Jack polynomials as fractional quantum Hall states and the Betti numbers of the (k+1)-equals ideal

TL;DR

This paper establishes vanishing properties of certain Jack polynomials at specific parameters, linking them to fractional quantum Hall states and exploring algebraic structures related to the (k+1)-equals ideal.

Contribution

It proves vanishing theorems for Jack polynomials at special parameters, confirms conjectures about their relation to quantum Hall wavefunctions, and conjectures graded resolutions of Cherednik algebra representations.

Findings

Jack polynomials vanish to order r when k+1 coordinates coincide at specific parameters.

Special cases include Laughlin and Read-Rezayi wavefunctions.

Proved conjectures on clustering properties of Jack polynomials.

Abstract

We show that for Jack parameter \alpha = -(k+1)/(r-1), certain Jack polynomials studied by Feigin-Jimbo-Miwa-Mukhin vanish to order r when k+1 of the coordinates coincide. This result was conjectured by Bernevig and Haldane, who proposed that these Jack polynomials are model wavefunctions for fractional quantum Hall states. Special cases of these Jack polynomials include the wavefunctions of Laughlin and Read-Rezayi. In fact, along these lines we prove several vanishing theorems known as clustering properties for Jack polynomials in the mathematical physics literature, special cases of which had previously been conjectured by Bernevig and Haldane. Motivated by the method of proof, which in case r = 2 identifies the span of the relevant Jack polynomials with the S_n-invariant part of a unitary representation of the rational Cherednik algebra, we conjecture that unitary representations of the type A Cherednik algebra have graded minimal free resolutions of Bernstein-Gelfand-Gelfand type; we prove this for the ideal of the (k+1)-equals arrangement in the case when the number of coordinates n is at most 2k+1. In general, our conjecture predicts the graded S_n-equivariant Betti numbers of the ideal of the (k+1)-equals arrangement with no restriction on the number of ambient dimensions.