VectorValued Jack Polynomials from Scratch
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.
 Publication:

SIGMA
 Pub Date:
 March 2011
 DOI:
 10.3842/SIGMA.2011.026
 arXiv:
 arXiv:1009.2366
 Bibcode:
 2011SIGMA...7..026D
 Keywords:

 Jack polynomials;
 YangBaxter graph;
 Hecke algebra;
 Mathematics  Combinatorics;
 Mathematics  Classical Analysis and ODEs;
 05E05;
 16T25;
 05C25;
 33C80
 EPrint:
 SIGMA 7 (2011), 026, 48 pages