Generalized Calogero–Moser systems from rational Cherednik algebras

Feigin, Misha

doi:10.1007/s00029-011-0074-y

Cited by 21 publications

(32 citation statements)

References 54 publications

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“…More precisely, in Sections 6.3 and 6.4 we show that they are given by particular partial differential operators of so-called deformed CMS type. The operators in question were previously considered by Feigin [11], Hallnäs and Langman [12]. They also appeared, in a disguised form, in the work of Guhr and Kohler [21], and more recently in the context of Random Matrix Theory [10].…”

Section: Description Of Main Resultsmentioning

confidence: 97%

Hermite and Laguerre Symmetric Functions Associated with Operators of Calogero-Moser-Sutherland Type

Desrosiers

Hallnäs

2012

SIGMA

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Abstract. We introduce and study natural generalisations of the Hermite and Laguerre polynomials in the ring of symmetric functions as eigenfunctions of infinite-dimensional analogues of partial differential operators of Calogero-Moser-Sutherland (CMS) type. In particular, we obtain generating functions, duality relations, limit transitions from Jacobi symmetric functions, and Pieri formulae, as well as the integrability of the corresponding operators. We also determine all ideals in the ring of symmetric functions that are spanned by either Hermite or Laguerre symmetric functions, and by restriction of the corresponding infinite-dimensional CMS operators onto quotient rings given by such ideals we obtain socalled deformed CMS operators. As a consequence of this restriction procedure, we deduce, in particular, infinite sets of polynomial eigenfunctions, which we shall refer to as super Hermite and super Laguerre polynomials, as well as the integrability, of these deformed CMS operators. We also introduce and study series of a generalised hypergeometric type, in the context of both symmetric functions and 'super' polynomials.

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Section: Description Of Main Resultsmentioning

confidence: 97%

Hermite and Laguerre Symmetric Functions Associated with Operators of Calogero-Moser-Sutherland Type

Desrosiers

Hallnäs

2012

SIGMA

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“…(1) 1 The integrability of the deformed CMS systems turned out to be a quite nontrivial question. The standard methods like Dunkl operator technique are not working in the general deformed case (for special values of parameters see recent M. Feigin's paper [9]). For the classical series A(n, m) and BC(n, m) the integrability was proved in [17] by explicit construction of the quantum integrals.…”

Section: Introductionmentioning

confidence: 99%

Dunkl Operators at Infinity and Calogero–Moser Systems

Сергеев¹,

Веселов²

2015

Int Math Res Notices

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We define the Dunkl and Dunkl-Heckman operators in infinite number of variables and use them to construct the quantum integrals of the Calogero-Moser-Sutherland problems at infinity. As a corollary we have a simple proof of integrability of the deformed quantum CMS systems related to classical Lie superalgebras. We show how this naturally leads to a quantum version of the Moser matrix, which in the deformed case was not known before.

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“…There are a number of other subspace arrangements whose ideals are unitary modules for the rational Cherednik algebra. For instance, see the papers of Feigin [Fei12] and Feigin-Shramov [FS12]. One might hope that BGG-style resolutions of these modules also exist, so that one could obtain a wider class of examples of linear subspace arrangements whose minimal free resolutions are explicitly known.…”

Section: Conjectural Bgg-style Resolutionmentioning

confidence: 99%

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

Zamaere

Griffeth

Sam

2014

Commun. Math. Phys.

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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 case r = 2 identifies the span of the relevant Jack polynomials with the S ninvariant 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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Generalized Calogero–Moser systems from rational Cherednik algebras

Cited by 21 publications

References 54 publications

Hermite and Laguerre Symmetric Functions Associated with Operators of Calogero-Moser-Sutherland Type

Hermite and Laguerre Symmetric Functions Associated with Operators of Calogero-Moser-Sutherland Type

Dunkl Operators at Infinity and Calogero–Moser Systems

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

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