# Symmetric and Antisymmetric Vector-valued Jack Polynomials

@article{Dunkl2010SymmetricAA, title={Symmetric and Antisymmetric Vector-valued Jack Polynomials}, author={Charles F. Dunkl}, journal={arXiv: Combinatorics}, year={2010} }

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…

## 8 Citations

### Orthogonality Measure on the Torus for Vector-Valued Jack Polynomials

- Mathematics
- 2016

For each irreducible module of the symmetric group on N objects there is a set of parametrized nonsymmetric Jack polynomials in N variables taking values in the module. These polynomials are…

### Vector-Valued Jack Polynomials from Scratch

- Mathematics
- 2011

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…

### Vector-valued Jack polynomials and wavefunctions on the torus

- Mathematics
- 2017

The Hamiltonian of the quantum Calogero–Sutherland model of N identical particles on the circle with 1/r2 interactions has eigenfunctions consisting of Jack polynomials times the base state. By use…

### 2 Yang – Baxter type graph associated to a partition 2 . 1 Sorting a vector

- Mathematics
- 2011

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…

### A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus

- Mathematics
- 2016

For each irreducible module of the symmetric group $\mathcal{S}_{N}$ there is a set of parametrized nonsymmetric Jack polynomials in $N$ variables taking values in the module. These polynomials are…

### Generalized Jack polynomials and the representation theory of rational Cherednik algebras

- Mathematics
- 2010

We apply the Dunkl–Opdam operators and generalized Jack polynomials to study category $${{\mathcal O}_c}$$ for the rational Cherednik algebra of type G(r, p, n). We determine the set of aspherical…

### Macdonald polynomials as characters of Cherednik algebra modules

- MathematicsMathematische Zeitschrift
- 2013

We prove that Macdonald polynomials are characters of irreducible Cherednik algebra modules.

### Macdonald polynomials as characters of Cherednik algebra modules

- Mathematics
- 2012

We prove that Macdonald polynomials are characters of irreducible Cherednik algebra modules.

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Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale, toute copie ou impressions de ce fichier doit contenir la présente mention de copyright.

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