Universal character and q-difference Painleve equations with affine Weyl groups

Uni v ersal character and q -di ff erenc e P ainle v ´ e equations with a ffi ne W e yl groups T eruhisa TSUD A Abstract The univ ersal charac ter is a poly nomial attach ed to a pair of partition s and is a generaliza tion of the Schur polynomial. In this paper , we introduc e an integrab le system of q -d i ff erence lattice equ ations satisfied by the uni vers al char acter , and call it the latt ice q-UC hier ar chy . W e reg ard it a s general izing both q -K P and q -UC hierarchie s. Suita ble similarity and periodic reduct ions of the hierarchy yield the q -di ff erence Painle v ´ e equations of types A (1) 2 g + 1 ( g ≥ 1), D (1) 5 , an d E (1) 6 . As its c onsequence , a c lass of al gebraic soluti ons of the q -Pai nle v ´ e equations is rapidl y obtained by means of the uni versal chara cter . In pa rticular , we demonstrate expli citly the reduction procedure for the case of type E (1) 6 , via the frame work of τ -functions based on the geometry of certai n ration al surf aces. 2000 Mathematics Subject Classification 34M55, 37K10, 39A13. K eywor ds : a ffi ne W eyl group, q -Painle v ´ e equation, UC hierarch y , univ er sal character . 1 1 Introduction The present article is aimed t o clarify the underlying relationsh ip between th e u niv ersal character and the q -di ff erence Painle v ´ e equations f rom the viewpoint of infinite integra ble systems. The u niv ersal character S [ λ,µ ] , de fined by K. K oike [6], is a po lynomial attached to a pair of partitions [ λ, µ ] and is a generalization of the Schur polynom ial S λ . The univ ersal character describes the character of an irreducible ratio nal representation of the general l inear group, wh ile the Sc h ur polynomial, as is well-known, does that of an irreducible polynomial re p resentation; see [6], for details. The algebraic theory of the KP hiera rchy of nonlinear partial-di ff erential equations is probably the most b eautiful one in the fie l d of classical integrable systems. It was dis cover ed by M. Sato that the t otality o f solutions of th e KP hierarchy forms an infinit e-dimensional Grassmann manifol d; in particular , the set of homo geneous pol ynomial so lutions coincides wit h the who le set of Schur polynomial s; see [9, 14]. W e say that the KP hierarchy is an infinite int egrable system which characterizes the Schur polyn omials. On the other hand, an extension o f the KP hierarchy called the UC hierar chy was proposed by the author [15]; it is an infinite inte g rable system characterizing the univer sal chara cters as its homog eneous polynomial solutions ( see the table below). Character polyno mials versus Infinite integrable systems Schur polynomial S λ KP hierarchy ∩ ∩ Univ ersal character S [ λ,µ ] UC hierarchy In this paper , we first introduce an integrable system of q -di ff erence equations defined on two- dimensional lattice, ca lled th e lattice q-UC hierar chy (see Definition 2.2 ). It is consi dered as generalizing bot h q -KP and q -UC hierarchies, whi ch are the q -analogues of the KP and UC hi- erarchies; cf. [5] and [17] (see Remark 2 .5). Next we show that suit able simil arity and periodic reductions of the lattice q -UC hierarchy yield the q -Pa i nlev ´ e equations with a ffi ne W eyl group symmetries. Let us refer each of q -Painlev ´ e equations by the Dynkin diagram of associated root system; for example, the q -Painle v ´ e VI equation is represented by D (1) 5 ; see [1, 13]. Then our main result is stated as follows: Theor em 1.1. The q-P ainlev ´ e equations of types A (1) 2 g + 1 ( g ≥ 1) , D (1) 5 , and E (1) 6 can be obtained as certain simila rity r eduction s of the lattice q-UC hierar chy with the period ic conditions of or der ( g + 1 , g + 1) , (2 , 2) , and (3 , 3) , r espectively . W e shall demonstrate the proof of the abov e theorem in detail, particularly for the case of type E (1) 6 ; the other cases are briefly studied in Appendix. Recall that the (high er order) q -Painle v ´ e equation of type A (1) N − 1 is a further generalization of q -Painle v ´ e IV and V equations which corre spond to the cases N = 3 and 4, respectiv ely; see [4, 8]. As sho wn in [5], it can also be obtained as a similarity reduction of the q -KP hierarchy with N -periodicity . W ith this fact in mind, we summ arize in t he following t able how the q -Painle v ´ e equations relate to the similarity reductions of q -KP or lattice q -UC hierarchies with periodic conditions: q -Painle v ´ e equation A (1) 2 g A (1) 2 g + 1 D (1) 5 E (1) 6 q -KP hierarchy 2 g + 1 2 g + 2 – – Lattice q -UC hierarchy – ( g + 1 , g + 1) (2 , 2) (3 , 3) 2 The unive rs al characters are h omogeneous sol utions of the l attice q -UC hierarchy (see Propo- sition 2.3). Hence we hav e imm ediately from Theorem 1 .1 a class of algebraic sol utions o f the q -Painle v ´ e equations in terms of the universal character . Corollary 1.2. T h e q-P ainlev ´ e equatio ns of types A (1) 2 g + 1 ( g ≥ 1) , D (1) 5 , and E (1) 6 admit a class of algebraic s olutions e xpr essed i n terms of the univer sa l char acters attached to pairs of ( g + 1) -, 2 -, and 3 -cor e parti tions, r espectively . Remark 1.3 . (i) In K. Kajiwara et al. [5], rati onal soluti ons of the q -Painle v ´ e equations of type A (1) N − 1 were constructed by means of t he Schur po lynomial att ached to an N -core partitio n, via t he similarity reduction of the q -KP hierarchy . (ii) W e in vestigated certain simil arity reductions of the q -UC hierarchy and already obtained th e same class of soluti ons as above for the cases A (1) 2 g + 1 and D (1) 5 ; see [17] and [18]. Also , for A (1) 3 (the q -Painle v ´ e V equ ation), the rati onal solution s were firstly found by T . Masud a [8] without concerning any relationship to the infinite integrable syst ems. (iii) It i s sti ll an interesting open prob lem to obtain the q -Painlev ´ e equations of typ es E (1) 7 and E (1) 8 as reductions of some integrable hierarchies such as KP , UC, or beyond. In Section 2, we introdu ce the lattice q -UC hierarchy , which is an integrable system of q - di ff erence lattice equ ations satisfied by the univer sal characters (Definition 2.2 and Proposi tion 2.3). In Section 3, we present a birational representation of a ffi ne W eyl group of type E (1) 6 defined over the field of τ -functions , starting from a certain configuration of nine points in t he complex projec- tiv e plane (Theorem 3.2) . Then we define the q -P ainlev ´ e equation o f type E (1) 6 ( q - P ( E 6 )) by means of the translat ion part o f t he a ffi ne W eyl g roup (Definition 3.3). Section 4 concerns the system of bilinear equations satis fied by τ -functions (Propositio n 4.2). In Section 5, we sh ow t hat the bili near form of q - P ( E 6 ) coincides with a similarity re d uction of the lattice q -UC hierarc hy . Consequently , in Section 6, we have a class of algebraic solu tions of q - P ( E 6 ) in terms of t he universal character (Theorem 6.2). Section 7 is dev ot ed to the p roof of Proposit ion 2.3. W e briefly s um up in Ap- pendix results on the reductions to the q -Painle v ´ e equations of types A (1) 2 g + 1 and D (1) 5 . Note. Throughout this paper , we shall use the following con vention of q-shifted factorials : ( a ; q ) ∞ = ∞ Y i = 0 (1 − aq i ) , ( a ; p , q ) ∞ = ∞ Y i , j = 0 (1 − a p i q j ) , and also ( a 1 , . . . , a r ; q ) ∞ = ( a 1 ; q ) ∞ · · · ( a r ; q ) ∞ . 3 2 Universal cha racters and lattice q -UC hierar c hy 2.1 Univ ersal characters For a pair of sequences of integers λ = ( λ 1 , λ 2 , . . . , λ l ) and µ = ( µ 1 , µ 2 , . . . , µ l ′ ), the universal character S [ λ,µ ] ( x , y ) is a polynom ial in ( x , y ) = ( x 1 , x 2 , . . . , y 1 , y 2 , . . . ) defined by the determinant formula of twisted Jacobi–Trudi typ e (see [6]): S [ λ,µ ] ( x , y ) = d et p µ l ′ − i + 1 + i − j ( y ) , 1 ≤ i ≤ l ′ p λ i − l ′ − i + j ( x ) , l ′ + 1 ≤ i ≤ l + l ′ ! 1 ≤ i , j ≤ l + l ′ , (2.1) where p n is a polynomial defined by the generating function: X k ∈ Z p k ( x ) z k = exp        ∞ X n = 1 x n z n        . (2.2) Schur polynomial S λ ( x ) (see [7]) is regarded as a special case of the unive rsal char acter: S λ ( x ) = det  p λ i − i + j ( x )  = S [ λ, ∅ ] ( x , y ) . If we count the degree of variables as deg x n = n and deg y n = − n , then the universal character S [ λ,µ ] ( x , y ) is a weighted homog eneous polyn omial of degree | λ | − | µ | , wh ere | λ | = λ 1 + · · · + λ l . Namely , we ha ve S [ λ,µ ] ( c x 1 , c 2 x 2 , . . . , c − 1 y 1 , c − 2 y 2 , . . . ) = c | λ |−| µ | S [ λ,µ ] ( x 1 , x 2 , . . . , y 1 , y 2 , . . . ) , (2.3) for any nonzero constant c . Example 2.1 . When λ = (2 , 1) and µ = (1), t he unive rsal character is giv en by S [(2 , 1) , (1)] ( x , y ) =         p 1 ( y ) p 0 ( y ) p − 1 ( y ) p 1 ( x ) p 2 ( x ) p 3 ( x ) p − 1 ( x ) p 0 ( x ) p 1 ( x )         = x 1 3 3 − x 3 ! y 1 − x 1 2 . 2.2 Lattice q -UC hierar chy Let I ⊂ Z > 0 and J ⊂ Z < 0 be fini te indexing sets and t i ( i ∈ I ∪ J ) th e independent va riables. Let T i = T i ; q be the q -shift operator defined by T i ; q ( t i ) = ( qt i ( i ∈ I ) , q − 1 t i ( i ∈ J ) , and T i ; q ( t j ) = t j ( i , j ). W e use also the notation: T i 1 T i 2 · · · T i n = T i 1 i 2 ... i n , for the sake of bre vity . Definition 2.2. The follo wing system o f q -di ff erence equatio ns for unknowns σ m , n ( t ) ( m , n ∈ Z ) is called the lattice q-UC hierar chy : t i T i ( σ m , n + 1 ) T j ( σ m + 1 , n ) − t j T j ( σ m , n + 1 ) T i ( σ m + 1 , n ) = ( t i − t j ) T i j ( σ m , n ) σ m + 1 , n + 1 , (2.4) where i , j ∈ I ∪ J . 4 Let us consider the change of variables x n = P i ∈ I t i n − q n P j ∈ J t j n n (1 − q n ) , y n = P i ∈ I t i − n − q − n P j ∈ J t j − n n (1 − q − n ) , (2.5) then define the symmetri c function s [ λ,µ ] = s [ λ,µ ] ( t ) in t i ( i ∈ I ∪ J ) by s [ λ,µ ] ( t ) = S [ λ,µ ] ( x , y ) . (2.6) The unive rsal char acters solve the lattice q -UC hierarchy in the follo wing sense. Pr oposition 2.3. W e have t i T i ( s [ λ, ( k ′ ,µ )] ) T j ( s [( k ,λ ) ,µ ] ) − t j T j ( s [ λ, ( k ′ ,µ )] ) T i ( s [( k ,λ ) ,µ ] ) = ( t i − t j ) T i j ( s [ λ,µ ] ) s [( k ,λ ) , ( k ′ ,µ )] , (2.7) for any inte gers k , k ′ and sequences of inte gers λ = ( λ 1 , . . . , λ l ) , µ = ( µ 1 , . . . , µ l ′ ) . The proof of the proposit ion a b ove w ill be giv en in Section 7. Remark 2.4 . Define the functions h n = h n ( t ) and H n = H n ( t ) by h n ( t ) = p n ( x ) , H n ( t ) = p n ( y ) , under (2.5). W e no te also the following expression by the generating functions: ∞ X k = 0 h k ( t ) z k = Y i ∈ I , j ∈ J ( qt j z ; q ) ∞ ( t i z ; q ) ∞ , ∞ X k = 0 H k ( t ) z k = Y i ∈ I , j ∈ J ( q − 1 t j − 1 z ; q − 1 ) ∞ ( t i − 1 z ; q − 1 ) ∞ . (2.8) Hence, function s [ λ,µ ] ( t ) can be expressed as s [ λ,µ ] ( t ) = det H µ l ′ − i + 1 + i − j ( t ) , 1 ≤ i ≤ l ′ h λ i − l ′ − i + j ( t ) , l ′ + 1 ≤ i ≤ l + l ′ ! 1 ≤ i , j ≤ l + l ′ . (2.9) Remark 2.5 . (i) One can easily deduce from (2.4) the following equation: ( t i − t j ) T i j ( σ m , n ) T k ( σ m + 1 , n ) + ( t j − t k ) T jk ( σ m , n ) T i ( σ m + 1 , n ) + ( t k − t i ) T ik ( σ m , n ) T j ( σ m + 1 , n ) = 0 , (2.10) where i , j , k ∈ I ∪ J , which is exactly the bilinear equation of the q -UC hierarchy; see [17]. (ii) If σ m , n ( t ) does not depend on n , that is, σ m , n = σ m , n + 1 for all m and n , then (2.4) is reduced to the q -KP hierarchy (see [5]): t i T i ( ρ m ) T j ( ρ m + 1 ) − t j T j ( ρ m ) T i ( ρ m + 1 ) = ( t i − t j ) T i j ( ρ m ) ρ m + 1 , (2.11) where ρ m : = σ m , n . 5 3 τ -functions of q -Painlev ´ e equ ation In t his section we pre s ent a geometric formul ation of the q -Painle v ´ e equation o f type E (1) 6 by means of τ -functions; cf. [13]. Cons ider the configuration of nin e points in t he compl ex projecti ve plane P 2 , which are divided into three tri ples of coll inear point s. Let [ x : y : z ] be the homogeneous coordinate of P 2 . W e can normalize, without loss of generality , the nine points p i (1 ≤ i ≤ 9) under consideration as follows: p 1 = [0 : − 1 : a 3 ] , p 2 = [0 : − 1 : a 3 a 6 3 ] , p 3 = [0 : − 1 : a 3 a 6 3 a 0 3 ] , p 4 = [ a 3 : 0 : − 1] , p 5 = [ a 2 3 a 3 : 0 : − 1] , p 6 = [ a 1 3 a 2 3 a 3 : 0 : − 1] , p 7 = [ − 1 : a 3 : 0] , p 8 = [ − 1 : a 3 a 4 3 : 0] , p 9 = [ − 1 : a 3 a 4 3 a 5 3 : 0] , (3.1) where a i ∈ C × are parameters such t hat a 0 a 1 a 2 2 a 3 3 a 4 2 a 5 a 6 2 = q . Let ψ : X = X a → P 2 be the blowing-up at the nine poin ts. Let e i = ψ − 1 ( p i ) b e the exceptional divisor and h the divisor class corresponding to a hyperplane. W e thu s ha ve the Picard lattice: Pic( X ) = Z h ⊕ Z e 1 ⊕ · · · ⊕ Z e 9 , of rational surface X , equipped wit h the in tersection form ( | ) defined by ( h | h ) = 1, ( e i | e j ) = − δ i , j and ( h | e j ) = 0. T he anti-canonical divisor − K X is uniquely decomposed into prime divisors: − K X = 3 h − X 1 ≤ i ≤ 9 e i = D 1 + D 2 + D 3 , where D 1 = h − e 1 − e 2 − e 3 , D 2 = h − e 4 − e 5 − e 6 and D 3 = h − e 7 − e 8 − e 9 . Since the dual graph of the i ntersections o f D i ’ s is of type A (1) 2 , we call X the A (1) 2 -surface following the classification of t he generalized Halph en surfa ces due to H. Sakai [13]. The orthogonal complement ( − K X ) ⊥ def = { v ∈ Pic( X ) | ( v | D i ) = 0 for i = 1 , 2 , 3 } is iso morphic to the root latti ce of type E (1) 6 . In fact, ( − K X ) ⊥ is generated by the vectors α i j = e i − e j (where both i and j belong to the same indexing set { 1 , 2 , 3 } , { 4 , 5 , 6 } , or { 7 , 8 , 9 } ) and α i jk = h − e i − e j − e k ( i ≤ 3 < j ≤ 6 < k ); hence we can choose a root basis B = { α 0 , . . . , α 6 } defined by α 0 = α 23 , α 1 = α 56 , α 2 = α 45 , α 3 = α 147 , α 4 = α 78 , α 5 = α 89 , α 6 = α 12 , whose Dynkin diagram is of type E (1) 6 and looks as follows (see, e.g. , [2 ]): ❝ 1 ❝ 2 ❝ 3 ❝ 4 ❝ 5 ❝ 6 ❝ 0 Note that the 72 roots of E 6 are represented by α i j (18 vectors) and ± α i jk (54 vectors). W e define the action of the reflection corresponding to a root α ∈ ( − K X ) ⊥ by r α ( v ) = v + ( v | α ) α, v ∈ Pic( X ) . 6 W e prepare the not ations, r i j : = r α i j , r i jk : = r α i jk and s i : = r α i ( i = 0 , . . . , 6), for con venience. Also, the diagram automorphism ι i ( i = 1 , 2) is defined by ι 1 ( e { 1 , 2 , 3 , 7 , 8 , 9 } ) = e { 7 , 8 , 9 , 1 , 2 , 3 , } , ι 2 ( e { 1 , 2 , 3 , 4 , 5 , 6 } ) = e { 4 , 5 , 6 , 1 , 2 , 3 , } . W e thus obtain the linear actio n of th e (extended) a ffi ne W eyl group e W ( E (1) 6 ) = h s 0 , . . . , s 6 , ι 1 , ι 2 i on Pic( X ). In parallel, we fix the action of e W ( E (1) 6 ) on the multiplicat ive root v ariables a = ( a 0 , . . . , a 6 ) as follows: s i ( a j ) = a j a − C i j i , ι 1 ( a { 0 , 1 , 2 , 3 , 4 , 5 , 6 } ) = a { 5 , 1 , 2 , 3 , 6 , 0 , 4 } − 1 , ι 2 ( a { 0 , 1 , 2 , 3 , 4 , 5 , 6 } ) = a { 1 , 0 , 6 , 3 , 4 , 5 , 2 } − 1 , (3.2) where ( C i j ) being the Cartan matrix of type E (1) 6 . Next we shall extend the linear action above to bi rational transformation s. T o this end, we in- troduce the n otion of τ -functions; cf . [3]. Consider the field L = K ( τ 1 , . . . , τ 9 ) of rational functi ons in indeterminates τ i (1 ≤ i ≤ 9) with the coe ffi cient field K = C ( a 1 / 3 ) = C ( a 0 1 / 3 , . . . , a 6 1 / 3 ). T ake a sub-lattice M = S i = 1 , 2 , 3 M i of Pic( X ), where M i = n v ∈ Pic( X )    ( v | v ) = − ( v | D i ) = − 1 , ( v | D j ) = 0 ( j , i ) o . Definition 3.1. A function τ : M → L is said to be a τ -function i ff it satisfies the conditi ons: (i) τ ( w . v ) = w .τ ( v ) for any v ∈ M and w ∈ e W ( E (1) 6 ); (ii) τ ( e i ) = τ i (1 ≤ i ≤ 9). Such functions and the action of e W ( E (1) 6 ) on them are explicitl y determi ned in the following way . Any divisor Λ = nh − e i 1 − · · · − e i n + 1 ∈ M corresponds t o a curve of degree n o n P 2 passing through n + 1 points p i 1 , . . . , p i n + 1 (with counting the mul tiplicity). W e can cho ose uniq uely the normalized defining polynom ial F Λ ( x , y , z ) = P i + j + k = n A i jk x i y j z k ∈ Q ( a )[ x , y , z ] of the curve, such that Q A i jk = 1. For example, we ha ve F h − e 1 − e 4 = a 3 − 1 x + a 3 y + z , F h − e 4 − e 7 = x + a 3 − 1 y + a 3 z , F h − e 1 − e 7 = a 3 x + y + a 3 − 1 z . Let x c x , y c y , z c z ! = ( τ 1 τ 2 τ 3 , τ 4 τ 5 τ 6 , τ 7 τ 8 τ 9 ) , (3.3) where c x = a 1 1 3 a 2 2 3 a 4 − 2 3 a 5 − 1 3 , c y = a 5 1 3 a 4 2 3 a 6 − 2 3 a 0 − 1 3 , c z = a 0 1 3 a 6 2 3 a 2 − 2 3 a 1 − 1 3 . Suppose that F Λ ( x , y , z ) = τ ( nh − e i 1 − · · · − e i n + 1 ) τ ( e i 1 ) · · · τ ( e i n + 1 ) . (3.4) Therefore we see that the linear action o f e W ( E (1) 6 ) on M yield s the actio n on τ -function s immedi- ately . For instance, by using r i jk ( e k ) = h − e i − e j , we can compute the action of r i jk : r i jk ( τ ( e k )) = τ ( h − e i − e j ) = F h − e i − e j ( x , y , z ) τ ( e i ) τ ( e j ) . Each actio n of r i j and diagram automorphis m ι i is realized as just a permutatio n of τ i ’ s. Summa- rizing above , we now arri ve at the following theorem . 7 Theor em 3.2. Define the birational transformatio ns s i (0 ≤ i ≤ 6) and ι j ( j = 1 , 2) on L = C ( a 1 / 3 )( τ 1 , . . . , τ 9 ) by s 1 ( τ { 5 , 6 } ) = τ { 6 , 5 } , s 2 ( τ { 4 , 5 } ) = τ { 5 , 4 } , s 4 ( τ { 7 , 8 } ) = τ { 8 , 7 } , s 5 ( τ { 8 , 9 } ) = τ { 9 , 8 } , s 6 ( τ { 1 , 2 } ) = τ { 2 , 1 } , s 0 ( τ { 2 , 3 } ) = τ { 3 , 2 } , ι 1 ( τ { 1 , 2 , 3 } ) = τ { 7 , 8 , 9 } , ι 2 ( τ { 1 , 2 , 3 } ) = τ { 4 , 5 , 6 } , s 3 ( τ 1 ) =  c x τ 1 τ 2 τ 3 + a 3 − 1 c y τ 4 τ 5 τ 6 + a 3 c z τ 7 τ 8 τ 9  / ( τ 4 τ 7 ) , s 3 ( τ 4 ) =  a 3 c x τ 1 τ 2 τ 3 + c y τ 4 τ 5 τ 6 + a 3 − 1 c z τ 7 τ 8 τ 9  / ( τ 1 τ 7 ) , s 3 ( τ 7 ) =  a 3 − 1 c x τ 1 τ 2 τ 3 + a 3 c y τ 4 τ 5 τ 6 + c z τ 7 τ 8 τ 9  / ( τ 1 τ 4 ) . (3.5) Then (3.5) with (3.2) pr ovide a r ealization of e W ( E (1) 6 ) = h s 0 , . . . , s 6 , ι 1 , ι 2 i . Let [ f : g : 1] = " x c x : y c y : z c z # = [ τ 1 τ 2 τ 3 : τ 4 τ 5 τ 6 : τ 7 τ 8 τ 9 ] . (3.6) By virtue of Theorem 3.2, we obtain t he following birational transform ations on the inhom oge- neous coordinate ( f , g ): s 3 ( f ) = f c x f + a 3 − 1 c y g + a 3 c z a 3 − 1 c x f + a 3 c y g + c z , s 3 ( g ) = g a 3 c x f + c y g + a 3 − 1 c z a 3 − 1 c x f + a 3 c y g + c z , ι 1 ( f ) = 1 f , ι 1 ( g ) = g f , ι 2 ( f ) = g , ι 2 ( g ) = f . (3.7) The birational actio n arisi ng from the t ranslation part of a ffi ne W eyl group can be regarded as a discrete d ynamical sys tem and is called a di screte Painlev ´ e equation; cf . [12]. Consider an element ℓ = r 258 r 369 r 258 r 147 = ( s 2 s 4 s 6 s 0 s 1 s 5 s 3 s 2 s 4 s 6 s 3 ) 2 ∈ W ( E (1) 6 ) , (3.8) acting on the parameters a = ( a 0 , . . . , a 6 ) as their q -shifts: ℓ ( a ) = a = ( a 0 , a 1 , q − 1 a 2 , q 2 a 3 , q − 1 a 4 , a 5 , q − 1 a 6 ) . (3.9) W e define rational functions F ( a ; f , g ) , G ( a ; f , g ) ∈ C ( a 1 / 3 ; f , g ) by ℓ ( f ) = F ( a ; f , g ) , ℓ ( g ) = G ( a ; f , g ) . (3.10) Definition 3.3. The system of functional equations f ( a ) = F ( a ; f ( a ) , g ( a )) , g ( a ) = G ( a ; f ( a ) , g ( a )) , (3.11) for unknowns f = f ( a ) and g = g ( a ) is called the q-P ainlev ´ e equation of type E (1) 6 . W e shall often denote (3.11) shortly by q - P ( E 6 ). Remark 3.4 . W e hav e th e following inclusion relation of a ffi ne W eyl groups: W ( E (1) 6 ) ⊃ W ( A (1) 5 ) ⊕ W ( A (1) 1 ). For instance, the sets of vectors B ′ = { α 158 , α 367 , α 248 , α 169 , α 257 , α 349 } and B ′′ = { α 147 , α 258 + α 369 } realize the root bases of t ypes A (1) 5 and A (1) 1 , re s pectiv ely . M oreover , the y are mutually ort hog- onal. The transformation ℓ , used to define the q -P ainlev ´ e equation (3.11), is exactly the translation in W ( A (1) 1 ); that is, r α 258 + α 369 r α 147 = ( r 258 r 369 r 258 ) r 147 = ℓ . 8 4 Bilinear equations among τ -funct ions Let us i ntroduce the transformations ℓ 2 = r 369 r 147 r 369 r 258 and ℓ 3 = r 147 r 258 r 147 r 369 , in parallel with ℓ 1 = ℓ = r 258 r 369 r 258 r 147 . These act on the root variables as their q -shifts: ℓ 1 ( a ) = ( a 0 , a 1 , q − 1 a 2 , q 2 a 3 , q − 1 a 4 , a 5 , q − 1 a 6 ) , ℓ 2 ( a ) = ( q − 1 a 0 , q − 1 a 1 , qa 2 , q − 1 a 3 , qa 4 , q − 1 a 5 , qa 6 ) , ℓ 3 ( a ) = ( qa 0 , qa 1 , a 2 , q − 1 a 3 , a 4 , qa 5 , a 6 ) . (4.1) Note that ℓ i ’ s are mutually commutable and ℓ 1 ℓ 2 ℓ 3 = id. Th e action of ℓ i on the auxiliary va ri ables a = ( a 0 a 1 a 5 ) 1 / 3 , b = ( a 2 a 4 a 6 q ) 1 / 3 , (4.2) is described as follows: ℓ 1 ( a , b ) = ( a , q − 1 b ) , ℓ 2 ( a , b ) = ( q − 1 a , qb ) , ℓ 3 ( a , b ) = ( qa , b ) . (4.3) Lemma 4.1. It holds that τ 3 ℓ 3 ( τ 6 ) − a 2 b ℓ 3 ( τ 3 ) τ 6 = a 1 2 a 2 a 0 2 a 6 ! 1 / 3 1 − a 6 b 3 a 2 b τ 7 τ 8 . (4.4) Pr oof. W e hav e (see Section 3) F h − e 3 − e 9 ( x , y , z ) = a 0 a 3 a 4 2 a 5 2 a 6 x + a 0 a 6 y a 4 a 5 + z a 0 2 a 3 a 4 a 5 a 6 2 , F h − e 6 − e 9 ( x , y , z ) = a 4 a 5 x a 1 a 2 + y a 1 a 2 a 3 a 4 2 a 5 2 + a 1 2 a 2 2 a 3 a 4 a 5 z . Eliminati ng x and y , we get F h − e 3 − e 9 − a 0 a 1 a 2 a 3 a 4 a 5 a 6 F h − e 6 − e 9 = 1 − ( a 0 a 1 a 2 a 3 a 4 a 5 a 6 ) 3 a 0 2 a 3 a 4 a 5 a 6 2 z . (4.5) Recall t hat z = c z τ 7 τ 8 τ 9 and F h − e i − e j = τ i τ j τ ( h − e i − e j ). By virtue of ℓ 3 ( e 6 ) = h − e 3 − e 9 and ℓ 3 ( e 3 ) = h − e 6 − e 9 , we thus obtain (4.4) from (4.5).  W e shall rename the τ -functions as follows: U { 1 , 2 , 3 } = τ { 1 , 4 , 7 } N ( a , b ) , V { 1 , 2 , 3 } = τ { 2 , 5 , 8 } N ( q 1 / 3 a , q − 2 / 3 b ) , W { 1 , 2 , 3 } = τ { 3 , 6 , 9 } N ( q − 1 / 3 a , q − 1 / 3 b ) , (4.6) where the normalization factor N ( a , b ) is defined by N ( a , b ) =  − aq b , − ab 2 q , − q a 2 b ; q , q  ∞ b 3 q 3 a 3 , q 3 a 3 b 6 , a 6 b 3 q 3 ; q 3 , q 3 ! ∞ b 2 q 2 a 2 , q 2 a 2 b 4 , a 4 b 2 q 2 ; q 2 , q 2 ! ∞ . (4.7) Equation (4.4) in Lemma 4.1 is then rewritten into 1 a W 1 ℓ 3 ( W 2 ) − ab ℓ 3 ( W 1 ) W 2 = a 1 2 a 2 a 0 2 a 6 ! 1 / 3 1 a − ab ! U 3 V 3 , (4.8) 9 by straig htforward comput ation. As seen below , all the other bil inear equ ations for U i , V i and W i can also be deriv ed from (4.8) by s uitable symm etries of e W ( E (1) 6 ). Applyi ng r 13 r 46 r 79 to (4.8) and viewing that ℓ 1 = r 13 r 46 r 79 ℓ 3 r 13 r 46 r 79 , we thus obtain abU 1 ℓ 1 ( U 2 ) − q b ℓ 1 ( U 1 ) U 2 = a 0 a 6 2 a 1 a 2 2 ! 1 / 3  ab − q b  V 3 W 3 . (4.9) Moreover , we consider an element π = s 0 s 1 s 5 ι 1 ι 2 ∈ e W ( E (1) 6 ) of o rder six whose action is give n as follows: π : ( a 0 , a 1 , a 2 , a 3 , a 4 , a 5 , a 6 ; τ { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } ) 7→ 1 a 5 , 1 a 0 , a 0 a 6 , a 3 , a 1 a 2 , 1 a 1 , a 4 a 5 ; τ { 7 , 9 , 8 , 1 , 3 , 2 , 4 , 6 , 5 } ! . Hence we see that π : ( a , b ; U i , V i , W i ) 7→ 1 a , ab ; U i − 1 , W i − 1 , V i − 1 ! , (4.10) and also that the commutation relations πℓ 1 = ℓ 1 π , πℓ 2 = ℓ 3 π and πℓ 3 = ℓ 2 π hold. Not e that π realizes the rotational diagram automorphis m of A (1) 5 , considered in Remark 3.4. Applying π to (4.8) and (4.9), we get the following proposition. Pr oposition 4.2. The following bili near equations among the τ -functions U i , V i and W i hold : abU i ℓ 1 ( U i + 1 ) − q b ℓ 1 ( U i ) U i + 1 = γ i  ab − q b  V i + 2 W i + 2 , (4.11a) 1 b V i ℓ 2 ( V i + 1 ) − 1 a ℓ 2 ( V i ) V i + 1 = δ i 1 b − 1 a ! W i + 2 U i + 2 , (4.11b) 1 a W i ℓ 3 ( W i + 1 ) − ab ℓ 3 ( W i ) W i + 1 = ǫ i 1 a − ab ! U i + 2 V i + 2 , (4.11c) for i ∈ Z / 3 Z . Her e γ i , δ i and ǫ i ar e the parameters defined by γ 1 = a 0 a 6 2 a 1 a 2 2 ! 1 / 3 , γ 2 = a 1 a 2 2 a 4 2 a 5 ! 1 / 3 , γ 3 = a 4 2 a 5 a 0 a 6 2 ! 1 / 3 , δ 1 = a 0 a 2 a 1 a 6 ! 1 / 3 , δ 2 = a 1 a 4 a 2 a 5 ! 1 / 3 , δ 3 = a 5 a 6 a 0 a 4 ! 1 / 3 , ǫ 1 = a 1 2 a 2 a 0 2 a 6 ! 1 / 3 , ǫ 2 = a 4 a 5 2 a 1 2 a 2 ! 1 / 3 , ǫ 3 = a 0 2 a 6 a 4 a 5 2 ! 1 / 3 . (4.12) W e call system (4.11) the bilinear form of the q-P a inlev ´ e e q uation of type E (1) 6 . Con versely , we can verify that, for an y function s U i , V i , W i ( i ∈ Z / 3 Z ) satisfying (4.11), the pair ( f , g ) defined by f = U 1 V 1 W 1 U 3 V 3 W 3 , g = U 2 V 2 W 2 U 3 V 3 W 3 , certainly solves the q -P ainl ev ´ e equation (3.11); here we recall (3.6) and (4.6). 10 5 Similarity r e duction of lattice q -UC hierarch y to q - P ( E 6 ) W e shall explain how the bilinear form of q - P ( E 6 ), (4.11), arises naturally from the lattice q -UC hierarchy , through certain periodi c and si milarity reductions. Let I = { 1 , 2 , 3 } and J = ∅ and consider the lattice q -UC hierarchy: t i T i ( σ m , n + 1 ) T j ( σ m + 1 , n ) − t j T j ( σ m , n + 1 ) T i ( σ m + 1 , n ) = ( t i − t j ) T i j ( σ m , n ) σ m + 1 , n + 1 . (5.1) W e impos e the (3 , 3)-periodic condition: σ m , n = σ m + 3 , n = σ m , n + 3 , (5.2) and the simil arity condition: σ m , n ( ct 1 , ct 2 , ct 3 ) = c d m , n σ m , n ( t 1 , t 2 , t 3 ) , (5.3) for any c ∈ C × . Here d m , n are const ant parameters such that d m , n + d m + 1 , n + 1 = d m + 1 , n + d m , n + 1 . W e introduce the functions e σ m , n ( a , b ) i n two variables defined by e σ m , n ( a , b ) = σ m , n ( t 1 , t 2 , t 3 ) under the substitut ion ( t 1 , t 2 , t 3 ) = ( a − 1 , b − 1 , ab ). W e thus ha ve the follo win g lemma. Lemma 5.1. Let U i ( a , b ) = e σ i , − i ( a , b ) , V i ( a , b ) = e σ i + 1 , − i + 1 ( q 1 / 3 a , q − 2 / 3 b ) , (5.4) W i ( a , b ) = e σ i + 2 , − i + 2 ( q − 1 / 3 a , q − 1 / 3 b ) , for i ∈ Z / 3 Z . Then these functions satisf y th e bi linear f orm of q- P ( E 6 ) , (4.11) , with t he parameters : γ i = q ( d i , − i + 2 − d i + 1 , − i ) / 3 , δ i = q ( d i + 1 , − i − d i + 2 , − i + 1 ) / 3 , ǫ i = q ( d i + 2 , − i + 1 − d i , − i + 2 ) / 3 . (5.5) Pr oof. B eing at tentive to the action of ℓ i ’ s on variables a and b (see (4.3)), one can deduce the bilinear form of q - P ( E 6 ) straightforwardly from th e l attice q -UC hierarchy (5. 1) by the s imilarity condition (5.3) together with the periodicity (5.2). For instance, we shall start from (5.1) with ( m , n ) = ( r + 1 , − r ) and ( i , j ) = (1 , 2): t 1 σ r + 1 , − r + 1 ( qt 1 , t 2 , t 3 ) σ r + 2 , − r ( t 1 , qt 2 , t 3 ) − t 2 σ r + 1 , − r + 1 ( t 1 , qt 2 , t 3 ) σ r + 2 , − r ( qt 1 , t 2 , t 3 ) = ( t 1 − t 2 ) σ r + 1 , − r ( qt 1 , qt 2 , t 3 ) σ r + 2 , − r + 1 ( t 1 , t 2 , t 3 ) . By using the homogeneity (5.3), we hav e q ( d r + 1 , − r + 1 + d r + 2 , − r ) / 3 t 1 σ r + 1 , − r + 1 ( q 2 / 3 t 1 , q − 1 / 3 t 2 , q − 1 / 3 t 3 ) σ r + 2 , − r ( q − 1 / 3 t 1 , q 2 / 3 t 2 , q − 1 / 3 t 3 ) − q ( d r + 1 , − r + 1 + d r + 2 , − r ) / 3 t 2 σ r + 1 , − r + 1 ( q − 1 / 3 t 1 , q 2 / 3 t 2 , q − 1 / 3 t 3 ) σ r + 2 , − r ( q 2 / 3 t 1 , q − 1 / 3 t 2 , q − 1 / 3 t 3 ) = q 2 d r + 1 , − r / 3 ( t 1 − t 2 ) σ r + 1 , − r ( q 1 / 3 t 1 , q 1 / 3 t 2 , q − 2 / 3 t 3 ) σ r + 2 , − r + 1 ( t 1 , t 2 , t 3 ) . Putting ( t 1 , t 2 , t 3 ) = ( a − 1 , b − 1 , ab ), therefore we obtain 1 a e σ r + 1 , − r + 1 ( q − 2 / 3 a , q 1 / 3 b ) e σ r + 2 , − r ( q 1 / 3 a , q − 2 / 3 b ) − 1 b e σ r + 1 , − r + 1 ( q 1 / 3 a , q − 2 / 3 b ) e σ r + 2 , − r ( q − 2 / 3 a , q 1 / 3 b ) = q ( d r + 1 , − r − d r + 2 , − r + 1 ) / 3 1 a − 1 b ! e σ r + 1 , − r ( q − 1 / 3 a , q − 1 / 3 b ) e σ r + 2 , − r + 1 ( a , b ) , which turns out to coincide wi th (4.11b ) in vie w of the action of ℓ 2 . In the same way , we can deri ve also (4.11a) and (4.11c). The p roof is now com plete.  11 6 Algebraic solutions of q -Painlev ´ e equatio n in term s of t he universal char acter As seen in t he preceding section, the q -Painle v ´ e equat ion of type E (1) 6 is in fact equiv alent to a similarity reduction of t he (periodic) lattice q -UC hierarchy . On the other hand, we have already known that t he lattice q -UC hierarchy admits the universal characters as i ts homogeneous soluti ons; see Propos ition 2.3. Consequently , we obtain in particular a class of alg ebraic soluti ons of t he q - Painle v ´ e equation in terms of the universal character . In order to s tate ou r result precisely , we first recall the notion of N -core partitions; see, e.g. , [10]. A subset M ⊂ Z is s aid to be a Maya diagram if m ∈ M ( m ≪ 0) and m < M ( m ≫ 0). Each Maya diagram M = { . . . , m 3 , m 2 , m 1 } corresponds to a unique partition λ = ( λ 1 , λ 2 , . . . ) such that m i − m i + 1 = λ i − λ i + 1 + 1. For a sequence of integers n = ( n 1 , n 2 , . . . , n N ) ∈ Z N , let us consider the Maya diagram M ( n ) = ( N Z < n 1 + 1) ∪ ( N Z < n 2 + 2) ∪ · · · ∪ ( N Z < n N + N ) , and denote by λ ( n ) the corresponding partition. Note that λ ( n ) = λ ( n + 1 ) where 1 = (1 , 1 , . . . , 1). W e call a partition of the form λ ( n ) an N -cor e partitio n . It is well-known t hat a partition λ is N -core if and o nly i f λ has n o hook with leng th of a multipl e of N . W e hav e a cyclic chain of th e univ ersal c haracters attached to N -core p artitions; see [16, Lemma 2.2]. Lemma 6.1. It holds that S   k i , λ ( n ( i − 1))  , µ  = ± S [ λ ( n ( i )) , µ ] , (6.1) for arbitrary n = ( n 1 , n 2 , . . . , n N ) ∈ Z N and partition µ . Her e n ( i ) = n + ( i z }| { 1 , . . . , 1 , N − i z }| { 0 , . . . , 0) an d k i = N n i − | n | with | n | = n 1 + n 2 + · · · + n N . Finally , by virtue of Proposition 2.3 and Lemmas 5.1 and 6 .1, we are led to the following expression o f algebraic s olutions by means of the univ ersal character at tached to a pair of th ree- core partitions. Define a rational fun ction R [ λ,µ ] = R [ λ,µ ] ( a , b ) by (recall (2.1) or (2.9)) R [ λ,µ ] ( a , b ) = S [ λ,µ ] ( x , y ) = s [ λ,µ ] ( t ) , (6.2) under the substit ution: x n = a − n + b − n + ( ab ) n n (1 − q n ) , y n = a n + b n + ( ab ) − n n (1 − q − n ) , (6.3) or ( t 1 , t 2 , t 3 ) = ( a − 1 , b − 1 , ab ) with I = { 1 , 2 , 3 } and J = ∅ . Theor em 6.2. F or any m = ( m 1 , m 2 , m 3 ) , n = ( n 1 , n 2 , n 3 ) ∈ Z 3 , let U i ( a , b ) = R [ λ ( m ( i )) , λ ( n ( − i ))] ( a , b ) , V i ( a , b ) = R [ λ ( m ( i + 1)) , λ ( n ( − i + 1))] ( q 1 / 3 a , q − 2 / 3 b ) , (6.4) W i ( a , b ) = R [ λ ( m ( i + 2)) , λ ( n ( − i + 2))] ( q − 1 / 3 a , q − 1 / 3 b ) . (i) These functions solve the system of bilinear equations (4.11) with the parameters : γ i = q n − i − m i + 1 + | m |−| n | 3 , δ i = q n − i + 1 − m i + 2 + | m |−| n | 3 , ǫ i = q n − i + 2 − m i + | m |−| n | 3 . (6.5) 12 (ii) Consequently , the pair of functions f = U 1 V 1 W 1 U 3 V 3 W 3 , g = U 2 V 2 W 2 U 3 V 3 W 3 , (6.6) gives an algebraic solution of the q-P ainlev ´ e equation of type E (1) 6 , (3.11) , when a 1 = aq | m | + | n | 3 − m 1 − n 3 , a 5 = aq | m | + | n | 3 − m 2 − n 2 , a 0 = aq | m | + | n | 3 − m 3 − n 1 , a 2 = bq | m | + | n |− 1 3 − m 3 − n 2 , a 4 = bq | m | + | n |− 1 3 − m 1 − n 1 , a 6 = bq | m | + | n |− 1 3 − m 2 − n 3 . (6.7) Example 6.3 . Let us consider the function P [ λ,µ ] ( a , b ; q ) = ( ab ) | λ | + | µ | q −| ν | Y ( i , j ) ∈ λ  1 − q h ( i , j )  Y ( k , l ) ∈ µ  q h ( k , l ) − 1  R [ λ,µ ] ( a , b ) , associated wit h the algebraic soluti ons given in Theorem 6.2 . Here we denote by h ( i , j ) t he hook- length , that is, h ( i , j ) = λ i + λ ′ j − i − j + 1 (see [7]) and let ν = ( ν 1 , ν 2 , . . . ) be a sequence of integers defined by ν i = max { 0 , µ ′ i − λ i } . It is interesti ng that P [ λ,µ ] ( a , b ; q ) forms a pol ynomial whose coe ffi cients are all posi tive integers. A fe w examples of the special polynomia ls are given below: λ µ P [ λ,µ ] ( a , b ; q ) ∅ ∅ 1 (1) ∅ a + b + a 2 b 2 (2) ∅ a 2 + b 2 + a 4 b 4 + (1 + q ) ab (1 + a 2 b + ab 2 ) (1 , 1) ∅ q ( a 2 + b 2 + a 4 b 4 ) + (1 + q ) ab (1 + a 2 b + ab 2 ) ∅ (1) 1 + a 2 b + ab 2 ∅ (2) q (1 + a 4 b 2 + a 2 b 4 ) + (1 + q ) ab ( a + b + a 2 b 2 ) (1) (1) (1 + q + q 2 ) a 2 b 2 + qab ( a 2 + b 2 ) + q ( a + b )(1 + a 3 b 3 ) (1) (2) (1 + q + 2 q 2 + q 3 ) a 2 b 2 (1 + a 2 b + ab 2 ) + q (1 + q ) ab ( a 2 + b 2 + a 4 b 4 ) + q 2 ( a + b + a 2 b 2 ( a 3 + b 3 ) + a 4 b 4 ( a 2 + b 2 )) This po lynomial is t hought of an analog ue of the Um emura polyn omials which arise from algebraic solutions of the Painle v ´ e di ff erential equations; cf. [11]. 7 V erification of Pr oposition 2.3 T ake an ( l + l ′ + 2) × ( l + l ′ + 2) matri x of the form: X = ( X a , b ) 1 ≤ a , b ≤ l + l ′ + 2 = − t i − 1 T j ( H µ l ′ − a + 1 + a − 1 ) − t j − 1 T i ( H µ l ′ − a + 1 + a − 1 ) T i j ( H µ l ′ − a + 1 + a − b + 2 ) T j ( h λ a − l ′ − 2 − a + 2 ) T i ( h λ a − l ′ − 2 − a + 2 ) T i j ( h λ a − l ′ − 2 − a + b ) !  l ′ + 1  l + 1 . (7.1) | {z } 1 | {z } 1 | {z } l + l ′ Let D = det X and denot e by D [ i 1 , i 2 , . . . ; j 1 , j 2 , . . . ] i ts m inor determi nant removing rows { i a } and columns { j a } . W e pu t λ 0 = k and µ 0 = k ′ . 13 Lemma 7.1. It holds that ( t i − t j ) s [( k ,λ ) , ( k ′ ,µ )] ( t ) = ( t i t j ) l ′ + 1 D , (7.2a) T i j ( s [ λ,µ ] ( t )) = D [ l ′ + 1 , l ′ + 2; 1 , 2] , (7.2b) T i ( s [( k ,λ ) ,µ ] ( t )) = ( − t j ) l ′ D [ l ′ + 1; 1] , (7.2c) T j ( s [ λ, ( k ′ ,µ )] ( t )) = ( − t i ) l ′ + 1 D [ l ′ + 2; 2] , (7.2d) T j ( s [( k ,λ ) ,µ ] ( t )) = ( − t i ) l ′ D [ l ′ + 1; 2] , (7.2e) T i ( s [ λ, ( k ′ ,µ )] ( t )) = ( − t j ) l ′ + 1 D [ l ′ + 2; 1] . (7. 2f ) Pr oof. Let us prov e o nly (7.2a ) in t he fol lowing; the others (7.2b)–(7.2f) can be verified i n a similar manner . It is easy to see that T i ( h n ) = h n − t i h n − 1 , (7.3a) T i ( H n ) = H n − t i − 1 H n − 1 . (7.3b) W e shall appl y elementary transformations successively to t he row v ector ( h n , h n + 1 , . . . , h n + r − 1 ) of si ze r = l + l ′ + 2 . First we add the b th column m ultiplied by − t i to the ( b + 1) th column for 1 ≤ b ≤ r − 1. W e th en obtain by (7.3a), ( h n , T i ( h n + 1 ) , T i ( h n + 2 ) , . . . , T i ( h n + r − 1 ) ) . Secondly adding the b th column multiplied by − t j to the ( b + 1) th column for 2 ≤ b ≤ r − 1, we get  h n , T i ( h n + 1 ) , T i j ( h n + 2 ) , . . . , T i j ( h n + r − 1 )  . Adding the second column multip lied by ( t i − t j ) − 1 to the fir st column, we finally obtain t he v ector:  ( t i − t j ) − 1 T j ( h n + 1 ) , T i ( h n + 1 ) , T i j ( h n + 2 ) , . . . , T i j ( h n + r − 1 )  . By the same procedure as above, the low vector ( H n , H n − 1 , . . . , H n − r + 1 ) is also con verted to  − ( t i − t j ) − 1 t j T j ( H n ) , − t i T i ( H n ) , t i t j T i j ( H n ) , . . . , t i t j T i j ( H n − r + 3 )  , via (7.3b). Therefore, remembering (2.9), we arriv e at the expression (7.2a).  Pr oof of Pr opositi on 2.3. By the us e of Jacobi’ s identity: DD [ l ′ + 1 , l ′ + 2; 1 , 2] = D [ l ′ + 1; 1] D [ l ′ + 2; 2] − D [ l ′ + 1; 2] D [ l ′ + 2; 1] , we see that (2.7) follows immediatel y fr om Lemma 7.1.  14 A Reductio ns to q -Painle v ´ e equ ations of types A (1) 2 g + 1 and D (1) 5 Recall that the q -Painlev ´ e equations of types A (1) 2 g + 1 and D (1) 5 can be derive d as reductions from t he q -UC hierarchy; see [17] and [18]. Accordingly , they can be deriv ed also from the lattice q -UC hierarchy , as the lat ter hi erarchy i ncludes the former one; s ee Remark 2.5. W e ve ri fy that the equations of ty pes A (1) 2 g + 1 and D (1) 5 are in fact simil arity reductio ns of the lattice q -UC hierarchy together with periodi c conditi ons of o rder ( g + 1 , g + 1) and (2 , 2), respectively . In this appendix, we de monstrate ho w to obtain the q -Painlev ´ e equation only for the case of ty pe D (1) 5 ; the oth er case is simpler , so it may be left to the reader; cf. [17]. Let I = { 1 , 2 } and J = {− 1 , − 2 } . Suppose t hat σ m , n = σ m , n ( t ) is a so lution of the latt ice q -UC hi- erarchy (2.4), satisfying the periodic condition σ m , n = σ m + 2 , n = σ m , n + 2 and the similarity condition σ m , n ( c t ) = c d m , n σ m , n ( t ). Here d m , n are constants balanced as d m , n + d m + 1 , n + 1 = d m + 1 , n + d m , n + 1 . Now let us introdu ce the function ρ m , n ( α, β ; x ) in x , equipped with constant parameters α and β , defined by ρ m , n ( α, β ; x ) = σ m , n ( t ) under the s ubstituti on t = ( t 1 , t 2 , t − 1 , t − 2 ) = ( α, α − 1 , − q − 1 β x , − q − 1 β − 1 x ). Let Φ ( − ) i ( x ) = ρ i , i ( α, β ; x ) , Φ ( + ) i ( x ) = ρ i , i ( q 1 / 2 α, q 1 / 2 β ; x ) , Ψ ( − ) i ( x ) = ρ i , i + 1 ( α, q 1 / 2 β ; q 1 / 2 x ) , Ψ ( + ) i ( x ) = ρ i , i + 1 ( q 1 / 2 α, β ; q 1 / 2 x ) , (A.1) for i ∈ Z / 2 Z . As s imilar t o the case of E (1) 6 (see Section 5), we t herefore obtain, from (2.4) with the above constraints, the following system of bilinear equations: α ± 1 q ( d i , i − d i , i + 1 ) / 2 Φ ( ± ) i ( x ) Φ ( ∓ ) i + 1 ( x ) + β ± 1 xq ( d i + 1 , i − d i , i ) / 2 Φ ( ∓ ) i ( x ) Φ ( ± ) i + 1 ( x ) =  α ± 1 + β ± 1 x  Ψ ( ± ) i ( q − 1 x ) Ψ ( ∓ ) i + 1 ( x ) , (A.2a) α ± 1 q ( d i , i + 1 − d i , i ) / 2 Ψ ( ± ) i ( x ) Ψ ( ∓ ) i + 1 ( x ) +  q 1 / 2 β  ∓ 1  q 1 / 2 x  q ( d i + 1 , i − d i , i ) / 2 Ψ ( ∓ ) i ( x ) Ψ ( ± ) i + 1 ( x ) =  α ± 1 + ( q 1 / 2 β ) ∓ 1 q 1 / 2 x  Φ ( ± ) i ( x ) Φ ( ∓ ) i + 1 ( q x ) , (A.2b) where i ∈ Z / 2 Z . W e take the variables f ( x ) = Φ ( + ) 1 ( x ) Φ ( − ) 2 ( x ) Φ ( − ) 1 ( x ) Φ ( + ) 2 ( x ) , g ( x ) = Ψ ( + ) 1 ( x ) Ψ ( − ) 2 ( x ) Ψ ( − ) 1 ( x ) Ψ ( + ) 2 ( x ) , (A.3) and let γ = q ( d 1 , 1 − d 1 , 2 ) / 2 and δ = q ( d 2 , 1 − d 1 , 1 ) / 2 . Hence it fol lows from (A.2) that f f = ( g + α − 1 β − 1 γδ x )( g + αβγ − 1 δ − 1 q x ) ( xg + αβγ δ )( q xg + α − 1 β − 1 γ − 1 δ − 1 ) , (A.4a) g g = ( f + α − 1 βγ − 1 δ x )( f + αβ − 1 γδ − 1 x ) ( x f + αβ − 1 γ − 1 δ )( x f + α − 1 βγδ − 1 ) , (A.4b) where th e symbo ls f and g stand for f ( q x ) and g ( q − 1 x ), respective ly . This s ystem is equiv alent to the q -Painle v ´ e equation of type D (1) 5 , known a s the q -Painle v ´ e VI equation; see [1]. Ack nowledgement s. The author wishes to thank T etsu Masuda, Masatoshi Noumi, Y asuhiro Ohta, T omoy uki T akena wa, and Y asuhik o Y amada for v aluable discussio ns. This work is p artially supporte d by a fello wship of the Japan Society for the Promotion of Science (JSPS). 15 References [1] Jimbo, M., Sakai, H.: A q -analo g of the sixth Painl ev ´ e equation . Lett. Math . Phys. 3 8 , 145 –154 (1996) [2] Kac, V . 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