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

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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.

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Authors and Affiliations

  1. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Christine Berkesch Zamaere

  2. Instituto de Matemática y Física, Universidad de Talca, Talca, Chile

    Stephen Griffeth

  3. Department of Mathematics, University of California, Berkeley, CA, USA

    Steven V Sam

Authors
  1. Christine Berkesch Zamaere
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  2. Stephen Griffeth
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  3. Steven V Sam
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Correspondence to Steven V Sam.

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Communicated by N. Reshetikhin

SG acknowledges the financial support of Fondecyt Proyecto Regular 1110072. SS was supported by a Miller research fellowship.

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Zamaere, C.B., Griffeth, S. & Sam, S.V. Jack Polynomials as Fractional Quantum Hall States and the Betti Numbers of the (k + 1)-Equals Ideal. Commun. Math. Phys. 330, 415–434 (2014). https://doi.org/10.1007/s00220-014-2010-4

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  • DOI: https://doi.org/10.1007/s00220-014-2010-4

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