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Character formulae of \(\widehat{\text{sl}}_n\)-modules and inhomogeneous paths. (English) Zbl 0952.17013

Summary: Let \(B_{(l)}\) be the perfect crystal for the \(l\)-symmetric tensor representation of the quantum affine algebra \(U^\prime_q(\widehat{sl}_n)\). For a partition \(\mu=(\mu_1, \cdots, \mu_m)\), elements of the tensor product \(B_{(\mu_1)}\otimes\cdots\otimes B_{(\mu_m)}\) can be regarded as inhomogeneous paths. We establish a bijection between a certain large \(\mu\) limit of this crystal and the crystal of an (generally reducible) integrable \(U_q(\widehat{sl}_n)\)-module, which forms a large family depending on the inhomogeneity of \(\mu\) kept in the limit. For the associated one-dimensional sums, relations with the Kostka-Foulkes polynomials are clarified, and new fermionic formulae are presented. By combining their limits with the bijection, we prove or conjecture several formulae for the string functions, branching functions, coset branching functions and spinon character formula of both vertex and RSOS types.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
82B23 Exactly solvable models; Bethe ansatz
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
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[1] Andrews, G. E.; Baxter, R. J.; Forrester, P. J., Eight vertex SOS model and generalized Rogers-Ramanujan-type identities, J. Stat. Phys., 35, 193 (1984) · Zbl 0589.60093
[2] Arakawa, T.; Nakanishi, T.; Oshima, K.; Tsuchiya, A., Spectral decomposition of path space in solvable lattice model, Comm. Math. Phys., 181, 159 (1996)
[3] A. Berkovich, B.M. McCoy and A. Schilling, Rogers-Schur-Ramanujan type identities for the \(Mpp\); A. Berkovich, B.M. McCoy and A. Schilling, Rogers-Schur-Ramanujan type identities for the \(Mpp\) · Zbl 0901.60091
[4] Bernard, D.; Pasquier, V.; Serban, D., Spinons in conformal field theory, Nucl. Phys. B, 428, 612 (1994), [FS] · Zbl 1049.81535
[5] Bethe, H. A., Zur Theorie der Metalle, I. Eigenwerte und Eigenfunktionen der linearen Atomkette, Z. Phys., 71, 205 (1931) · Zbl 0002.37205
[6] Bouwknegt, P.; Ludwig, A.; Schoutens, K., Spinon basis for higher level SU(2) WZW models, Phys. Lett. B, 359, 304 (1995)
[7] Date, E.; Jimbo, M.; Kuniba, A.; Miwa, T.; Okado, M., Exactly solvable SOS models II: Proof of the star-triangle relation and combinatorial identities, Adv. Stud. Pure Math., 16, 17 (1988) · Zbl 0679.17011
[8] Date, E.; Jimbo, M.; Kuniba, A.; Miwa, T.; Okado, M., One-dimensional configuration sums in vertex models and affine Lie algebra characters, Lett. Math. Phys., 17, 69 (1989) · Zbl 0681.17016
[9] B.L. Feigin and A.V. Stoyanovsky, Quasi-particle models for the representations of Lie algebras and geometry of flag manifold, hep-th/9308079.; B.L. Feigin and A.V. Stoyanovsky, Quasi-particle models for the representations of Lie algebras and geometry of flag manifold, hep-th/9308079.
[10] G. Georgiev, Combinatorial constructions of modules for infinite-dimensional Lie algebras, (I) Principal subspace, hep-th/9412054.; G. Georgiev, Combinatorial constructions of modules for infinite-dimensional Lie algebras, (I) Principal subspace, hep-th/9412054.
[11] G. Georgiev, Combinatorial constructions of modules for infinite-dimensional Lie algebras, (II) Parafermionic space, q-alg/9504024.; G. Georgiev, Combinatorial constructions of modules for infinite-dimensional Lie algebras, (II) Parafermionic space, q-alg/9504024.
[12] Gupta, R., Generalized exponents via Hall-Littlewood symmetric functions, Bull. AMS, 16, 287 (1987) · Zbl 0648.22011
[13] Jimbo, M.; Miwa, T.; Okado, M., Solvable lattice models whose states are dominant integral weights of \(A_{n−1}^{(1)}\), Lett. Math. Phys., 14, 123 (1987) · Zbl 0642.17015
[14] Kac, V. G., Infinite-Dimensional Lie Algebras (1990), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0425.17009
[15] Kac, V. G.; Peterson, D. H., Infinite-dimensional Lie algebras, theta functions and modular forms, Adv. Math., 53, 125 (1984) · Zbl 0584.17007
[16] S-J. Kang and M. Kashiwara, Quantized affine algebras and crystals with head, q-alg/9710008.; S-J. Kang and M. Kashiwara, Quantized affine algebras and crystals with head, q-alg/9710008.
[17] Kang, S-J.; Kashiwara, M.; Misra, K. C.; Miwa, T.; Nakashima, T.; Nakayashiki, A., Affine crystals and vertex models, Int. J. Mod. Phys. A, 7, Suppl. 1A, 449 (1992) · Zbl 0925.17005
[18] Kang, S-J.; Kashiwara, M.; Misra, K. C.; Miwa, T.; Nakashima, T.; Nakayashiki, A., Perfect crystals of quantum affine Lie algebras, Duke Math. J., 68, 499 (1992) · Zbl 0774.17017
[19] Kashiwara, M., On crystal bases of the \(q\)-analogue of universal enveloping algebras, Duke Math. J., 63, 465 (1991) · Zbl 0739.17005
[20] Kirillov, A. N., Dilogarithm identities, (Lectures in Mathematical Sciences, 7 (1995), The University of Tokyo) · Zbl 0831.33008
[21] Kirillov, A. N., On the Kostka-Green-Foulkes polynomials and Clebsh-Gordan numbers, J. Geom. and Phys., 5, 365 (1988) · Zbl 0695.22009
[22] A.N. Kirillov, New combinatorial formula for modified Hall-Littlewood polynomials, math.QA/9803006.; A.N. Kirillov, New combinatorial formula for modified Hall-Littlewood polynomials, math.QA/9803006. · Zbl 0956.05101
[23] A.N. Kirillov, A. Kuniba and N. Nakanishi, Skew Young diagram method in spectral decomposition of integrable lattice models II: Higher levels, q-alg/9711009.; A.N. Kirillov, A. Kuniba and N. Nakanishi, Skew Young diagram method in spectral decomposition of integrable lattice models II: Higher levels, q-alg/9711009. · Zbl 1033.17018
[24] Kirillov, A. N.; Reshetikhin, N. Yu., The Bethe ansatz and the combinatorics of Young tableaux, J. Sov. Math., 41, 925 (1988) · Zbl 0639.20029
[25] A.N. Kirillov and M. Shimozono, A generalization of the Kostka-Foulkes polynomials, math.QA/9803062.; A.N. Kirillov and M. Shimozono, A generalization of the Kostka-Foulkes polynomials, math.QA/9803062. · Zbl 0995.05146
[26] Kuniba, A.; Misra, K. C.; Okado, M.; Takagi, T.; Uchiyama, J., Paths, Demazure crystals and symmetric functions, (Nankai-CRM Proc. of “Extended and Quantum Algebras and their Applications to Physics”. Nankai-CRM Proc. of “Extended and Quantum Algebras and their Applications to Physics”, Tianjin (1996)), q-alg.9612018, to appear in
[27] Kuniba, A.; Misra, K. C.; Okado, M.; Takagi, T.; Uchiyama, J., Characters of Demazure modules and solvable lattice models, Nucl. Phys. B, 510, 555 (1998), [PM] · Zbl 0953.17007
[28] Kuniba, A.; Nakanishi, T.; Suzuki, J., Characters of conformal field theories from thermodynamic Bethe ansatz, Mod. Phys. Lett. A, 8, 1649 (1993) · Zbl 1021.81843
[29] Lascoux, A.; Schützenberger, M. P., Sur une conjecture de H.O. Foulkes, C.R. Acad. Sc. Paris, 288A, 323 (1978) · Zbl 0374.20010
[30] Macdonald, I., Symmetric functions and Hall polynomials (1995), Oxford Univ. Press: Oxford Univ. Press New York · Zbl 0824.05059
[31] Nakayashiki, A.; Yamada, Y., Kostka polynomials and energy functions in solvable lattice models, Selecta Mathematica, New Ser., 3, 547 (1997) · Zbl 0915.17016
[32] Nakayashiki, A.; Yamada, Y., On spinon character formulas, (Itoyama; etal., Frontiers in Quantum Field Theory (1996), World Scientific: World Scientific Singapore), 367-371
[33] A. Schilling and S.O. Warnaar, Supernomial coefficients, polynomial identities and \(q\); A. Schilling and S.O. Warnaar, Supernomial coefficients, polynomial identities and \(q\) · Zbl 0921.05007
[34] Stanley, R., The stable behaviour of some characters of \(SL (n)\), Linear and Multilinear Algebra, 16, 3 (1984) · Zbl 0573.20042
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