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Communications in Mathematical Physics

, Volume 330, Issue 1, pp 415–434 | Cite as

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

  • Christine Berkesch Zamaere
  • Stephen Griffeth
  • Steven V SamEmail author
Article

Abstract

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 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.

Keywords

Hilbert Series Degree Sequence Free Resolution Coordinate Ring Algebra Module 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Christine Berkesch Zamaere
    • 1
  • Stephen Griffeth
    • 2
  • Steven V Sam
    • 3
    Email author
  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  2. 2.Instituto de Matemática y FísicaUniversidad de TalcaTalcaChile
  3. 3.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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