Advertisement

Theoretical and Mathematical Physics

, Volume 154, Issue 3, pp 331–348 | Cite as

Quantum Knizhnik-Zamolodchikov equation, totally symmetric self-complementary plane partitions, and alternating sign matrices

  • P. Zinn-JustinEmail author
  • P. Di Francesco
Article

Abstract

We present multiple-residue integral formulas for partial sums in the basis of link patterns of the polynomial solution of the level-1 \(U_q (\widehat{\mathfrak{s}\mathfrak{l}_2 })\) quantum Knizhnik-Zamolodchikov equation at arbitrary values of the quantum parameter q. These formulas allow rewriting and generalizing a recent conjecture of Di Francesco connecting these sums to generating polynomials for weighted totally symmetric self-complementary plane partitions. We reduce the corresponding conjectures to a single integral identity, yet to be proved.

Keywords

loop model combinatorics quantum integrability 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    P. Di Francesco, J. Stat. Mech., 0609, P09008 (2006); arXiv:cond-mat/0607499v1 (2006).CrossRefGoogle Scholar
  2. 2.
    P. Di Francesco, J. Stat. Mech., 0701, P01024 (2007); arXiv:math-ph/0611012v2 (2006).CrossRefGoogle Scholar
  3. 3.
    M. T. Batchelor, J. de Gier, and B. Nienhuis, J. Phys. A, 34, L265–L270 (2001); arXiv:cond-mat/0101385v1 (2001).zbMATHCrossRefADSGoogle Scholar
  4. 4.
    D. Bressoud, Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, Cambridge Univ. Press., Cambridge (1999).zbMATHGoogle Scholar
  5. 5.
    A. V. Razumov and Yu. G. Stroganov, Theor. Math. Phys., 138, 333–337 (2004); arXiv:math/0104216v2 [math.CO] (2001).CrossRefMathSciNetGoogle Scholar
  6. 6.
    P. Di Francesco and P. Zinn-Justin, Electron. J. Comb., 12, No. 1, R6 (2005); arXiv:math-ph/0410061v4 (2004).Google Scholar
  7. 7.
    A. Izergin, Sov. Phys. Dokl., 32, 878–879 (1987); V. Korepin, Comm. Math. Phys., 86, 391–418 (1982).ADSGoogle Scholar
  8. 8.
    G. Kuperberg, Ann. Math. (2), 156, 835–866 (2002); arXiv:math/0008184v3 (2000).CrossRefMathSciNetGoogle Scholar
  9. 9.
    I. B. Frenkel and N. Yu. Reshetikhin, Comm. Math. Phys., 146, 1–60 (1992).zbMATHCrossRefADSMathSciNetGoogle Scholar
  10. 10.
    V. Pasquier, Ann. Henri Poincaré, 7, 397–421 (2006); arXiv:cond-mat/0506075v1 (2005).zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    P. Di Francesco and P. Zinn-Justin, J. Phys. A, 38, L815–L822 (2005); arXiv:math-ph/0508059v3 (2005).zbMATHCrossRefGoogle Scholar
  12. 12.
    M. Kasatani and V. Pasquier, Comm. Math. Phys., 276, 397–435 (2007); arXiv:cond-mat/0608160v3 (2006).zbMATHCrossRefADSMathSciNetGoogle Scholar
  13. 13.
    P. Di Francesco, J. Phys. A, 38, 6091–6120 (2005); arXiv:math-ph/0504032v2 (2005); J. Stat. Mech., 0511, P11003 (2005); arXiv:math-ph/0509011v3 (2005).zbMATHCrossRefADSMathSciNetGoogle Scholar
  14. 14.
    D. Robbins, Math. Intelligencer, 13, No. 2, 12–19 (1991).zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    B. Lindström, Bull. London Math. Soc., 5, 85–90 (1973); I. Gessel and G. Viennot, Adv. Math., 58, 300–321 (1985).zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    F. A. Smirnov, J. Phys. A, 19, L575–L578 (1986).CrossRefADSGoogle Scholar
  17. 17.
    M. Jimbo and T. Miwa, Algebraic Analysis of Solvable Lattice Models (CBMS Reg. Conf. Ser. Math., Vol. 85), Amer. Math. Soc., Providence, R. I. (1995).zbMATHGoogle Scholar
  18. 18.
    D. Zeilberger, Electron. J. Combin., 3, No. 2, R13 (1996); arXiv:math/9407211v1 [math.CO] (1994).MathSciNetGoogle Scholar
  19. 19.
    P. Zinn-Justin, Electron. J. Combin., 13, No. 1, R110 (2006); arXiv:math/0607183v1 [math.CO] (2006).MathSciNetGoogle Scholar
  20. 20.
    P. Di Francesco and P. Zinn-Justin, “From orbital varieties to alternating sign matrices,” arXiv:math-ph/0512047v1 (2005).Google Scholar
  21. 21.
    P. Zinn-Justin, “Combinatorial point for higher spin loop models,” arXiv:math-ph/0603018v3 (2006).Google Scholar
  22. 22.
    A. V. Razumov and Yu. G. Stroganov, J. Stat. Mech., 0607, P07004 (2006); arXiv:math-ph/0605004v2 (2006).CrossRefMathSciNetGoogle Scholar
  23. 23.
    A. V. Razumov, Yu. G. Stroganov, and P. Zinn-Justin, J. Phys. A, 40, 11827–11847 (2007); arXiv:0704.3542v3 (2007).zbMATHCrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.Laboratoire de Physique Théorique et Modèles StatistiquesUniv Paris-SudOrsayFrance
  2. 2.Service de Physique Théorique de SaclayGif sur Yvette CedexFrance

Personalised recommendations