ⇒ a 5 , a 1 { 1 , 2 , 3 , 4 , 5 } : ⇒ a { 4 , 3 , 2 , 1 , 0 } , a 2 { 1 , 2 } : ⇒ a { 6 , 7 } , a 3 1 : ⇒ a 8 . Define the action of simple reflections s i on the root variables by s i ( a j ) = a j − a i C i j , where C i j is the Cartan matrix of E (1) 8 . First we blow up t he two po ints P 2 1 , P 3 1 and blow down (th e proper transform) of the l ine { F h − e 2 1 − e 3 1 = 0 } . This yields the birational map ϕ 1 : P 2 → P 1 × P 1 of the form ϕ 1 : [ x 1 : x 2 : x 3 ] 7→ ( u 1 , v 1 ) = F h − e 2 1 − e 3 1 ( a ; x ) x 1 − x 2 , F h − e 2 1 − e 3 1 ( a ; x ) x 1 − x 3 ! . Secondly we bl ow up P 2 2 : ( u 1 , v 1 ) = ( − a 6 , ∞ ) and P 3 2 : ( u 1 , v 1 ) = ( ∞ , − a 8 ), and blo w do wn the lines { u 1 = − a 6 } and { u 1 = ∞} . Then we ha ve the birational map ϕ 2 : P 1 × P 1  writ ten as ϕ 2 : ( u 1 , v 1 ) 7→ ( u 2 , v 2 ) = ( u 1 + a 6 , ( u 1 + a 6 )( v 1 + a 8 )) . Thirdly , by blowing up ( u 2 , v 2 ) = (0 , 0) and P 2 3 : ( u 2 , v 2 ) = ( − a 7 , ∞ ) and contracting { v 2 = 0 } and { v 2 = ∞} , we obtain the birational map ϕ 3 : P 1 × P 1  , ϕ 3 : ( u 2 , v 2 ) 7→ ( u 3 , v 3 ) = v 2 ( u 2 + a 7 ) u 2 , v 2 ! . Finally we apply the linear transformation ϕ 4 : ( u 3 , v 3 ) 7→ ( u , v ) =  4 u 3 + ( a 6 + a 7 − a 8 ) 2 , 4 v 3 + ( a 6 − a 8 ) 2  . W e see t hat L 1 ∪ L 2 ∪ L 3 is generically transformed b y ϕ = ϕ 4 ◦ ϕ 3 ◦ ϕ 2 ◦ ϕ 1 into the curve { ( u − v ) 2 − 2 a 7 2 ( u + v ) + a 7 4 = 0 } ⊂ P 1 × P 1 of bidegree (2 , 2) with a cusp at ( ∞ , ∞ ). By virtue of Theorem 1.1, we hav e the following realization of W ( E (1) 8 ) on ( u , v ): s 6 ( u ) = ˜ u , s 7 ( u ) = v , s 7 ( v ) = u , where ˜ u = ˜ u ( u , v ) is a rational function determined by ˜ u − ( a 7 + a 8 ) 2 ˜ u − ( a 7 − a 8 ) 2 = u − ( a 6 + a 7 + a 8 ) 2 u − ( a 6 + a 7 − a 8 ) 2 · v − ( a 6 − a 8 ) 2 v − ( a 6 + a 8 ) 2 . Alternative ly , wit h respect to the v ariable (cf. [14]) Γ = v − u + a 7 2 2 a 7 , we can describe simply the action of s 6 as a linear fractional transformation s 6 ( Γ ) = ( v + ( a 6 − a 8 )( a 6 + a 8 )) Γ + 2 a 6 v 2 a 6 Γ + v + ( a 6 − a 8 )( a 6 + a 8 ) . 10 Consider an element T = ( s 6 s 5 s 4 s 3 s 2 s 1 s 0 s 8 s 5 s 4 s 3 s 2 s 1 s 6 s 5 s 4 s 3 s 2 s 8 s 5 s 4 s 3 s 6 s 5 s 4 s 8 s 5 s 6 s 7 ) 2 ∈ W ( E (1) 8 ) acting on the root variables as T ( a 0 , a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , a 8 ) = ( a 0 , a 1 , a 2 , a 3 , a 4 , a 5 , a 6 + δ , a 7 − 2 δ , a 8 ), where δ = a 0 + 2 a 1 + 3 a 2 + 4 a 3 + 5 a 4 + 6 a 5 + 4 a 6 + 2 a 7 + 3 a 8 . Let u = T ( u ) and v = T − 1 ( v ). Then we hav e the system of functional equations:  u − v − ( a 7 − δ ) 2   u − v − a 7 2  + 4 a 7 ( a 7 − δ ) v 2 a 7  u − v − ( a 7 − δ ) 2  + 2( a 7 − δ )  u − v − a 7 2  = G 4 ( θ ; v ) G 3 ( θ ; v ) , (2.3a)  u − v + ( a 7 + δ ) 2   u − v + a 7 2  + 4 a 7 ( a 7 + δ ) u 2 a 7  u − v + ( a 7 + δ ) 2  + 2( a 7 + δ )  u − v + a 7 2  = G 4 ( θ + a 7 1 ; u ) G 3 ( θ + a 7 1 ; u ) , (2.3b) called the di ff er ence P ainlev ´ e equation of type E (1) 8 . Here we introduce the polynom ials G 4 ( θ ; v ) = v 4 + σ 2 v 3 + σ 4 v 2 + σ 6 v + σ 8 , G 3 ( θ ; v ) = σ 1 v 3 + σ 3 v 2 + σ 5 v + σ 7 , where σ i is the i -th elem entary symmetric function of t he eight variables θ = ( θ 1 , . . . , θ 8 ) with θ 1 = a 6 − a 8 , θ 2 = s 8 ( θ 1 )( = a 6 + a 8 ), θ 3 = s 5 ( θ 2 ), θ 4 = s 4 ( θ 3 ), θ 5 = s 3 ( θ 4 ), θ 6 = s 2 ( θ 5 ), θ 7 = s 1 ( θ 6 ) and θ 8 = s 0 ( θ 7 ), and the symbol 1 denotes (1 , . . . , 1 ). Remark 2 .1 . For reference, we m ention some known results about the di ff erence Painle v ´ e equa- tions of type E (1) r ( r = 6 , 7 , 8); these equations were origin ally discovere d by Grammaticos-Ohta- Ramani [15] (see also good re view articles [6, 18] wit h a list of discrete Painle v ´ e equation s). The associated Lax formalisms (or linear problems) were established by Ari nkin-Borodin [1] for E (1) 6 and by Boalch [3] for E (1) 7 and E (1) 8 , respectively . It was recently reported by Kajiwara [8] th at for all E (1) r cases they admi t special so lutions in t erms of the hypergeometric functions in Askey’ s scheme as well as the q -di ff erence ones [10]. It often occurs in discrete / continuous Painlev ´ e equations that an in teresting class of solution s (written by means of Schur function and its variations) is generated by the W eyl group action from a fixed point of diagram automorphism s. In connection with integrable system s like KP hierarchy , we will study elsewhere such s olutions based on the frame work of tau-functions (cf. [13, 1 9]). Ack nowledg ements . The author wis hes to thank Ken ji Kajiwara ,T etsu Masuda, Mi kio Mur ata an d Y asuhi ko Y amada for fru itful discussions . Research of the author is support ed in part by JSPS Grant 198400 39. References [1] Arinkin, D., Borodin , A.: Moduli spaces of d -conne ctions and di ff er ence Pain le v ´ e equ ations . Duke Math. J. 134 , 515–5 56 (2006 ) [2] Barth, W ., Hulek, K., Peters, C., V an de V en, A.: Compact complex surfaces . 2nd edn. Berlin: Springer , 2004 [3] Boalch, P .: Quiv ers and di ff erence Painle v ´ e equa tions. arXiv:07 06.263 4 (preprint) 11 [4] Coble, A . B.: Algebraic geometry and theta function s. A mer . Math. Soc., Provide nce, RI, 1929 [5] Dolgache v , I., Ortland, D. : Point sets in projecti ve spaces and theta function s. 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