Skip to main content
SpringerLink
Log in

Polynomial Tau Functions of Symplectic KP and Multi-component Symplectic KP Hierarchies

  • Published:
Annals of Combinatorics Aims and scope Submit manuscript

Abstract

In this paper, we prove that a zero mode of an appropriate combinatorial generating function is a polynomial tau-function of a form of symplectic KP (SKP) hierarchy and extend it to the multi-component SKP hierarchy. Moreover, we write the polynomial tau-function of SKP hierarchy as determinant form in terms of the Vandermonde-like identity.

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

Similar content being viewed by others

References

  1. I. G. Macdonald, Symmetric functions and Hall polynomials. Oxford Mathematical Monographs, Clarendon Press, Oxford, 1979.

    MATH  Google Scholar 

  2. W. Fulton, J. Harris, Representation theory, A first course. Springer-Verlag, New York, 1991.

    MATH  Google Scholar 

  3. R. P. Stanley, Enumerative Combinatorics, Volume II. Cambridge University Press, Cambridge, 1999.

    Book  Google Scholar 

  4. H. Weyl, The classical groups; their invariants and representations, Princeton Univ. Press, Princeton, 1946.

    MATH  Google Scholar 

  5. I. Schur, \(\ddot{U}\)ber die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrocheme lineare Substitionen. J. Reine Angew. Math. 139, (1911), 155-250.

    MathSciNet  MATH  Google Scholar 

  6. P. Hoffman, J. Humphreys, Projective representations of the symmetric groups. Q-functions and shifted tableaux. The Clarendon Press, Oxford University Press, New York, 1992.

    MATH  Google Scholar 

  7. J. R. Stembridge, Shifted tableaux and the projective representations of symmetric groups. Adv. in Math. 74, (1989), 87-134.

    Article  MathSciNet  Google Scholar 

  8. T. Miwa, M. Jimbo, and E. Date, Solitons: Differential equations, symmetries and infinite dimensional algebras. Cambridge University Press, Cambridge, 2000.

    MATH  Google Scholar 

  9. Y. You, Polynomial solutions of the BKP hierarchy and projective representations of symmetric groups, in Infinite-Dimensional Lie Algebras and Groups (Luminy-Marseille, 1988), Adv. Ser. Math. Phys., Vol. 7, World Sci. Publ., Teaneck, NJ, (1989) 449-464.

  10. J. Nimmo, Hall-Littlewood symmetric functions and the BKP equation. J. Phys, A: Mark Gen. 23 (1990), 751-760.

  11. M. Sato, Soliton equations as dynamical systems on infinite-dimensional Grassmann manifold, RIMS Kokyuroku, 439 (1981), 30-46.

    Google Scholar 

  12. V. G. Kac, J. W. van de Leur, Equivalence of formulations of the MKP hierarchy and its polynomial tau-functions, Jpn. J. Math. 13 (2018), 235-271.

    Article  MathSciNet  Google Scholar 

  13. V. G. Kac, J. W. van de Leur, Polynomial tau-functions for the multi-component KP hierarchy, PRIMS (2020). arXiv:1901.07763.

  14. N. Rozhkovskaya, Multiparameter Schur Q-functions are solutions of the BKP hierarchy, Symmetry Integrability Geom. Methods Appl. 15 (2019), 065.

    MathSciNet  MATH  Google Scholar 

  15. V. G. Kac, N. Rozhkovskaya, Johan van de Leur, Polynomial tau-functions of the KP, BKP, and the \(s\)-component KP hierarchies, J. Math. Phys. 62(2021), 021702.

    Article  Google Scholar 

  16. E. Date, M. Jimbo, M. Kashiwara, T. Miwa, Operator approach to the Kadomtsev-Petviashvili equation-transformation groups for soliton equations III, J. Phys. Soc. Japan 50 (1981), 3806-3812.

    Article  MathSciNet  Google Scholar 

  17. E. Date, M. Kashiwara, M. Jimbo and T. Miwa, Transformation groups for soliton equations, in Nonlinear Integrable Systems-Classical Theory and Quantum Theory (Kyoto, 1981) (World Sci. Publishing, Singapore, 1983), 39-119.

    Google Scholar 

  18. C. Li, Multi-component universal character hierarchy and its polynomial tau-functions, Physica D 432(2022), 133166.

    Article  MathSciNet  Google Scholar 

  19. K. Koike, I. Terada, Young diagrammatic methods for the representation theory of the classical groups of type \(B_n\), \(C_n\), \(D_n\), J. Algebra 107 (1987), 466-511.

    Article  MathSciNet  Google Scholar 

  20. T. H. Baker, Vertex operator realization of symplectic and orthogonal S-functions, J. Phys. A: Math. Gen, 29 (1996), 3099-3117.

    Article  MathSciNet  Google Scholar 

  21. N. Jing, B. Nie, Vertex operators, Weyl determinant formulae and Littlewood duality. Ann. Combin. 19 (2015), 427-442.

    Article  MathSciNet  Google Scholar 

  22. F. Huang, N. Wang, Generalized symplectic Schur functions and SUC hierarchy, J. Math. Phys. 61 (2020), 061508.

    Article  MathSciNet  Google Scholar 

  23. C. Li, Two-component symplectic universal characters and integrable hierarchies, Int. J. Math. 32(2021), 2150045.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

Chuanzhong Li is supported by the National Natural Science Foundation of China under Grant No. 12071237.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Henan University, Kaifeng, 475004, China

    Fang Huang

  2. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, China

    Chuanzhong Li

Authors
  1. Fang Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chuanzhong Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chuanzhong Li.

Additional information

Communicated by Ilse Fischer

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, F., Li, C. Polynomial Tau Functions of Symplectic KP and Multi-component Symplectic KP Hierarchies. Ann. Comb. 26, 593–611 (2022). https://doi.org/10.1007/s00026-022-00580-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00026-022-00580-9

Keywords

Mathematics Subject Classification

Search

Navigation