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On Polynomials Interpolating Between the Stationary State of a O(n) Model and a Q.H.E. Ground State

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Abstract

We obtain a family of polynomials defined by vanishing conditions and associated to tangles. We study more specifically the case where they are related to a O(n) loop model. We conjecture that their specializations at z i  = 1 are positive in n. At n = 1, they coincide with the Razumov-Stroganov integers counting alternating sign matrices.

We derive the CFT modular invariant partition functions labelled by Coxeter-Dynkin diagrams using the representation theory of the affine Hecke algebras.

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References

  1. Razumov A.V. and Stroganov Y.G. (2001). Spin chains and combinatorics. J. Phys. A 34: 3185–3190

    Article  MATH  ADS  MathSciNet  Google Scholar 

  2. Pearce P.A., Rittenberg V., Nienhuis B. and Gier J. (2002). Temperley-Lieb Stochastic Processes. J. Phys. A 35: L661–668

    Article  MATH  ADS  Google Scholar 

  3. Batchelor M.T., Nienhuis B. and Gier J. (2001). J. Phys. A 34: L265–270

    Article  MATH  ADS  Google Scholar 

  4. Razumov A.V. and Stroganov Y.G. (2004). Combinatorial nature of ground state vector of O(1) loop models. Theor. Math. Phys. 138: 333–337

    Article  MathSciNet  Google Scholar 

  5. Di Francesco P. and Zinn-Justin P. (2005). Around the Razumov-Stroganov conjecture: proof of a multiparameter sum rule. Electr. J. Combin. 12: R6

    MathSciNet  Google Scholar 

  6. Kasatani M. (2005). Subrepresentations in the polynomial representation of the double affine Hecke algebra of type GL n at t k+1 q r-1 = 1. Int. Math. Res. Not. 2005(28): 1717–1742

    Article  MATH  MathSciNet  Google Scholar 

  7. Pasquier V. (2006). Quantum incompressibility and Razumov stroganov type conjectures. Ann. Henri Poincaré 7: 397–421

    Article  MATH  MathSciNet  Google Scholar 

  8. Cherednik I. (2005). Double Affine Hecke Algebras. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  9. Bernard D., Gaudin M., Haldane D. and Pasquier V. (1993). Yang-Baxter equation in spin chains with long range interactions. J. Phys. A 26: 5219–5236

    Article  MATH  ADS  MathSciNet  Google Scholar 

  10. Feigin B., Jimbo M., Miwa T. and Mukhin E. (2003). Symmetric polynomials vanishing on the shifted diagonal and Macdonald polynomials. Int. Math. Res. Not. 18: 101

    Google Scholar 

  11. Macdonald, I.G.: A new class of symmetric functions. Actes 20e Seminaire Lotharingien, p 131–171, Publications I.R.M.A. Strasbourg (1988), 372/S-20

  12. Macdonald I.G. (1995). Symmetric Functions and Hall Polynomials. Oxford University Press, Oxford

    MATH  Google Scholar 

  13. Prange E. and Girvin S. (1987). The Quantum Hall effect. Springer-Verlag, Berlin-Heidelberg-New York

    Google Scholar 

  14. Haldane, F.D.M., Rezayi, E.H.: Spin-singlet wave function for the half-integral quantum Hall effect. Phys. Rev. Lett. 60, 956, and E60, 1886 (1988)

    Google Scholar 

  15. Kazhdan D. and Lusztig G. (1979). Representation of Coxeter groups and Hecke algebras. Invent. Math. 53: 165–184

    Article  MATH  ADS  MathSciNet  Google Scholar 

  16. Lascoux A. and Schützenberger M.-P. (1981). Polynômes de Kazhdan & Lusztig pour les grassmaniennes. Asterisque 87–88: 249–266

    Google Scholar 

  17. Lascoux, A., Schützenberger, M.-P.: Symmetry and Flag manifolds. In: Invariant Theory, Springer L.N 996, Berlin-Heidelberg-New York, 1983, pp. 118–144

  18. Lascoux A. and Kirillov A. (2000). Factorization of Kazhdan-Lusztig elements for Grassmanians. Adv. Stud. 28: 143–154

    MathSciNet  Google Scholar 

  19. Fulton W. (2003). Young Tableaux. London Mathematical Society Student Texts 35. Cambridge University Press, Cambridge

    Google Scholar 

  20. Mitra, S., Nienhuis, B.: Osculating Random Walks on Cylinders. In: Discrete Random Walks, C. Banderier, C. Krattenthaler (eds.), Discrete Math. Theor. Computer Sci. Proceed, 2003, pp. 259–264

  21. Mitra, S., Nienhuis, B.: Exact conjectured expressions for correlations in the dense O(1) loop model on cylinders. JSTAT, P10006, 2004

  22. Razumov, A.V., Stroganov, Y.G.: Enumeration of half-turn symmetric alternating sign matrices of odd order. http://arxiv.org/list/math-phys/0504022, 2005

  23. Lusztig G. (1989). Affine Hecke algebras and their graded version. J. Amer. Math. Soc. 2: 599–635

    Article  MATH  MathSciNet  Google Scholar 

  24. Lusztig G. (1993). Introduction to Quantum Groups. Birkhauser, Boston

    MATH  Google Scholar 

  25. Cherednik I. (1995). Nonsymmetric Macdonald Polynomials. Internat. Math. Res. Notices 10: 483–515

    Article  MathSciNet  Google Scholar 

  26. Di Francesco, P., Zinn-Justin, P.: In homogenous model of ossing loops and multidegrees of some algebraic varieties. http://arxiv.org/list/math-ph/0412031, 2001

  27. Di Francesco, P. Zinn-Justin, P., Zuber, J.B.: Sum rules for the ground states of the O(1) loop model on a cylinder and the XXZ spin chain. J. Stat. Mech. p08011 (2006)

  28. Goodman F.M., Jones V.F.R. and Harpe P. (1989). Coxeter Graphs and Towers of Algebras. Springer-Verlag, Berlin- Heidelberg-New York

    MATH  Google Scholar 

  29. Graham J.J. and Lehrer G.I. (1998). Enseign. Math. The representation theory of affine Temperley-Lieb algebras 44: 173–218

    MATH  MathSciNet  Google Scholar 

  30. Pasquier V. (1987). Two-dimentional itical systems labelled by Dynkin diagrams. Nucl. Phys. B FS19: 162–172

    Article  ADS  MathSciNet  Google Scholar 

  31. Pasquier V. (1987). Lattice derivation of modular invarient partition functions on the torus. J. Phys. A 20: L1229

    Article  ADS  MathSciNet  Google Scholar 

  32. Pasquier V. and Saleur H. (1990). Common structures between finite systems and conformal field theories through quantum groups. Nucl. Phys. B 330: 525–556

    Article  ADS  MathSciNet  Google Scholar 

  33. Cappelli A., Itzykson C. and Zuber J.B. (1987). Modular invariant partition functions in two dimensions. Nucl. Phys. B 280([FS18]): 445–465

    Article  MATH  ADS  MathSciNet  Google Scholar 

  34. Gepner D. and Qiu Z. (1987). Modular invariant partition functions for parafermionic field theories. Nucl. Phys. B 285([FS19]): 423–453

    Article  ADS  MathSciNet  Google Scholar 

  35. Prange E. and Girvin S. (1987). The Quantum Hall effect. Springer-Verlag, Berlin-Heidelberg-New York

    Google Scholar 

  36. Knop F. and Sahi S. (1997). A recursion and a combinatorial formula for Jack polynomials. Invent. Math. 128(1): 9–22

    Article  MATH  MathSciNet  Google Scholar 

  37. Di Francesco P. and Zinn-Justin P. (2002). The quantum Knizhnik-Zamolodchikov equation, generalized Razumov-Stroganov sum rules and extended Joseph polynomials. J. Phys. A 38: L815–822

    Article  MathSciNet  Google Scholar 

  38. Ram A. (2003). Affine Hecke algebras and generalized standard Young tableaux. J. Algebra 260(1): 367–415

    Article  MATH  MathSciNet  Google Scholar 

  39. Suzuki T. and Vazirani M. (2005). Tableaux on periodic skew diagrams and irreducible representations of the double affine Hecke algebra of type A. Int. Math. Res. Not. 2005(27): 1621–1656

    Article  MATH  MathSciNet  Google Scholar 

  40. Walker, K.: Private communication

  41. Gaudin M. (1981). La Fonction d’Onde de Bethe. Masson, Paris

    Google Scholar 

  42. Baxter R.J. (1982). Exactly solved Models in Statistical Mechanics. Academic, London

    MATH  Google Scholar 

  43. Wenzl H. (1988). Hecke Algebras of type A n and subfactors. Invent. Math. 92: 349–383

    Article  MATH  ADS  MathSciNet  Google Scholar 

  44. Pasquier V. (1996). Scattering Matrices and Affine Hecke Algebras. Schladming School 1995, Nucl. Phys. B (Proc. Suppl.) 45A: 62–73

    Article  MATH  MathSciNet  Google Scholar 

  45. Pasquier V. (2006). Incompressible representations of the Birman Wenzl algebra. Ann. Henri Poincaré 7: 603–619

    Article  MATH  MathSciNet  Google Scholar 

  46. Pasquier, V.: A lecture on the Calogero Sutherland models. The third Baltic Rim Student Seminar, Saclay preprint, Spht-94060 (1994), avaialble at http://arxiv.org/list/hep-th/9405104, 1994

  47. Haldane F.D.M., Ha Z.N.C., Talstra J.C., Bernard D. and Pasquier V. (1992). Yangian symmetry of integrable quantum chains with long-range interactions and a new desiption of states in conformal field theory. Phys. Rev. Lett. 69: 2021–2025

    Article  MATH  ADS  MathSciNet  Google Scholar 

  48. Wilczek F. (1990). Fractional Statistics and Anyon Superconductivity. World Scientific, Singapore

    Google Scholar 

  49. Moore G. and Read N. (1991). Nonabelions in the fractional quantum hall effect. Nucl. Phys. B 360: 362–91

    Article  ADS  MathSciNet  Google Scholar 

  50. Sutherland B. (1972). Phys. Rev. A 5: 1372

    Article  ADS  Google Scholar 

  51. Di Francesco, P.: Totally Symmetric Self-Complementary Plane Partitions and Quantum Knizhnik-Zamolodchikov equation: a conjecture. J. Stat. Mech. P09008 (2006)

  52. Lascoux, A., Pasquier, V.: In preparation

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Authors and Affiliations

  1. Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan

    M. Kasatani

  2. Service de Physique Théorique, C.E.A/ Saclay, 91191, Gif-sur-Yvette, France

    V. Pasquier

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  1. M. Kasatani
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  2. V. Pasquier
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Correspondence to V. Pasquier.

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Communicated by L. Takhtajan

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Kasatani, M., Pasquier, V. On Polynomials Interpolating Between the Stationary State of a O(n) Model and a Q.H.E. Ground State. Commun. Math. Phys. 276, 397–435 (2007). https://doi.org/10.1007/s00220-007-0341-0

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