Séminaire Lotharingien de Combinatoire, B64a (2010), 31 pp.

Charles F. Dunkl

Symmetric and Antisymmetric Vector-valued Jack Polynomials

Abstract. Polynomials with values in an irreducible module of the symmetric group can be given the structure of a module for the rational Cherednik algebra, called a standard module. This algebra has one free parameter and is generated by differential-difference ("Dunkl") operators, multiplication by coordinate functions and the group algebra. By specializing Griffeth's (arXiv:0707.0251) results for the G(r,p,N) setting, one obtains norm formulae for symmetric and antisymmetric polynomials in the standard module. Such polynomials of minimum degree have norms which involve hook-lengths and generalize the norm of the alternating polynomial.

Received: June 7, 2010. Revised: September 30, 2010. Accepted: September 30, 2010.

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