
SIGMA 7 (2011), 026, 48 pages arXiv:1009.2366
http://dx.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.
pdf (716 Kb)
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