Annals of Combinatorics 3 (1999) 159-170


Symmetric Jack Polynomials from Non-Symmetric Theory

T.H. Baker and P.J. Forrester

Department of Mathematics and Statistics, University of Melbourne, Parkville, Victoria 3052, Australia
{tbaker, matpjf}@maths.mu.oz.au

Received Feburary 25, 1999

AMS Subject Classification: 33C80

Abstract. We show how a number of fundamental properties of the symmetric and anti-symmetric Jack polynomials can be derived from knowledge of the corresponding properties of the non-symmetric Jack polynomials. These properties include orthogonality relations, normalization formulas, a specialization formula and the evaluation of a proportionality constant relating the anti-symmetric Jack polynomials to a product of differences multiplied by the symmetric Jack polynomials with a shifted parameter.

Keywords: Jack polynomials, symmetric functions


References

1.  T.H. Baker, C.F. Dunkl, and P.J. Forrester, Polynomial eigenfunctions of the Calogero--Sutherland--Moser models with exchange terms, In: CRM Proc. Lecture Notes, Amer. Math. Soc., Providence RI, to appear.

2.  T.H. Baker and P.J. Forrester, Non-symmetric Jack polynomials and integral kernels, Duke J. Math. 95 (1998) 1–50.

3.  T.H. Baker and P.J. Forrester, The Calogero--Sutherland model and generalized classical polynomials, Commun. Math. Phys. 188 (1997) 175–216.

4.  T.H. Baker and P.J. Forrester, The Calogero--Sutherland model and polynomials with prescribed symmetry, Nucl. Phys. B 492 (1997) 682–716.

5.  C.F. Dunkl, Intertwining operators and polynomials associated with the symmetric group, Monat. Math. 126 (1998) 181–209.

6.  C.F. Dunkl, Orthogonal polynomials of types A and B and related Calogero models, Commun. Math. Phys. 197 (1998) 451–487.

7.  F. Knop and S. Sahi, A recursion and combinatorial formula for {Jack} polynomials, Inv. Math. 128 (1997) 9–22.

8.  M. Lassalle, Polynômes de Jacobi gènèralisès, C. R. Acad. Sci. Paris, t. Sèries I 312 (1991) 425–428.

9.  I.G. Macdonald, Hypergeometric functions, unpublished.

10.  I.G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd Ed., Oxford University Press, Oxford, 1995.

11.  E.M. Opdam, Harmonic analysis for certain representations of graded Hecke algebras, Acta Math. 175 (1995) 75–121.

12.  S. Sahi, A new scalar product for nonsymmetric {Jack} polynomials, Int. Math. Res. Not. 20 (1996) 997–1004.

13.  R.P. Stanley, Some combinatorial properties of {Jack} symmetric functions, Adv. Math. 77 (1989) 76–115.


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