Periodicities of T-systems and Y-systems

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1 Periodicities of T-systes and Y-systes (arxiv: ) Atsuo Kuniba (Univ. Tokyo) joint with R. Inoue, O. Iyaa, B. Keller, T.Nakanishi and J. Suzuki Quantu Integrable Discrete Systes 23 March 2009 I.N.I. Cabridge

2 Contents What are T-systes and Y-systes? Restriction and periodicity conjecture Quivers and Cluster algebra forulation Outlook

3 What are T-systes and Y-systes? Systes of difference equations aong couting variables related to root syste. T (u) and Y (u) a {nodes of Dynkin diagra of g} (g = A n, B n, C n, D n, E 6, E 7, E 8, F 4, G 2 ) Z 1 u C (spectral paraeter)

4 t a := long root 2 / α a 2 (= 1 for ADE) B n C n F G > < > > n 1 n n 1 n T (u 1 )T t (u + 1 ) = T 1 (u)t +1 (u) + product of T s, a t a Y (u 1 )Y t (u + 1 ) = a t a product of (1 + Y ) s (1 + Y 1 (u) 1 )(1 + Y +1 (u) 1 ). Structure of products in the RHS is dependent on od t a Z.

5 g = A n, D n, E n case C = (C ab ) 1 a,b n : Cartan atrix T-syste T (u 1)T (u + 1) = T 1 (u)t +1 (u) + b: C ab = 1 T (b) (u) Exaple A n T (u 1)T (u + 1) = T (T 1 (u)t +1 (u) + T (a 1) 0 (u) = T (0) (n+1) (u) = T (u) = 1.) (u)t (a+1) (u) A version of Hirota-Miwa or Toda-field equation on discrete space-tie.

6 B n T (a 1) (u 1)T (u + 1) = T 1 (u)t +1 (u) + T T (n 1) (u 1)T (n 1) (u + 1) = T (n 1) 1 (n 1) (u)t +1 T (n) 2 (u 1 (n) )T 2 2 (u ) = T (n) (n) (n 1) 2 1 (u)t 2+1 (u) + T T (n) 2+1 (u 1 (n) )T (u ) = T (n) (n) 2 (u)t 2+2 (u)t (a+1) (u) (1 a n 2), (n 2) (u) + T (u)t (n) 2 (u), (u) + T (n 1) (u 1 2 (u)t (n 1) )T (n 1) (u ), +1 (u). C n T (u 1 )T 2 (u ) = T (a 1) 1 (u)t +1 (u) + T T (n 1) 2 (u 1 2 )T (n 1) 2 (u ) = T (n 1) (n 1) 2 1 (u)t 2+1 T (n 1) 2+1 (u 1 (n 1) )T (u ) = T (n 1) 2 T (n) (n) (n) (n) (u 1)T (u + 1) = T (u)t 1 (n 1) (u)t (u)t (a+1) (n 2) (n) (u) + T 2 (u)t (u 1 2 (u) (1 a n 2), (n 2) (n) (n) (u) + T 2+1 (u)t (u)t +1 (u), (n 1) (u) + T (u). 2 )T (n) (u ),

7 F 4 (1) (1) (1) (2) (u 1)T (u + 1) = T 1 (u)t +1 (u) + T (u), T (2) (2) (2) (u 1)T (u + 1) = T 1 (u)t (2) +1 (1) (3) (u) + T (u)t 2 (u), T (3) 2 (u 1 (3) )T 2 2 (u ) = T (3) (3) (2) 2 1 (u)t 2+1 (u) + T (u 1 (2) )T 2 (u + 1 (4) )T 2 2 (u), T (3) 2+1 (u 1 (3) )T (u ) = T (3) (3) 2 (u)t 2+2 (2) (2) (u) + T (u)t T (4) (u 1 (4) )T 2 (u ) = T (4) (4) (3) 1 (u)t +1 (u) + T (u). +1 (u)t (4) 2+1 (u), G 2 (1) (1) (1) (2) (u 1)T (u + 1) = T 1 (u)t +1 (u) + T 3 (u), T (2) 3 (u 1 (2) )T 3 3 (u ) = T (2) (2) (1) 3 1 (u)t 3+1 (u) + T (u 2 (1) (1) )T (u)t 3 (u ), T (2) 3+1 (u 1 (2) )T (u ) = T (2) (2) 3 (u)t 3+2 (1) (u) + T (u 1 (1) )T 3 (u + 1 (1) )T +1 3 (u), T (2) 3+2 (u 1 (2) )T (u ) = T (2) (2) (1) (1) 3+1 (u)t 3+3 (u) + T (u)t +1 (u 1 (1) )T +1 3 (u ).

8 Origin of T-syste: T (u) stands for Phys: couting transfer atrices in Yang-Baxter solvable lattice odels. T (u) = Tr ( W (u) u ), [T (u), T (a ) (u )] = 0. Math: q-characters of Kirillov-Reshetikhin odules W (u) of quantu affine algebra U q (ĝ). For A W n (u) a 0 W (a 1) (u) W (a+1) (u) W (u 1) W (u+1) W 1 (u) W +1 (u) 0 Proposed in the forer context by K-Nakanishi-Suzuki (1994). Proved in the latter context by Nakajia for ADE (2003) and Hernandez for g (2006).

9 Solutions in A n case (Bazhanov-Reshetikhin 1991) Jacobi-Trudi type forula T (u) = Tableau-su type forula det (a i+j) (T 1 (u + i + j 1)). 1 i,j A 3 exaple : T (2) 3 (u) = Generalizations to g each tableaux = ratio of products of Baxter Q functions. BCD analogue of Jacobi-Trudi including Pfaffians: K-Nakaura-Hirota (1996). Casorati like deterinants (A type): Krichever-Lipan-Wiegann-Zabrodin (1997). (C type): K-Okado-Suzuki-Yaada (2002). Tableau-su forula for various g Chari-Pressley, K-Ohta-Suzuki, Kleber, Frenkel-Reshetikhin, Nakajia, Nakai-Nakanishi, Hernandez, etc.

10 Y-syste g = A n, D n, E n case (Y 0 (u) 1 = 0) (b) Y (u 1)Y (u + 1) = b: C ab = 1 (1 + Y (u)) (1 + Y 1 (u) 1 )(1 + Y +1 (u) 1 ) B n Y (a 1) (1 + Y (u))(1 + Y (a+1) (u)) (u 1)Y (u + 1) = (1 + Y 1 (u) 1 )(1 + Y +1 (u) 1 ) (1 a n 2), Y (n 1) (u 1)Y (n 1) (u + 1) (n 2) (1 + Y (u))(1 + Y (n) = (n) ))(1 + Y 2 (u 1 (n) (n) ))(1 + Y (u))(1 + Y 2 (u (1 + Y (n 1) 1 (u) 1 )(1 + Y (n 1) +1 (u) 1 ) 2+1 (u)), Y (n) 2 (u 1 (n) )Y 2 2 (u ) = 1 + Y (n 1) (u) (1 + Y (n) 2 1 (u) 1 )(1 + Y (n) 2+1 (u) 1 ), Y (n) 2+1 (u 1 (n) )Y (u ) = 1 (1 + Y (n) 2 (u) 1 )(1 + Y (n) 2+2 (u) 1 ).

11 C n Y (u 1 )Y 2 (u Y (n 1) 2 (u 1 (n 1) )Y 2 (a 1) (1 + Y (u))(1 + Y (a+1) (u)) ) = (1 a n 2), (1 + Y 1 (u) 1 )(1 + Y +1 (u) 1 ) 2 (u ) = (1 + Y (n 2) (n) 2 (u))(1 + Y (u)) (1 + Y (n 1) 2 1 (u) 1 )(1 + Y (n 1) 2+1 (u) 1 ), Y (n 1) 2+1 (u 1 (n 1) )Y (u ) = 1 + Y (n 2) 2+1 (u) (1 + Y (n) 2 (u) 1 )(1 + Y (n) 2+2 (u) 1 ), Y (n) (n) (u 1)Y (u + 1) = (1 + Y (n 1) 2 (u (n 1) ))(1 + Y 2 (u 1 2 (1 + Y (n) 1 (u) 1 )(1 + Y (n) +1 (u) 1 ) (n 1) (n 1) ))(1 + Y 2 1 (u))(1 + Y 2+1 (u)).

12 F 4 Y (1) (1) (u 1)Y (u + 1) = 1 + Y (2)(u) (1 + Y (1) 1 (u) 1 )(1 + Y (1) +1 (u) 1 ), Y (2) = (2) (u 1)Y (u + 1) (1 + Y (1) Y (3) 2 (u 1 (3) )Y 2 Y (3) 2+1 (u 1 (3) )Y 2 (3) (u))(1 + Y 2 (u 1 (3) ))(1 + Y 2 2 (u + 1 (3) (3) ))(1 + Y (u))(1 + Y (1 + Y (2) 1 (u) 1 )(1 + Y (2) +1 (u) 1 ) 2 (u ) = (1 + Y (2) (4) (u))(1 + Y 2 (u)) (1 + Y (3) 2 1 (u) 1 )(1 + Y (3) 2+1 (u) 1 ), 2+1 (u ) = 1 + Y (4) 2+1 (u) (1 + Y (3) 2 (u) 1 )(1 + Y (3) 2+2 (u) 1 ), Y (4) (u 1 (4) )Y 2 (u ) = 1 + Y (3)(u) (1 + Y (4) 1 (u) 1 )(1 + Y (4) +1 (u) 1 ). 2+1 (u)),

13 G 2 Y (1) (1) (2) (u 1)Y (u + 1) = (1 + Y 3 (u 2 3 (2) (2) ))(1 + Y 3 (u))(1 + Y 3 (u )) (1 + Y (2) 3 1 (u 1 (2) ))(1 + Y (u )) (1 + Y (2) 3+1 (u 1 (2) ))(1 + Y (u )) (1 + Y (2) (2) 3 2 (u))(1 + Y 3+2 ( (u)) ) 1 (1 + Y (1) 1 (u) 1 )(1 + Y (1) +1 (u) 1 ) Y (2) 3 (u 1 (2) )Y 3 3 (u ) = 1 + Y (1)(u) (1 + Y (2) 3 1 (u) 1 )(1 + Y (2) 3+1 (u) 1 ), Y (2) 3+1 (u 1 (2) )Y (u ) = 1 (1 + Y (2) 3 (u) 1 )(1 + Y (2) 3+2 (u) 1 ), Y (2) 3+2 (u 1 (2) )Y (u ) = 1 (1 + Y (2) 3+1 (u) 1 )(1 + Y (2) 3+3 (u) 1 ).

14 Y-syste is an algebraic for of therodynaic Bethe ansatz equation of type g under string hypothesis. Y (u) Boltzann factor of string/hole excitation with color a, length, rapidity u. A 1 exaple: (Y (u) = Y (1) (u) 1 ) log Y (u) = known fcn. + log(1 + Y 1 (v))(1 + Y +1 (v)) dv 4 cosh π(u v) 2 Y (u i)y (u + i) = (1 + Y 1 (u))(1 + Y +1 (u)). Y-syste was proposed by ADE: Al. Zaolodchikov (1991), Ravanini-Tateo-Valleriani (1993). g: K-Nakanishi (1992).

15 Relation of T and Y-systes A 1 exaple Y (u 1)Y (u + 1) = (1 + Y 1 (u))(1 + Y +1 (u)), T (u 1)T (u + 1) = T 1 (u)t +1 (u) + 1. Forally setting Y (u) = T 1 (u)t +1 (u), Y (u 1)Y (u + 1) = T 1 (u 1)T +1 (u 1)T 1 (u + 1)T +1 (u + 1) = T +1 (u 1)T +1 (u + 1)T 1 (u 1)T 1 (u + 1) = (T +2 (u)t (u) + 1)(T 2 (u)t (u) + 1) = (Y +1 (u) + 1)(Y 1 (u) + 1). Siilarly for g, T-syste solves Y-syste. (Yet to be understood why.)

16 Restriction and Periodicity conjecture Introduce l Z 2 called level. Level l restricted T and Y-syste are those closing aong T (u) and Y (u) with 1 t al 1, obtained respectively by iposing T t a l (u) = 1 and Y t a l (u) 1 = 0.

17 C 2 exaple. (t 1, t 2 ) = (2, 1) 2+1 (u 1 (1) )T (u ) = (1) 2 (u)t 2+2 (2) (2) (u) + T (u)t +1 (u), 2 (u 1 (1) )T 2 2 (u ) = (1) (2) 2 1 (u)t 2+1 (u) + T (u 1 (2) )T 2 (u ), (2) (2) (2) (1) (u 1)T (u + 1) = T 1 (u)t +1 (u) + T 2 (u). T (2) Level 2 restriction: 4 (u) = T (2) 2 (u) = 1. (b.c. T 0 (u) = 1) 1 (u 1 (1) )T 1 (u ) = 2 (u) + T (2) 1 (u), 2 (u 1 (1) )T 2 (u ) = 1 (u) 3 (u) + 1 (u 1 (1) )T 1 (u ), 3 (u 1 (1) )T 3 (u ) = 2 (u) + T (2) 1 (u), T (2) 1 (u 1)T (2) 1 (u + 1) = (u), which closes aong 1 (u), 2 (u), 3 (u), T (2) 1 (u). Restricted T and Y-systes evolution eqs. in the u direction.

18 Periodicity conjecture Level l restricted T -syste and Y -syste obey T (u + h + l) = T (ω) t a l (u) and Y (u + h + l) = Y (ω) t a l (u). ω is an involution whose only non-trivial cases are A n D n (n:odd) E 6 h = dual Coxeter nuber = g A n B n C n D n E 6 E 7 E 8 F 4 G 2 h n+1 2n 1 n+1 2n (Full periodicity: T (u+2(h +l)) = T (u) and sae for Y (u).)

19 Exaple: (A 2, 2) Write T (u) = T 1 (u). Periodicity reads (u 1) (u + 1) = 1 + T (2) (u), T (2) (u 1)T (2) (u + 1) = 1 + (u). (u + 5) = T (2) (u), T (2) (u + 5) = (u). (0)= a T (2) (1)= b (2) = 1 + T (2) (1) (0) = 1 + b a T (2) (3) = 1 + (2) T (2) (1) = b a b = 1 + a + b ab (4) = 1 + T (2) (3) (2) = a+b ab 1+b a = 1 + a b T (2) (5) = 1 + (4) T (2) (3) = a b 1+a+b ab = a = (0) (6) = 1 + T (2) (5) (4) = 1 + a 1+a b = b = T (2) (1)

20 0{10, 30, 50, 70} (E 8, 2) : { 1 (u), T (3) 1 (u), T (5) 1 (u), T (7) 1 (u)} 32 { , , , 31 } { , , , } 1525 { , , , } { , , , } { , , , } { , , , } { , , , } { , , , } { , , , } { , , , } { , , , } { , , } , { , , , } { , , , } { , , , 1781 } {10, 30, 50, 70} u=0

21 Periodicity of Y-syste for (g, l) was proposed: Al. Zaolodchikov (1991) (ADE, 2), Ravanini-Tateo-Valleriani (1993) (ADE, l), K-Nakanishi-Suzuki (1994) (g, l). It has been proved: (A n, 2): Frenkel-Szenes (1995), Gliozzi-Tateo (1996), (A n, l): Volkov, Henriques (2007), (ADE, 2): Foin-Zelevinsky (2003) Cluster algebra, (ADE, l): Keller (arxiv: ) Cluster algebra/category. Periodicity of T-syste: Proposed in Inoue-Iyaa-K-Nakanishi-Suzuki (arxiv: ), where (ACDE, l) case was proved. (A n, l) case: proof also contained in Henriques (2007). Periodicities of T and Y-systes do not follow fro each other in general.

22 Origin of the periodicity conjecture Phys: Level l restricted solid-on-solid odel [T-sys] Transfer at. T (u) is 2(h + l)-periodic by construction. [Y-sys] String hypothesis works by assuing t a l. Math: 2(h + l)-periodicity of q-characters holds in the quotient Ring of q-characters Ideal including l (u) 1 for type A.

23 Quivers and Cluster algebra forulation Q: quiver (finite oriented graph without loop and 2-cycle ) I = {1,..., N}: vertex set, x = (x 1,..., x N ): I-tuple of variables x i : cluster variable, (Q, x): seed. Cluster algebra A Q is defined by (i) (iv). Foin-Zelevinsky (2002) (i) Start fro the (initial) seed (Q, x) as above. (ii) For each k I, define another seed (R, y) by (R, y) = µ k (Q, x) ( utation at k, def. next page). (iii) Iterate utations for every new seed at every k, and collect all (possibly infinite) seeds. (iv) A Q = Z-subalgebra of Q(x 1,..., x N ) gen. by cluster variables.

24 Mutation at k: µ k (Q, x) = (R, y) A new quiver R is obtained fro Q by reversing arrows incident with k and Q R r r st s t s t k k r eans an r -fold arrows { (r 0) (r < 0) (s, t 0, r 0) New cluster variables y = (y 1,..., y n ) are given by x i i k, y i = 1 x j + x j i = k, x k arrows j k of Q arrows k j of Q µ 2 k = id, µ jµ k = µ k µ j for j, k not connected by an arrow.

25 Exaple. I = {1, 2}. Initial seed (Q, x) = (1 2, {a, b}). Seeds denoted by (a b), and utation µ 1, µ 2 by. (a b) Exchange graph (a 1+a b ) (1+b a b) ( 1+a+b 1+a ab b ) (1+b a 1+a+b ab ) Foin-Zelevinsky theore (2003) { (1) Laurent phenoenon, (2) Finite type classification. (1) cluster variables are Laurent polynoials. (2) {cluster var.} < Q utations orientation on ADE Dynkin diag.

26 Exaple with initial seed (a b c) closes aong 14 seeds. (a 1+ac b c) (a b c) (a b 1+b c ) ( 1+b a b c) Exchange graph is Stasheff associahedron.

27 Cluster algebra forulation of T-syste. (A 2, 4) exaple 1 (u 1) 1 (u+1) = 2 (u) + T (2) 1 (u), T (2) 2 (u 1)T (2) 2 (u+1) = T (2) 1 (u)t (2) 3 (u) + 2 (u), 3 (u 1) 3 (u+1) = 2 (u) + T (2) 3 (u). x 1 x 4 x 2 x 5 x 3 x 6 := 1 (0) T (2) 1 (1) 2 (1) T (2) 2 (0) 3 (0) T (2) 3 (1) µ 1 µ 3 = 1 (2) T (2) 1 (1) 2 (1) T (2) 2 (0) 3 (2) T (2) 3 (1) µ 5 1 (2) T (2) 1 (3) 2 (3) T (2) 2 (2) 3 (2) T (2) 3 (3) µ 2 = 1 (2) T (2) 1 (3) 2 (1) T (2) 2 (2) 3 (2) T (2) 3 (3) µ 4 µ 6 = 1 (2) T (2) 1 (1) 2 (1) T (2) 2 (2) 3 (2) T (2) 3 (1) (Q, x(u + 2)) = µ 2 µ 4 µ 6 µ 5 µ 3 µ 1 (Q, x(u)) for x = {T }.

28 Siilarly, T-syste for any (g, l) is forulated as (Q, x(u + 2)) = µ(q, x(u)) by an appropriate choice of Q : quiver, x(u) = {x i (u)} : cluster variables suitably identified with T (u) s, µ = µ i1 µ is : coposite utation. For g = ADE, Q = Q g Q Al 1 : (Keller s square product), Q g, Q Al 1 = Dynkin quivers with alternating source/sink. Exaple (g, l) = (D 5, 5) Q D5 Q A4 Q D5 Q A4

29 (Full) periodicity for any (g, l) is forulated as (Q, x(u)) = µ h +l (Q, x(u)). Theore (Inoue-Iyaa-K-Nakanishi-Suzuki 2008) Periodicity conjecture of T-syste is true for g = ACDE, l. Our proof is by (Full periodicity for ADE is also proved by Keller.) C : direct ethod using deterinant expressions, ADE : Cluster category. Sketch of the proof for (g, l) = (ADE, 2). Q = Q g Q A1 = Q g = alternating Dynkin quiver, µ = i:sink µ i j:source µ j, h = h (Coxeter nuber). To show µ h+2 (Q, x) = (Q, x).

30 KQ: Path algebra gen. by paths on Q. Product = coposition. D Q = (bounded) derived category of finite di. KQ-odules. A 3 exaple: Q = Auslander-Reiten (AR) quiver of D Q P 3 τ 1 (P 3 ) P 1 [1] P i : projective KQ-odule P 2 τ 1 (P 2 ) P 2 [1] τ 1 : AR translation (right shift) P 1 τ 1 (P 1 ) P 3 [1] [1]: suspention functor (upside down rightshift 2 ) Cluster category (Buan-Marsh-Reineke-Reiten-Todorov 2006) C Q := D Q /(τ 1 [1]). In C Q, τ h 2 id. ( ) categorical periodicity

31 Cluster tilting object rankg a=1 (indecoposables) satisfying certain conditions. In the present exaple, rank g = 3, and there are 14 cluster tilting objects like P 1 P 2 P 3 µ 1 µ 2 µ 3 τ 1 (P 1 ) P 2 P 3 P 1 P 2 [1] P 3 P 1 P 2 τ 1 (P 3 ) µ a : cluster tilting utation Cluster tilting objects are connected by cluster tilting utations. P 1 P 2 P 3 P 1 P 2 τ 1 (P 3 )

32 Theore (Buan-Marsh-Reineke-Reiten-Todorov 2006) bijection X s.t. the following diagra is coutative: Cluster algebra {seeds in A Q } utation µ a {seeds in A Q } X Cluster category {cluster tilting objects in C Q } µ a cluster tilting utation X {cluster tilting objects in C Q } This is an exaple of Categorification of cluster algebra. Proof of full periodicity. Take a cluster tilting object P = rankg a=1 P a and set (Q, x) = X P. µ h+2 (Q, x) = µ h+2 ( X P ) Th = X µ h+2 (P ) easy = X τ h 2 (P ) ( ) = X P = (Q, x). Higher level case: Idea is parallel although technically ore involved.

33 Outlook Y-syste can be incorporated in Cluster algebra with coefficients introduced by Foin-Zelevinsky (2007). T,Y-systes, periodicity conjectures for twisted affine Lie alg. Q-syste (a degeneration of T-syste) and its application are also discussed by Kede-Di Francesco (2008). Besides periodicity, T,Y-systes have various aspects related to; Ferionic character forula and Dilogarith identities in CFT, Crystal base of quantu affine algebras, Integrable cellular autoata and Ultradiscrete τ -functions, etc. (talk at Glasgow)

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