## Symmetric and antisymmetric vector-valued Jack polynomials

### Summary

Summary: 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.

### Mathematics Subject Classification

05E05, 20C30, 33C80, 05E35

### Keywords/Phrases

Jack polynomials, standard modules, Dunkl operators, hook-lengths