Jack Polynomials as Fractional Quantum Hall States and the Betti Numbers of the (k + 1)-Equals Ideal
Jack Polynomials as Fractional Quantum Hall States and the Betti Numbers of the (k + 1)-Equals Ideal
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作为分数量子霍尔态的杰克多项式和 (k 1) 等于理想值的贝蒂数
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发表时间：
2013
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通讯作者：
Steven V. Sam
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作者：
Christine Berkesch Zamaere;Stephen Griffeth;Steven V. Sam
我们证明了对于Jack参数α = −（k + 1）/（r − 1），Feigin-Jimbo-Miwa-Mukhin研究的某些Jack多项式在k + 1个坐标重合时消失到r阶。这一结果由Bernevig和Halfman证明，他们提出这些Jack多项式是分数量子霍尔态的模型波函数。这些杰克多项式的特殊情况包括Laughlin和Read-Rezayi的波函数。事实上，沿着这些线，我们证明了几个消失定理被称为集群性质的杰克多项式在数学物理文献中，特殊情况下，其中以前已被证明的Bernevig和Halflian。在证明方法的启发下，在r = 2的情况下，我们将相关Jack多项式的跨距与有理Cherednik代数的酉表示的Sn-不变部分等同起来，我们猜想A型Cherednik代数的酉表示具有Bernstein-Gelfand-Gelfand型的分次极小自由归结;我们证明了这一点的理想的（k + 1）-等于安排的情况下，坐标的数目n是最多2k + 1。在一般情况下，我们的猜想预测的分次Sn-等变贝蒂数的理想的（k + 1）-等于安排，没有限制的环境维数。
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 Sn-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 Sn-equivariant Betti numbers of the ideal of the (k + 1)-equals arrangement with no restriction on the number of ambient dimensions.