Jack Polynomials as Fractional Quantum Hall States and the Betti Numbers of the (k + 1)-Equals Ideal

• Published in: Communications in mathematical physics Vol. 330; no. 1; pp. 415 - 434
• Main Authors: Zamaere, Christine Berkesch, Griffeth, Stephen, Sam, Steven V
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2014
• Subjects: Classical and Quantum Gravitation; Complex Systems; Mathematical and Computational Physics; Mathematical Physics; Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Theoretical; Hilbert Series; Free Resolution; Degree Sequence; Coordinate Ring; Algebra Module
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-014-2010-4

We show that for Jack parameter α = −( 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 the 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 2 k + 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.