# Vector-Valued Jack Polynomials from Scratch

@article{Dunkl2010VectorValuedJP, title={Vector-Valued Jack Polynomials from Scratch}, author={Charles F. Dunkl and Jean-Gabriel Luque}, journal={Symmetry Integrability and Geometry-methods and Applications}, year={2010}, volume={7}, pages={026} }

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 operators with some additional properties concerning the leading monomial. These polynomials were introduced by Griffeth in the general setting of the complex reflections groups G(r;p;N) and studied by one of the authors (C. Dunkl) in the specialization r = p = 1 (i.e. for the symmetric group). By…

## 11 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…

### Connections between vector-valued and highest weight Jack and Macdonald polynomials

- Mathematics
- 2019

We analyze conditions under which a projection from the vector-valued Jack or Macdonald polynomials to scalar polynomials has useful properties, especially commuting with the actions of the symmetric…

### Singular Nonsymmetric Jack Polynomials for Some Rectangular Tableaux

- MathematicsSymmetry
- 2020

The main ingredient is the proof of uniqueness of certain spectral vectors, namely, the list of eigenvalues of the Jack polynomials for the Cherednik-Dunkl operators, when specialized to $\kappa=1/\left( m+2\right) $.

### Vector valued Macdonald polynomials

- Mathematics
- 2011

This paper defines and investigates nonsymmetric Macdonald polynomials with values in an irreducible module of the Hecke algebra of type $A_{N-1}$. These polynomials appear as simultaneous…

### A superpolynomial version of nonsymmetric Jack polynomials

- MathematicsThe Ramanujan Journal
- 2021

Superpolynomials consist of commuting and anti-commuting variables. By considering the anti-commuting variables as a module of the symmetric group the theory of vector-valued nonsymmetric Jack…

### 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…

### 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…

### The Smallest Singular Values and Vector-Valued Jack Polynomials

- MathematicsSymmetry, Integrability and Geometry: Methods and Applications
- 2018

There is a space of vector-valued nonsymmetric Jack polynomials associated with any irreducible representation of a symmetric group. Singular polynomials for the smallest singular values are…

### Singular Nonsymmetric Macdonald Polynomials and Quasistaircases

- MathematicsSymmetry, Integrability and Geometry: Methods and Applications
- 2020

Singular nonsymmetric Macdonald polynomials are constructed by use of the representation theory of the Hecke algebras of the symmetric groups. These polynomials are labeled by quasistaircase…

### Orthogonal Polynomials of Several Variables

- Mathematics
- 2014

Preface to the second edition Preface to the first edition 1. Background 2. Orthogonal polynomials in two variables 3. General properties of orthogonal polynomials in several variables 4. Orthogonal…

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