Skip to main content
SpringerLink
Log in

The six-vertex model and deformations of the Weyl character formula

  • Published:
Journal of Algebraic Combinatorics Aims and scope Submit manuscript

Abstract

We use statistical mechanics—variants of the six-vertex model in the plane studied by means of the Yang–Baxter equation—to give new deformations of Weyl’s character formula for classical groups of Cartan type B, C,  and D, and a character formula of Proctor for type BC. In each case, the corresponding Boltzmann weights are associated with the free fermion point of the six-vertex model. These deformations add to the earlier known examples in types A and C by Tokuyama and Hamel-King, respectively. A special case for classical types recovers deformations of the Weyl denominator formula due to Okada.

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.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

Similar content being viewed by others

References

  1. Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press (Harcourt Brace Jovanovich Publishers), London (1989, Reprint of the 1982 original)

  2. Brubaker, B., Bump, D., Friedberg, S.: Schur polynomials and the Yang–Baxter equation. Commun. Math. Phys. 308(2), 281–301 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  3. Brubaker, B., Bump, D., Friedberg, S.: Weyl group multiple Dirichlet series, Eisenstein series and crystal bases. Ann. Math. 173(2), 1081–1120 (2011)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  5. Hamel, A.M., King, R.C.: Symplectic shifted tableaux and deformations of Weyl’s denominator formula for \({\rm sp}(2n)\). J. Algebr. Comb. 16(3), 269–300 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Hamel, A.M., King, R.C.: Half-turn symmetric alternating sign matrices and Tokuyama type factorisation for orthogonal group characters. J. Combin. Theory Ser. A 131, 1–31 (2015)

  8. Ivanov, D.: Symplectic ice. In: Multiple Dirichlet series, L-functions and automorphic forms. Progr. Math. 300, 205–222. Birkhäuser/Springer, New York (2012)

  9. Korepin, V.E., Bogoliubov, N.M., Izergin, A.G.: Quantum Inverse Scattering Method and Correlation Functions. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (1993)

    Book  Google Scholar 

  10. Kuperberg, G.: Symmetry classes of alternating-sign matrices under one roof. Ann. Math. 156(3), 835–866 (2002)

    Article  MathSciNet  Google Scholar 

  11. McNamara, P.J.: Metaplectic Whittaker functions and crystal bases. Duke Math. J. 156(1), 1–31 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  12. Mirković, I., Vilonen, K.: Geometric Langlands duality and representations of algebraic groups over commutative rings. Ann. Math. 166(1), 95–143 (2007)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  14. Proctor, R.A.: Odd symplectic groups. Invent. Math. 92(2), 307–332 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  15. Razumov, A.V., Stroganov, Y.G.: Enumeration of odd-order alternating-sign half-turn-symmetric matrices. Teoret. Mat. Fiz. 148(3), 357–386 (2006)

    Article  MathSciNet  Google Scholar 

  16. Simpson, T.: Another deformation of Weyl’s denominator formula. J. Comb. Theory Ser. A 77(2), 349–356 (1997)

    Article  MATH  Google Scholar 

  17. Stein, W., et al.: Sage Mathematics Software (Version 5.8). The Sage Development Team (2012). http://www.sagemath.org

  18. Tabony, S.J.: Deformations of characters, metaplectic Whittaker functions, and the Yang–Baxter equation. Ph.D. thesis, Massachusetts Institute of Technology (2010)

  19. Tokuyama, T.: A generating function of strict Gel’fand patterns and some formulas on characters of general linear groups. J. Math. Soc. Jpn. 40(4), 671–685 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  20. Yamamoto, T., Tsuchiya, O.: Integrable \(1/r^2\) spin chain with reflecting end. J. Phys. A 29(14), 3977–3984 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  21. Želobenko, D.P.: Classical groups. Spectral analysis of finite-dimensional representations. Uspehi Mat. Nauk 17(1 (103)), 27–120 (1962)

    MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors thank Hamel and King for helpful conversations and their willingness to openly share portions of their work in progress. The computer algebra software Sage [17] was used extensively to perform supporting computations.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church St. SE, Minneapolis, MN, 55455, USA

    Ben Brubaker

  2. Department of Mathematics, Wellesley College, 106 Central Street, Wellesley, MA, 02482, USA

    Andrew Schultz

Authors
  1. Ben Brubaker
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andrew Schultz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ben Brubaker.

Additional information

This work was partially supported by NSF Grant DMS-1258675.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Brubaker, B., Schultz, A. The six-vertex model and deformations of the Weyl character formula. J Algebr Comb 42, 917–958 (2015). https://doi.org/10.1007/s10801-015-0611-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10801-015-0611-4

Keywords

Search

Navigation