
SIGMA 7 (2011), 026, 48 pages arXiv:1009.2366
https://doi.org/10.3842/SIGMA.2011.026
VectorValued Jack Polynomials from Scratch
Charles F. Dunkl ^{a} and JeanGabriel Luque ^{b}
^{a)} Dept. of Mathematics, University of Virginia, Charlottesville VA 229044137, USA
^{b)} Université de Rouen, LITIS SaintEtienne du Rouvray, France
Received September 21, 2010, in final form March 11, 2011; Published online March 16, 2011
Abstract
Vectorvalued Jack polynomials associated to the symmetric group S_{N}
are polynomials with
multiplicities in an irreducible module of S_{N} and which are
simultaneous eigenfunctions of the CherednikDunkl 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
YangBaxter 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; YangBaxter graph; Hecke algebra.
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References
 Baratta W., Forrester P.J.,
Jack polynomials fractional quantum Hall states and their generalizations,
Nuclear Phys. B 843 (2011), 362381,
arXiv:1007.2692.
 Baker T.H., Forrester P.J.,
Symmetric Jack polynomials from nonsymmetric theory,
Ann. Comb. 3 (1999), 159170,
qalg/9707001.
 Bervenig B.A., Haldane F.D.M.,
Clustering properties and model wave functions for nonabelian fractional quantum Hall quasielectrons,
Phys. Rev. Lett. 102 (2009), 066802, 4 pages,
arXiv:0810.2366.
 Dunkl C.F.,
Symmetric and antisymmetric vectorvalued Jack polynomials,
Sém. Lothar. Combin. 64 (2010), Art. B64a, 31 pages,
arXiv:1001.4485.
 Dunkl C., Griffeth S.,
Generalized Jack polynomials and the representation theory of rational Cherednik algebras,
Selecta Math. (N.S.) 16 (2010), 791818,
arXiv:1002.4607.
 Etingof P., Stoica E.,
Unitary representation of rational Cherednik algebras,
With an appendix by Stephen Griffeth,
Represent. Theory 13 (2009), 349370,
arXiv:0901.4595.
 Griffeth S.,
Jack polynomials and the coinvariant ring of G(r,p,n),
Proc. Amer. Math. Soc. 137 (2009), 16211629,
arXiv:0806.3292.
 Griffeth S.,
Orthogonal functions generalizing Jack polynomials,
Trans. Amer. Math. Soc. 362 (2010), 61316157,
arXiv:0707.0251.
 Knop F., Sahi S.,
A recursion and a combinatorial formula for Jack polynomials,
Invent. Math. 128 (1997), 922,
qalg/9610016.
 Jolicoeur T., Luque J.G.,
Highest weight Macdonald and Jack polynomials,
J. Phys. A: Math. Theor. 44 (2011), 055204, 21 pages,
arXiv:1003.4858.
 Jucys A.A.A.,
Symmetric polynomials and the center of the symmetric group ring,
Rep. Math. Phys. 5 (1974), 107112.
 Laughlin R.B.,
Anomalous quantum hall effect: an incompressible quantum fluid with fractionally charged excitations,
Phys. Rev. Lett. 50 (1983), 13951398.
 Lascoux A., Young's representation of the symmetric group, available at
http://wwwigm.univmlv.fr/~al/ARTICLES/ProcCrac.ps.gz.
 Lascoux A.,
YangBaxter graphs, Jack and Macdonald polynomials,
Ann. Comb. 5 (2001), 397424.
 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.
 Macdonald I.G.,
Affine Hecke algebras and orthogonal polynomials,
Cambridge Tracts in Mathematics, Vol. 157, Cambridge University Press, Cambridge, 2003.
 Murphy G.E.,
A new construction of Young's seminormal representation of the symmetric groups,
J. Algebra 69 (1981), 287297.
 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, 54715494,
math.RT/0503040.
Okounkov A., Vershik A.,
A new approach to representation theory of symmetric groups,
Selecta Math. (N.S.) 2 (1996), 581605.

