Jack polynomial fractional quantum Hall states and their generalizations

TL;DR

This paper proves that certain Jack polynomials with specific parameters satisfy a clustering condition relevant to fractional quantum Hall states, using translational invariance, and explores related polynomials.

Contribution

It provides a simple proof linking Jack polynomials with fractional quantum Hall clustering conditions, expanding understanding of their mathematical structure.

Findings

Jack polynomials with specified parameters satisfy the clustering condition

Translational invariance is key to the proof

Connections to generalized Hermite, Laguerre, and Macdonald polynomials are explored

Abstract

In the the study of fractional quantum Hall states, a certain clustering condition involving up to four integers has been identified. We give a simple proof that particular Jack polynomials with $\alpha = - (r-1)/(k+1)$, $(r-1)$ and $(k+1)$ relatively prime, and with partition given in terms of its frequencies by $[n_00^{(r-1)s}k 0 ^{r-1}k 0 ^{r-1}k \cdots 0 ^{r-1} m]$ satisfy this clustering condition. Our proof makes essential use of the fact that these Jack polynomials are translationally invariant. We also consider nonsymmetric Jack polynomials, symmetric and nonsymmetric generalized Hermite and Laguerre polynomials, and Macdonald polynomials from the viewpoint of the clustering.