Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 7 (2011), 026, 48 pages      arXiv:1009.2366      http://dx.doi.org/10.3842/SIGMA.2011.026

Vector-Valued Jack Polynomials from Scratch

Charles F. Dunkl a and Jean-Gabriel Luque b
a) Dept. of Mathematics, University of Virginia, Charlottesville VA 22904-4137, USA
b) Université de Rouen, LITIS Saint-Etienne du Rouvray, France

Received September 21, 2010, in final form March 11, 2011; Published online March 16, 2011

Abstract
Vector-valued Jack polynomials associated to the symmetric group SN are polynomials with multiplicities in an irreducible module of SN and which are simultaneous eigenfunctions of the Cherednik-Dunkl operators with some additional properties concerning the leading monomial. These polynomials were introduced by Griffeth in the general setting of the complex reflections groups G(r,p,N) and studied by one of the authors (C. Dunkl) in the specialization r=p=1 (i.e. for the symmetric group). By adapting a construction due to Lascoux, we describe an algorithm allowing us to compute explicitly the Jack polynomials following a Yang-Baxter graph. We recover some properties already studied by C. Dunkl and restate them in terms of graphs together with additional new results. In particular, we investigate normalization, symmetrization and antisymmetrization, polynomials with minimal degree, restriction etc. We give also a shifted version of the construction and we discuss vanishing properties of the associated polynomials.

Key words: Jack polynomials; Yang-Baxter graph; Hecke algebra.

pdf (716 Kb)   tex (58 Kb)

References

  1. Baratta W., Forrester P.J., Jack polynomials fractional quantum Hall states and their generalizations, Nuclear Phys. B 843 (2011), 362-381, arXiv:1007.2692.
  2. Baker T.H., Forrester P.J., Symmetric Jack polynomials from non-symmetric theory, Ann. Comb. 3 (1999), 159-170, q-alg/9707001.
  3. Bervenig B.A., Haldane F.D.M., Clustering properties and model wave functions for non-abelian fractional quantum Hall quasielectrons, Phys. Rev. Lett. 102 (2009), 066802, 4 pages, arXiv:0810.2366.
  4. Dunkl C.F., Symmetric and antisymmetric vector-valued Jack polynomials, Sém. Lothar. Combin. 64 (2010), Art. B64a, 31 pages, arXiv:1001.4485.
  5. Dunkl C., Griffeth S., Generalized Jack polynomials and the representation theory of rational Cherednik algebras, Selecta Math. (N.S.) 16 (2010), 791-818, arXiv:1002.4607.
  6. Etingof P., Stoica E., Unitary representation of rational Cherednik algebras, With an appendix by Stephen Griffeth, Represent. Theory 13 (2009), 349-370, arXiv:0901.4595.
  7. Griffeth S., Jack polynomials and the coinvariant ring of G(r,p,n), Proc. Amer. Math. Soc. 137 (2009), 1621-1629, arXiv:0806.3292.
  8. Griffeth S., Orthogonal functions generalizing Jack polynomials, Trans. Amer. Math. Soc. 362 (2010), 6131-6157, arXiv:0707.0251.
  9. Knop F., Sahi S., A recursion and a combinatorial formula for Jack polynomials, Invent. Math. 128 (1997), 9-22, q-alg/9610016.
  10. Jolicoeur T., Luque J.-G., Highest weight Macdonald and Jack polynomials, J. Phys. A: Math. Theor. 44 (2011), 055204, 21 pages, arXiv:1003.4858.
  11. Jucys A.-A.A., Symmetric polynomials and the center of the symmetric group ring, Rep. Math. Phys. 5 (1974), 107-112.
  12. Laughlin R.B., Anomalous quantum hall effect: an incompressible quantum fluid with fractionally charged excitations, Phys. Rev. Lett. 50 (1983), 1395-1398.
  13. Lascoux A., Young's representation of the symmetric group, available at http://www-igm.univ-mlv.fr/~al/ARTICLES/ProcCrac.ps.gz.
  14. Lascoux A., Yang-Baxter graphs, Jack and Macdonald polynomials, Ann. Comb. 5 (2001), 397-424.
  15. Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.
  16. Macdonald I.G., Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, Vol. 157, Cambridge University Press, Cambridge, 2003.
  17. Murphy G.E., A new construction of Young's seminormal representation of the symmetric groups, J. Algebra 69 (1981), 287-297.
  18. Okounkov A., Vershik A., A new approach to the representation theory of the symmetric groups. II, J. Math. Sci. (N.Y.) 131 (2005), no. 2, 5471-5494, math.RT/0503040.
    Okounkov A., Vershik A., A new approach to representation theory of symmetric groups, Selecta Math. (N.S.) 2 (1996), 581-605.

Previous article   Next article   Contents of Volume 7 (2011)