Highest weight Macdonald and Jack Polynomials

TL;DR

This paper investigates highest weight Macdonald and Jack polynomials, providing a classification based on their symmetry properties and parameter specialization, with applications to fractional quantum Hall states.

Contribution

It introduces a classification of highest weight Macdonald and Jack polynomials using parameter specialization and symmetry conditions.

Findings

Classification of highest weight Macdonald polynomials via (q,t)-deformation.

Specialization of parameters yields a classification of highest weight Jack polynomials.

Results applicable to staircase and rectangular partition indexing.

Abstract

Fractional quantum Hall states of particles in the lowest Landau levels are described by multivariate polynomials. The incompressible liquid states when described on a sphere are fully invariant under the rotation group. Excited quasiparticle/quasihole states are member of multiplets under the rotation group and generically there is a nontrivial highest weight member of the multiplet from which all states can be constructed. Some of the trial states proposed in the literature belong to classical families of symmetric polynomials. In this paper we study Macdonald and Jack polynomials that are highest weight states. For Macdonald polynomials it is a (q,t)-deformation of the raising angular momentum operator that defines the highest weight condition. By specialization of the parameters we obtain a classification of the highest weight Jack polynomials. Our results are valid in the case of staircase and rectangular partition indexing the polynomials.