Skip to main content
SpringerLink
Log in

Schur Polynomials and The Yang-Baxter Equation

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We describe a parametrized Yang-Baxter equation with nonabelian parameter group. That is, we show that there is an injective map \({g \mapsto R (g)}\) from \({ \rm{GL}(2, \mathbb{C}) \times \rm{GL}(1, \mathbb{C})}\) to End \({(V \otimes V)}\) , where V is a two-dimensional vector space such that if \({g, h \in G}\) then R 12(g)R 13(gh) R 23(h) = R 23(h) R 13(gh)R 12(g). Here R i j denotes R applied to the i, j components of \({V \otimes V \otimes V}\) . The image of this map consists of matrices whose nonzero coefficients a 1, a 2, b 1, b 2, c 1, c 2 are the Boltzmann weights for the non-field-free six-vertex model, constrained to satisfy a 1 a 2 + b 1 b 2 − c 1 c 2 = 0. This is the exact center of the disordered regime, and is contained within the free fermionic eight-vertex models of Fan and Wu. As an application, we show that with boundary conditions corresponding to integer partitions λ, the six-vertex model is exactly solvable and equal to a Schur polynomial s λ times a deformation of the Weyl denominator. This generalizes and gives a new proof of results of Tokuyama and Hamel and King.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Baxter R.J.: The inversion relation method for some two-dimensional exactly solved models in lattice statistics. J. Stat. Phys 28(1), 1–41 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  2. Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], 1982

  3. Brubaker B., Bump D., Friedberg S.: Weyl Group Multiple Dirichlet Series: Type A Combinatorial Theory Annals of Mathematics Studies, Vol. 175. NJ: Princeton University Press, Princeton (2011)

    Google Scholar 

  4. Brubaker B., Bump D., Friedberg S.: Weyl group multiple Dirichlet series, Eisenstein series and crystal bases. Ann. of Math 173, 1081–1120 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brubaker, B., Bump, D., Chinta, G., Friedberg, S., Gunnells, P.: Metaplectic ice. http://arxiv.org/abs/1009.1741v1 [math.RT], 2010

  6. Drinfeld, V.G.: Quantum groups. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Providence, RI: Amer. Math. Soc., 1987, pp. 798–820

  7. Faddeev, L.D., Reshetikhin, N.Yu., Takhtajan, L.A.: Quantization of Lie groups and Lie algebras. In: Algebraic Analysis, Vol. I, Boston, MA: Academic Press, 1988, pp. 129–139

  8. Fan C., Wu F.Y.: Ising model with next-neighbor interactions. I. Some exact results and an approximate solution Phys. Rev 179, 560–570 (1969)

    Article  Google Scholar 

  9. Fan C., Wu F.Y.: General lattice model of phase transitions. Phys. Revi. B 2(3), 723–733 (1970)

    Article  ADS  Google Scholar 

  10. Fomin, S., Kirillov, A.N.: The Yang-Baxter equation, symmetric functions, and Schubert polynomials. In: Proc. of the 5th Conference on Formal Power Series and Algebraic Combinatorics (Florence, 1993), Disc. Math. 153, 123–143 (1996)

  11. Fomin, S., Kirillov, A.N.: Grothendieck polynomials and the Yang-Baxter equation. In: Formal power series and algebraic combinatorics/Séries formelles et combinatoire algébrique, Piscataway, NJ: DIMACS, 1994, pp. 183–189

  12. Hamel A.M., King R.C.: U-turn alternating sign matrices, symplectic shifted tableaux and their weighted enumeration. J. Alg. Comb 21(4), 395–421 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hamel A.M., King R.C.: Bijective proofs of shifted tableau and alternating sign matrix identities. J. Alg. Comb 25(4), 417–458 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Izergin A.G.: Partition function of a six-vertex model in a finite volume. Dokl. Akad. Nauk SSSR 297(2), 331–333 (1987)

    MathSciNet  Google Scholar 

  15. Jimbo M.: Introduction to the Yang-Baxter equation. Internat. J. Modern Phys. A 4(15), 3759–3777 (1989)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. Jimbo M., Miwa T.: Solitons and infinite-dimensional Lie algebras. Publ. Res. Inst. Math. Sci 19(3), 943–1001 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kirillov A.N., Reshetikhin N.Yu.: The Bethe ansatz and the combinatorics of Young tableaux. J. Soviet Math 41(2), 925–955 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  18. Korepin V.E.: Calculation of norms of Bethe wave functions. Commun. Math. Phys 86(3), 391–418 (1982)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. Kuperberg G.: Another proof of the alternating-sign matrix conjecture. Int. Math. Res. Notices 1996(3), 139–150 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lascoux, A.: Chern and Yang through ice. Preprint, 2002

  21. Lascoux, A.: The 6 vertex model and Schubert polynomials. SIGMA Symmetry Integrability Geom. Methods Appl. 3, Paper 029, 12 pp. (electronic) (2007)

  22. Lascoux, A., Schützenberger, M.-P.: Symmetry and flag manifolds. In: Invariant theory (Montecatini, 1982), Volume 996 of Lecture Notes in Math., Berlin: Springer, 1983, pp. 118–144

  23. Lieb E.: Exact solution of the problem of entropy in two-dimensional ice. Phys. Rev. Lett 18, 692–694 (1967)

    Article  ADS  Google Scholar 

  24. Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs. New York: The Clarendon Press/Oxford University Press, Second edition, 1995, (with contributions by A. Zelevinsky)

  25. Majid S.: Quasitriangular Hopf algebras and Yang-Baxter equations. Int. J. Mod. Phys. A 5(1), 1–91 (1990)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. McNamara, P.J.: Factorial Schur functions via the six-vertex model. http://arxiv.org/abs/0910.5288v2 [math.co], 2009

  27. Mills W.H., Robbins D.P., Rumsey H. Jr.: Alternating sign matrices and descending plane partitions. J. Comb. Th. Ser. A 34(3), 340–359 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  28. Okada S.: Alternating sign matrices and some deformations of Weyl’s denominator formulas. J. Alg. Comb 2(2), 155–176 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  29. Robbins D.P., Rumsey H. Jr.: Determinants and alternating sign matrices. Adv. in Math. 62(2), 169–184 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  30. Stein, W., et al.: SAGE Mathematical Software, Version 4.1. http://www.sagemath.org, 2009

  31. Stroganov Yu.G.: The Izergin-Korepin determinant at a cube root of unity. Teoret. Mat. Fiz 146(1), 65–76 (2006)

    MathSciNet  Google Scholar 

  32. Sutherland B.: Exact solution for a model for hydrogen-bonded crystals. Phys. Rev. Lett 19(3), 103–104 (1967)

    Article  ADS  Google Scholar 

  33. Tokuyama T.: A generating function of strict Gelfand patterns and some formulas on characters of general linear groups. J. Math. Soc. Japan 40(4), 671–685 (1988)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  34. Tsilevich N.V.: The quantum inverse scattering problem method for the q-boson model, and symmetric functions. Funkt. Anal. i Priloz. 40(3), 53–65, 96 (2006)

    MathSciNet  Google Scholar 

  35. Zinn-Justin, P.: Six-vertex loop and tiling models: Integrability and combinatorics. Habilitation thesis http://arxiv.org/abs/0901.0665v2 [math-ph], 2009

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, MIT, Cambridge, MA, 02139-4307, USA

    Ben Brubaker

  2. Department of Mathematics, Stanford University, Stanford, CA, 94305-212, USA

    Daniel Bump

  3. Department of Mathematics, Boston College, Chestnut Hill, MA, 02467-3806, USA

    Solomon Friedberg

Authors
  1. Ben Brubaker
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniel Bump
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Solomon Friedberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniel Bump.

Additional information

Communicated by P. Forrester

Rights and permissions

Reprints and permissions

About this article

Cite this article

Brubaker, B., Bump, D. & Friedberg, S. Schur Polynomials and The Yang-Baxter Equation. Commun. Math. Phys. 308, 281–301 (2011). https://doi.org/10.1007/s00220-011-1345-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-011-1345-3

Keywords

Search

Navigation