S-J. Kang, M. Kashiwara, K. C. Misra, T. Miwa, T. Nakashima and A. Nakayashiki, Affine crystals and vertex models, Int. J. Mod. Phys. A 7 (suppl. 1A), 449-484 (1992).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....not been long before Nakayashiki and Yamada [31] solved this conjecture. Their idea was to relate Lascoux Schutzenberger s charge of a tableau with the so called energy of a path. Once this correspondence is established, the conjecture is found to be a corollary of the theory of perfect crystals [18, 19]. The purpose of this paper is to extend their result to more general setting and elucidate an interplay among the theory of crystals, the Kostka Foulkes polynomials, one dimensional sums, their fermionic formulae and affine Lie algebra characters. In a sense this is a far reaching application of ....

....weight lattice P cl = P=Zffi is also needed. We further define the following subsets of P cl : P = f 2 P cl j h; h i i 0 for any ig, P ) l = f 2 P j h; ci = lg. We introduce an element i 2 P cl by i = i mod Zffi, and fix the map af : P cl P by af ( i ) i . See Section 3. 1 of [18] for the details of P cl ; af , etc. The irreducible highest weight module V ( with highest weight 2 P has a crystal base (L( B( 20] We denote the highest weight vector in B( by u . On the crystal B = B( the actions of Kashiwara operators e i ; f i (i = 0; 1; Delta Delta Delta ....

[Article contains additional citation context not shown here]

S-J. Kang, M. Kashiwara, K. C. Misra, T. Miwa, T. Nakashima and A. Nakayashiki, Affine crystals and vertex models, Int. J. Mod. Phys. A 7 (suppl. 1A), 449-484 (1992).

....the character theory of sl n , we regard e as a power of the variables x 1 = e ; x 2 = e with the relation x 1 x 2 Delta Delta Delta x n = 1. There is a remarkable connection between the partition function of SK and an affine Lie algebra character. Theorem 2. 1 (DJKMO correspondence [4, 8]) For a given K 2 K l , let L( K) be the integrable module of sl n with the highest weight (K ) Then the following equality holds: s2S K ; 2.4) where chL( K) is the (unnormalized) character of L( K) 7] 2.3. Energy functions and nonmovable tableaux. Let us describe the energy ....

....set f1; 2; ng. We identify v a 1 : a l 2 B l with a semistandard tableau as a 1 a 2 a l v a 1 : a l = We shall construct the function H l 1 ;l 2 : B l B l 2 Z such that H l in (2.1) is realized as H l = H l;l under the above identification. Next we construct the maps (cf. [8, 9]) f i : B l B l [ f0g; e e i : B l B l [ f0g; 1 i n Gamma 1: Let b 2 B l be a semistandard tableau and i 2 f1; n Gamma 1g, we define i (b) number of i in b. For each i, 1 i n Gamma 1, we define a map f i : B l B l [ f0g by the following rule: Let b 2 B l , ....

[Article contains additional citation context not shown here]

S.-J. Kang, M. Kashiwara, K. Misra, T. Miwa, T. Nakashima, A. Nakayashiki, Affine crystals and vertex models, Int. J. Mod. Phys. A7, Suppl. 1A (1992) 449-484.

....with this paper and [24] 2. NOTATION AND STATEMENT OF MAIN RESULT 2.1. Quantized universal enveloping algebras. For this paper we only require the following three algebras: Uq(sl) C U(sl) C Uq( Let us recall some definitions for quantized universal enveloping algebras taken from [5] and [6]. Consider the following data: a rinkely generated Z module P (weight lattice) a set I (index set for Dynkin diagram) elements ai[ i C I (basic roots) and hi C P = Hornet(P, Z) i I (basic coroots) such that ( hi, is a generalized Cartan matrix, and a symmetric form ( P x P Q ....

....STRUCTURE ON TENSOR PRODUCTS OF RECTANGLES The goal of this section is to give explicit descriptions of the classical , crystal structure on tensor products of rectangular crystals and their energy functions. This is accomplished by translating the theory of sl crystals and classical crystals in [6] [7] 11] 22] into the language of Young tableaux and the RobinsonSchensted Knuth (RSK) correspondence. 3.1. Crystals. This section reviews the definition of a weighted crystal [6] and gives the convention used here for the tensor product of crystals. A P weighted crystal is a a ....

[Article contains additional citation context not shown here]

S.-J. Kang, M. Kashiwara, K. Misra, T. Miwa, T. Nakashima, and A. Nakayashiki, Affine crystals and vertex models, Int. J. Modern Phys. A Suppl. 1A (1992) 449 484.

....element of S . Then wt( s) s i : Proof. Since m j k modulo n, i=1 s i = k . Thus s i = k i ) k i ) wt( s) There is a remarkable connection between the partition function of S and an affine Lie algebra character. Theorem 2. 2 (DJKMO correspondence [8, 15]) For k = 0; 1; n Gamma 1, let L( k ) be the level 1 integrable module of the untwisted affine Lie algebra b sl n whose highest weight is the kth fundamental weight k of sl n . Then the following equality holds: chL( k ) q (2.6a) s2S (n Gammak) Gammawt( s) ....

....) Let S be the set of all the spin configurations. For each s 2 S we define iH(s i ; s i 1 ) wt( s) Gamma n s i : 8.2) Phi Phi H H f f f f f f 0 1 2 n Figure 4. The Dynkin diagram of A 2n . The Dynkin diagram of B n is obtained by removing the node 0. Theorem 8. 1 ([15]) Let ch L( n ) be the unnormalized character of the (unique) level 1 integrable module of A 2n . Then ffl Sigmai = x Sigma1 i ; e = 1: 8.3) See Appendix C for the explicit expression of chL( n ) q; x) The local energy map h : s 7 h = h i ) h i = H(s i ; s i 1 ) has the ....

S-J. Kang, M. Kashiwara, K. Misra, T. Miwa, T. Nakashima, A. Nakayashiki, Affine crystals and vertex models, Int. J. Mod. Phys. A7, Suppl. 1A (1992) 449-484.

.... Thus, one has chV (2 0 ) e 2 0 e 2 1 Gammaffi e 2 0 Gammaffi e 4 1 Gamma2 0 Gamma2ffi 2e 2 1 Gamma2ffi Delta Delta Delta Pi The relationship between paths and highest weight representations of b sl n was later clarified using the crystal basis theory of Kashiwara [33, 16, 20, 21]. This involves a q deformation of b sl n . Let U q ( b sl n ) be the quantized enveloping algebra of b sl n . We denote by V q ( the irreducible U q ( b sl n ) module with highest weight . We shall follow [16] and construct V q ( as a submodule of a q deformed level l Fock space F q ( As a ....

S-J. Kang, M. Kashiwara, K.C. Misra, T. Miwa, T. Nakashima and A. Nakayashiki, Affine crystals and vertex models, Int. J. Mod. Phys. A 7 (1992), suppl. 1A, 449--484.

.... f k i f j f 3 Gammak i = 0: Throughout the paper we will use the standard notation [n] q n Gamma q Gamman ) q Gamma q Gamma1 ) and [k] 1] 2] Delta Delta Delta [k] Let V = v v Gamma be a two dimensional space and V z = V Omega [z; z Gamma1 ] be its affinization [14]. Define the action of the generators t i ; t Gamma1 i ; e i ; f i on the space V z as follows z (e i )v z n = e i )v z n ffi i0 ; z (f i )v z n = f i )v z n Gammaffi i0 ; z (t i )v z n = t i )v z n ; where the action on the space V is given (e 0 ) f 1 ....

S.-J. Kang, M. Kashiwara, K. Misra, T. Miwa, T. Nakashima and A. Nakayashiki. Affine crystals and vertex models. Inter. J. Modern Phys., A 7(Suppl. 1A) 449--484, (1992).

....crystal bases by Kashiwara [K1,KMN1 2] The weight function describing the 1DCS is called the energy function and it is determined from the q = 0 behavior of the R matrix. The theory of crystal bases gives a simple characterization of energy functions without need for explicit forms of R matrices [KMN1]. Obtaining this characterization played an essential role in proving the relations between 1DCS and string functions [KMN1] and the branching coefficients [DJO] of affine Lie algebras. In this paper we add further application of the energy function to combinatorics and representation theory. In ....

....from the q = 0 behavior of the R matrix. The theory of crystal bases gives a simple characterization of energy functions without need for explicit forms of R matrices [KMN1] Obtaining this characterization played an essential role in proving the relations between 1DCS and string functions [KMN1] and the branching coefficients [DJO] of affine Lie algebras. In this paper we add further application of the energy function to combinatorics and representation theory. In particular we obtain a new expression of the Kostka polynomials in terms of energy functions. for instance see (1.1) ....

[Article contains additional citation context not shown here]

Kang, S-J., Kashiwara, M., Misra, K., Miwa, T., Nakashima, T., Nakayashiki, A. Affine crystals and vertex models, Int. J. Mod. Phys. A 7, Suppl. 1A, 449-484 (1992)

....and Jun Uchiyama Abstract We review the path realization of Demazure crystals and discuss Demazure characters in the light of symmetric functions. 1 Introduction Let U q (g) be a quantum affine Lie algebra. Representation of U q (g) at q = 0 is well described by the crystal base theory [Ka1] [KMN1], KMN2] For example, consider the irreducible highest weight U q (g) module V ( for any dominant integral weight of level l. At q = 0 its crystal B( admits a parametrization in terms of paths. The latter is the combinatorial object that arose in the studies of solvable lattice models ....

....is an element of the semi infinite tensor product Delta Delta Delta B Omega B. It must obey some boundary condition on the left tail, which is uniquely specified from and B. Letting P( B) denote the set of such paths, one has an isomorphism of crystals : B( P( B) These features [KMN1], KMN2] will be summarized in section 2. In [Ka2] Kashiwara showed that for each Weyl group element w there exists a finite subset Bw ( ae B( that corresponds to the crystal of the Demazure module Vw ( ae V ( Then a natural question arises; What kind of paths are contained in the image ....

[Article contains additional citation context not shown here]

S-J. Kang, M. Kashiwara, K. C. Misra, T. Miwa, T. Nakashima and A. Nakayashiki, Affine crystals and vertex models, Int. J. Mod. Phys. A 7 (suppl. 1A), 449-484 (1992).

....for the irreducible highest weight U q (g) module V ( at q = 0, we have a combinatorial object path. This path is different from Littelmann s path. It has emerged in the study of solvable lattice models (cf. DJKMO1] DJKMO2] Its study is accomplished with the aid of crystal base theory (cf. [KMN1], KMN2] Let B be a perfect crystal of level l. Then, for any dominant integral weight of level l, the crystal B( of V ( is represented as a set of paths. Roughly speaking, a path is an element of the semi infinite tensor product Delta Delta Delta Omega B Omega B with some stability ....

....for Scientific Research on Priority Areas, the Ministry of Education, Science and Culture, Japan. M.O. is supported in part by the Australian Research Council. 1 Perfect crystal 1. 1 Notation We follow the notations of the quantized universal enveloping algebra and the crystal base in [KMN1]. In particular, fff i g i2I is the set of simple roots, fh i g i2I is the simple coroots, P is the weight lattice and P = f 2 P j h; h i i 0 for any ig. U q (g) is the quantized universal enveloping algebra of an affine Lie algebra g. V ( is the irreducible highest weight module of highest ....

[Article contains additional citation context not shown here]

S-J. Kang, M. Kashiwara, K. C. Misra, T. Miwa, T. Nakashima and A. Nakayashiki, Affine crystals and vertex models, Int. J. Mod. Phys. A 7 (suppl. 1A), 449-484 (1992).

No context found.

S.-J.Kang, M.Kashiwara, K.Misra, T.Miwa, T.Nakashima and A. Nakayashiki, Affine crystals and Vertex models, International Journal of Modern Physics A 7, Suppl.1A (1992) 449--484

No context found.

Kang, S-J., Kashiwara, M., Misra, K., Miwa, T., Nakashima, T. and Nakayashiki, A. Affine crystals and vertex models, Int. J. Mod. Phys. A 7, Suppl. 1A, 449-484 (1992).

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC