Abstract
We study Jack polynomials in N variables, with parameter α, and having a prescribed symmetry with respect to two disjoint subsets of variables. For instance, these polynomials can exhibit a symmetry of type AS, which means that they are antisymmetric in the first m variables and symmetric in the remaining N − m variables. One of our main goals is to extend recent works on symmetric Jack polynomials (Baratta and Forrester in Nucl Phys B 843:362–381, 2011; Berkesch et al. in Jack polynomials as fractional quantum Hall states and the Betti numbers of the (k + 1)-equals ideal, 2013; Bernevig and Haldane in Phys Rev Lett 101:1–4, 2008) and prove that the Jack polynomials with prescribed symmetry also admit clusters of size k and order r, that is, the polynomials vanish to order r when k + 1 variables coincide. We first prove some general properties for generic α, such as their uniqueness as triangular eigenfunctions of operators of Sutherland type, and the existence of their analogues in infinity many variables. We then turn our attention to the case with α = −(k + 1)/(r − 1). We show that for each triplet (k, r, N), there exist admissibility conditions on the indexing sets, called superpartitions, that guaranty both the regularity and the uniqueness of the polynomials. These conditions are also used to establish similar properties for non-symmetric Jack polynomials. As a result, we prove that the Jack polynomials with arbitrary prescribed symmetry, indexed by (k, r, N)-admissible superpartitions, admit clusters of size k = 1 and order r ≥ 2. In the last part of the article, we find necessary and sufficient conditions for the invariance under translation of the Jack polynomials with prescribed symmetry AS. This allows to find special families of superpartitions that imply the existence of clusters of size k > 1 and order r ≥ 2.
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Baker, T.H., Dunkl, C.F., Forrester, P.J.: Polynomial eigenfunctions of the Calogero-Sutherland-Moser models with exchange terms In: van Diejen, J.F., Vinet, L. (eds.) Calogero-Sutherland-Moser Models. CRM Series in Mathematical Physics, pp. 37–42. Springer, Berlin (2000)
Baker T.H., Forrester P.J.: The Calogero-Sutherland model and polynomials with prescribed symmetry. Nucl. Phys. B 492, 682–716 (1997)
Baratta W.: Some properties of Macdonald polynomials with prescribed symmetry. Kyushu J. Math. 64, 323–343 (2010)
Baratta W., Forrester P.J.: Jack polynomial fractional quantum Hall states and their generalizations. Nucl. Phys. B 843, 362–381 (2011)
Berkesch, C., Griffeth, S., Sam, S.V.: Jack polynomials as fractional quantum Hall states and the Betti numbers of the (k + 1)-equals ideal. arXiv:1303.4126 (2013)
Bernard D., Gaudin M., Haldane F.D., Pasquier V.: Yang-Baxter equation in long range interacting system. J. Phys. A 26, 5219–5236 (1993)
Bernevig B.A., Haldane F.D.: Fractional quantum Hall states and Jack polynomials. Phys. Rev. Lett. 101(246806), 1–4 (2008)
Bernevig B.A., Haldane F.D.: Generalized clustering conditions of Jack polynomials at negative Jack parameter α. Phys. Rev. B 77(184502), 1–10 (2008)
Corteel S., Lovejoy J.: Overpartitions. Trans. Amer. Math. Soc. 356, 1623–1635 (2004)
Desrosiers P., Lapointe L., Mathieu P.: Jack polynomials in superspace. Commun. Math. Phys. 242, 331–360 (2003)
Desrosiers P., Lapointe L., Mathieu P.: Classical symmetric functions in superspace. J. Alg. Comb. 24, 209–238 (2006)
Desrosiers P., Lapointe L., Mathieu P.: Evaluation and normalization of Jack polynomials in superspace. Int. Math. Res. Not. 23, 5267–5327 (2012)
Desrosiers P., Lapointe L., Mathieu P.: Jack superpolynomials with negative fractional parameter: clustering properties and super-Virasoro ideals. Commun. Math. Phys. 316, 395–440 (2012)
Dunkl C.F.: Orthogonal polynomials of types A and B and related Calogero models. Commun. Math. Phys. 197, 451–487 (1998)
Dunkl, C.F., Luque, J.-G.: Clustering properties of rectangular Macdonald polynomials. arXiv:1204.5117 (2012)
Feigin B., Jimbo M., Miwa T., Mukhin E.: A differential ideal of symmetric polynomials spanned by Jack polynomials at β = −(r−1)/(k + 1). Int. Math. Res. Not. 23, 1223–1237 (2002)
Forrester, P.J.: Log-gases and random matrices. In: London Mathematical Society Monographs, vol. 34. Princeton University Press, Princeton (2010)
Forrester P.J., McAnally D.S., Nikoyalevsky Y.: On the evaluation formula for Jack polynomials with prescribed symmetry. J. Phys. A 34, 8407–8424 (2001)
Jolicoeur Th., Luque J.-G.: Highest weight Macdonald and Jack polynomials. J. Phys. A Math. Theor. 44, 055204 (2011)
Kato Y., Kuramoto Y.: Exact solution of the sutherland model with arbitrary internal symmetry. Phys. Rev. Lett. 74, 1222–1225 (1995)
Kato Y., Kuramoto Y.: Dynamics of One-Dimensional Quantum Systems: Inverse-Square Interaction Models. Cambridge University Press, Cambridge (2009)
Kato Y., Yamamoto T.: Jack polynomials with prescribed symmetry and hole propagator of spin Calogero-Sutherland model. J. Phys. A 31, 9171–9184 (1998)
Knop F., Sahi S.: A recursion and a combinatorial formula for Jack polynomials. Invent. Math. 128, 9–22 (1997)
Lassalle M.: Une formule du binôme généralisée pour les polynômes de Jack. C. R. Acad. Sci. Paris Série I 310, 253–256 (1990)
Lassalle M.: Coefficients binomiaux génétalisés et polynômes de Macdonald. J. Funct. Anal. 158, 289–324 (1998)
Macdonald I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press Inc, New York (1995)
Opdam E.M.: Harmonic analysis for certain representations of graded Hecke algebras. Acta Math. 175, 75–121 (1995)
Polychronakos A.P.: Exchange operator formalism for integrable systems of particles. Phys. Rev. Lett. 69, 702–705 (1992)
Polychronakos A.P.: The physics and mathematics of Calogero particles. J. Phys. A 39, 12793–12845 (2006)
Stanley R.P.: Some combinatorial properties of Jack symmetric functions. Adv. Math. 77, 76–115 (1989)
Sutherland B.: Exact results for a quantum many-body problem in one dimension. Phys. Rev. A 4, 2019–2021 (1971)
Sutherland B.: Exact results for a quantum many-body problem in one dimension. II. Phys. Rev. A 5, 1372–1376 (1972)
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Communicated by Jean-Michel Maillet.
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Desrosiers, P., Gatica, J. Jack Polynomials with Prescribed Symmetry and Some of Their Clustering Properties. Ann. Henri Poincaré 16, 2399–2463 (2015). https://doi.org/10.1007/s00023-014-0376-7
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DOI: https://doi.org/10.1007/s00023-014-0376-7