Skip to main content
SpringerLink
Log in

Jucys–Murphy elements, orthogonal matrix integrals, and Jack measures

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

We study symmetric polynomials whose variables are odd-numbered Jucys–Murphy elements. They define elements of the Hecke algebra associated to the Gelfand pair of the symmetric group with the hyperoctahedral group. We evaluate their expansions in zonal spherical functions and in double coset sums. These evaluations are related to integrals of polynomial functions over orthogonal groups. Furthermore, we give their extension based on Jack polynomials.

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. Borodin, A., Olshanski, G.: Z-measures on partitions and their scaling limits. Eur. J. Comb. 26(6), 795–834 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cheng, S.-E., Eu, S.-P., Fu, T.-S.: Area of Catalan paths on a checkerboard. Eur. J. Comb. 28(4), 1331–1344 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Collins, B.: Moments and cumulants of polynomial random variables on unitary groups, the Itzykson–Zuber integral, and free probability. Int. Math. Res. Not. 17, 953–982 (2003)

    Article  Google Scholar 

  4. Collins, B., Matsumoto, S.: On some properties of orthogonal Weingarten functions. J. Math. Phys. 50, 113516 (2009)

    Article  MathSciNet  Google Scholar 

  5. Collins, B., Śniady, P.: Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. Commun. Math. Phys. 264, 773–795 (2006)

    Article  MATH  Google Scholar 

  6. Fujii, S., Kanno, H., Moriyama, S., Okada, S.: Instanton calculus and chiral one-point functions in supersymmetric gauge theories. Adv. Theor. Math. Phys. 12(6), 1401–1428 (2008)

    MathSciNet  MATH  Google Scholar 

  7. Garsia, A.: Young’s Seminormal Representation and Murphy Elements and Content Evaluations. Lecture Notes in Algebraic Combinatorics. Available at: http://math.ucsd.edu/~garsia/

  8. Jucys, A.: Symmetric polynomials and the center of the symmetric group ring. Rep. Math. Phys. 5, 107–112 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kerov, S., Okounkov, A., Olshanski, G.: The boundary of the Young graph with Jack edge multiplicities. Int. Math. Res. Not. (4), 173–199 (1998)

  10. Lascoux, A., Thibon, J.-Y.: Vertex operators and the class algebras of symmetric groups. Zap. Nauč. Semin. St. Petersbg. Otd. Mat. Inst. Steklova 283 (2001)

  11. Lascoux, A., Thibon, J.-Y.: Vertex operators and the class algebras of symmetric groups. Teor. Predst. Din. Sist. Komb. Algoritm. Metody 6, 156–177, 261; translation in J. Math. Sci. 121(3), 2380–2392 (2004)

  12. Lassalle, M.: Some combinatorial conjectures for Jack polynomials. Ann. Comb. 2(1), 61–83 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lassalle, M.: Jack polynomials and some identities for partitions. Trans. Am. Math. Soc. 356(9), 3455–3476 (2004) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lassalle, M.: A positivity conjecture for Jack polynomials. Math. Res. Lett. 15(4), 661–681 (2008)

    MathSciNet  MATH  Google Scholar 

  15. Lassalle, M.: Jack polynomials and free cumulants. Adv. Math. 222, 2227–2269 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lassalle, M.: Table for Jack polynomials: Jack polynomials and alpha-contents. Available at http://igm.univ-mlv.fr/lassalle/jack.html

  17. Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, Oxford (1995)

    MATH  Google Scholar 

  18. Matsumoto, S.: Jack deformations of Plancherel measures and traceless Gaussian random matrices. Electr. J. Combin. 15, #R149 (2008)

    Google Scholar 

  19. Matsumoto, S., Novak, J.: Jucys–Murphy elements and unitary matrix integrals. arXiv:0905.1992v2

  20. Novak, J.: Jucys–Murphy elements and Weingarten function. Banach Cent. Publ. 89, 231–235 (2010)

    Article  Google Scholar 

  21. Olshanski, G.: Plancherel averages: remarks on a paper by Stanley. Electron. J. Comb. 17, R43 (2010)

    MathSciNet  Google Scholar 

  22. Sagan, B.E.: The Symmetric Group. Representations, Combinatorial Algorithms, and Symmetric Functions, 2nd edn. Graduate Texts in Mathematics, vol. 203. Springer, New York (2001)

    MATH  Google Scholar 

  23. Weingarten, D.: Asymptotic behavior of group integrals in the limit of infinite rank. J. Math. Phys. 19(5), 999–1001 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  24. Zinn-Justin, P.: Jucys–Murphy elements and Weingarten matrices. Lett. Math. Phys. 91(2), 119–127 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Graduate School of Mathematics, Nagoya University, Furocho, Chikusa-ku, Nagoya, 464-8602, Japan

    Sho Matsumoto

Authors
  1. Sho Matsumoto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sho Matsumoto.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Matsumoto, S. Jucys–Murphy elements, orthogonal matrix integrals, and Jack measures. Ramanujan J 26, 69–107 (2011). https://doi.org/10.1007/s11139-011-9317-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-011-9317-y

Keywords

Mathematics Subject Classification (2000)

Search

Navigation