Generalized Clustering Conditions of Jack Polynomials at Negative Jack Parameter $\alpha$

TL;DR

This paper investigates the clustering properties of Jack polynomials at negative parameters, proposing conjectures on their behavior and potential applications in fractional quantum Hall physics.

Contribution

It introduces new conjectures on the clustering behavior of Jack polynomials at negative parameters and their relation to admissibility rules and quantum Hall quasiparticles.

Findings

Jack polynomials at negative lpha exhibit specific clustering properties.

Explicit counting formulas for certain partitions are conjectured.

These properties have implications for fractional quantum Hall quasiparticles.

Abstract

We present several conjectures on the behavior and clustering properties of Jack polynomials at \emph{negative} parameter $\alpha=-\frac{k+1}{r-1}$, of partitions that violate the $(k,r,N)$ admissibility rule of Feigin \emph{et. al.} [\onlinecite{feigin2002}]. We find that "highest weight" Jack polynomials of specific partitions represent the minimum degree polynomials in $N$ variables that vanish when $s$ distinct clusters of $k+1$ particles are formed, with $s$ and $k$ positive integers. Explicit counting formulas are conjectured. The generalized clustering conditions are useful in a forthcoming description of fractional quantum Hall quasiparticles.