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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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