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.

22 pages, added remark about the Gordon-Stafford Theorem, corrected some typos

References in corpus (3)

• Unitary representations of rational Cherednik algebras
• Orthogonal functions generalizing Jack polynomials
• A New Approach to the Representation Thoery of the Symmetric Groups. 2

Cited by in corpus (6)

• Vector-Valued Jack Polynomials from Scratch
• Generalized Jack polynomials and the representation theory of rational Cherednik algebras
• Orthogonality Measure on the Torus for Vector-Valued Jack Polynomials
• Vector-valued Jack Polynomials and Wavefunctions on the Torus
• A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus
• Macdonald polynomials as characters of Cherednik algebra modules