Combinatorial R Matrices for a Family of Crystals: Cn(1) and A2n-1(2) Cases

  • Goro Hatayama
  • Atsuo Kuniba
  • Masato Okado
  • Taichiro Takagi
Part of the Progress in Mathematics book series (PM, volume 191)


The combinatorial R matrices are obtained for a family {B l { of crystals for U′ q (C n (1) ) and U′ q (A 2n-1 (2) ), where B l the crystal of the irreducible module corresponding to the one-row Young diagram of length l. The isomorphism B l B k B k B l and the energy function are described explicitly in terms of a C n -analogue of the Robinson- Schensted-Knuth-type insertion algorithm. As an application, a C n (1) -analogue of the Kostka polynomials is calculated for several cases.


Energy Function Young Tableau Leftmost Column High Element Left Region 
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  1. [ABF]
    G. E. Andrews, R. J. Baxter, and P. J. Forrester, Eight vertex SOS model and generalized Rogers-Ramanujan-type identities, J. Statist. Phys., 35 (1984), 193–266.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [B]
    T. H. Baker, An insertion scheme for C n crystals, in M. Kashiwara and T. Miwa, eds., Physical Combinatorics, Springer Science+Business Media New York, 2000,1–48 (this volume).CrossRefGoogle Scholar
  3. [Ba]
    R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, London, 1982.zbMATHGoogle Scholar
  4. [Be]
    H. A. Bethe, Zur Theorie der Metalle I: Eigenwerte und Eigenfunktionen der linearen Atomkette, Z. Phys., 71 (1931), 205–231.CrossRefGoogle Scholar
  5. [Ber]
    A. Berele, A Schensted-type correspondence for the symplectic group, J. Combin. Theory Ser. A, 43 (1986), 320–328MathSciNetzbMATHCrossRefGoogle Scholar
  6. [DJKMO]
    E. Date, M. Jimbo, A. Kuniba, T. Miwa, and M. Okado, One dimensional configuration sums in vertex models and affine Lie algebra characters, Lett Math. Phys., 17 (1989), 69–77.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [F]
    W. Fulton, Young Tableaux: With Applications to Representation Theory and Geometry, London Mathematical Society Student Texts 35, Cambridge University Press, Cambridge, 1997.zbMATHGoogle Scholar
  8. [FOY]
    K. Fukuda, M. Okado, and Y. Yamada, Energy functions in box ball systems, preprint math.QA/9908116.Google Scholar
  9. [HHIKTT]
    G. Hatayama, K. Hikami, R. Inoue, A. Kuniba, T. Takagi, and T. Tokihiro, The A M (1) automata related to crystals of symmetric tensors, preprint math.QA/9912209.Google Scholar
  10. [HKKOT]
    G. Hatayama, Y. Koga, A. Kuniba, M. Okado, and T. Takagi, Finite crystals and paths, preprint math.QA/9901082.Google Scholar
  11. [HKKOTY]
    G. Hatayama, A. N. Kirillov, A. Kuniba, M. Okado, T. Takagi, and Y. Yamada, Character formulae of 138-01-modules and inhomogeneous paths, Nuclear Phys. B, 536 (1999), 575–616.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [HKOTY]
    G. Hatayama, A. Kuniba, M. Okado, T. Takagi, and Y. Yamada, Remarks on fermionic formula, in N. Jing and K. C. Misra, eds., Recent Developments in Quantum Affine Algebras and Related Topics, Contemporary Mathematics 248, AMS, Providence, 1999, 243–291.Google Scholar
  13. [HKT]
    G. Hatayama, A. Kuniba, and T. Takagi, Soliton cellular automata associated with finite crystals, preprint solv-int/9907020.Google Scholar
  14. [KE]
    R. C. King and N. G. I. El-Sharkaway, Standard Young tableaux and weight multiplicities of the classical Lie groups, J. Phys. A, 16 (1983), 3153–3177.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [KKM]
    S-J. Kang, M. Kashiwara, and K. C. Misra, Crystal bases of Verma modules for quantum affine Lie algebras, Compositio Math., 92 (1994), 299–325.MathSciNetzbMATHGoogle Scholar
  16. [KMN1]
    S.-J. Kang, M. Kashiwara, K. C. Misra, T. Miwa, T. Nakashima, and A. Nakayashiki, Affine crystals and vertex models, Internat J. Modern Phys. A, 7-1A (1992), 449–484.MathSciNetCrossRefGoogle Scholar
  17. [KMN2]
    S-J. Kang, M. Kashiwara, K. C. Misra, T. Miwa, T. Nakashima, and A. Nakayashiki, Perfect crystals of quantum affine Lie algebras, Duke Math. J., 68 (1992), 499–607.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [KN]
    M. Kashiwara and T. Nakashima, Crystal graph for representations of the q-analogue of classical Lie algebras, J. Algebra, 165 (1994), 295–345.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [KR]
    A. N. Kirillov and N. Yu. Reshetikhin, The Bethe ansatz and the combinatorics of Young tableaux, J. Soviet Math., 41 (1988), 925–955.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [L]
    G. Lusztig, Singularities, character formulas, and a q-analogue of weight multiplicities, Astérisque, 101–102 (1983), 208–227.MathSciNetGoogle Scholar
  21. [Ma]
    I. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed.,Oxford University Press, New York, 1995.zbMATHGoogle Scholar
  22. [NY]
    A. Nakayashiki and Y. Yamada, Kostka polynomials and energy functions in solvable lattice models, Sel. Math., 3 (1997), 547–599.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [S]
    M. Shimozono, Affine type A crystal structure on tensor product of rectangles, Demazure characters, and nilpotent varieties, preprint math.QA/9804039.Google Scholar
  24. [SW]
    A. Schilling and S. O. Warnaar, Inhomogeneous lattice paths, generalized Kostka polynomials and A n − 1 supernomials, Comm. Math. Phys., 202 (1999), 359–401.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [T]
    I. Terada, A Robinson-Schensted-type correspondence for a dual pair on spinors, J. Combin. Theory Ser. A, 63 (1993), 90–109.MathSciNetzbMATHCrossRefGoogle Scholar
  26. [TS]
    D. Takahashi and J. Satsuma, A soliton cellular automaton, J. Phys. Soc. Japan, 59 (1990), 3514–3519.MathSciNetCrossRefGoogle Scholar
  27. [TTMS]
    T. Tokihiro, D. Takahashi, J. Matsukidaira, and J. Satsuma, From soliton equations to integrable cellular automata through a limiting procedure, Phys. Rev. Lett., 76 (1996), 3247–3250.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • Goro Hatayama
    • 1
  • Atsuo Kuniba
    • 1
  • Masato Okado
    • 2
  • Taichiro Takagi
    • 3
  1. 1.Institute of PhysicsUniversity of TokyoTokyoJapan
  2. 2.Department of Informatics and Mathematical Science Graduate School of Engineering ScienceOsaka UniversityOsakaJapan
  3. 3.Department of Mathematics and PhysicsNational Defense AcademyYokosukaJapan

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