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

SIGMA 7 (2011), 026, 48 pages      arXiv:1009.2366      https://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.

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