Abstract
The symmetric Macdonald polynomials may be constructed from the nonsymmetric Macdonald polynomials. This allows us to develop the theory of the symmetric Macdonald polynomials by first developing the theory of their nonsymmetric counterparts. In taking this approach we are able to obtain new results as well as simpler and more accessible derivations of a number of the known fundamental properties of both kinds of polynomials.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G.E. Andrews, Problems and prospects for basic hypergeometric functions, In: Theory and Applications of Special Functions, R. Askey, Ed., Academic Press, New York, 1977, pp. 191–224.
R. Askey, Some basic hypergeometric extensions of integrals of Selberg and Andrews, SIAM J. Math. Analysis11 (1980) 938–951.
T.H. Baker and P.J. Forrester, Isomorphisms of typeA-affine Hecke algebras and multivariable orthogonal polynomials, Pacific J. Math, submitted.
T.H. Baker and P.J. Forrester, Multivariable Al-Salam and Carlitz polynomials associated with the type aq-dunkl kernel, Math. Nachr., in press.
T.H. Baker and P.J. Forrester, Aq-analogue of the typeA Dunkl operator and integral kernel, Int. Math. Res. Not.14 (1997) 667–686.
T.H. Baker and P.J. Forrester, Symmetric Jack polynomials from nonsymmetric theory, q-alg/9707001, 1997, preprint.
T.H. Baker and P.J. Forrester, Generalizations of theq-Morris constant term identity, J. Combin. Theory Ser. A81 (1998) 69–87.
T.H. Baker, C.F. Dunkl, and P.J. Forrester, Polynomial eigenfunctions of the Colegero-Sutherland-Moser models with exchange terms, In: Proc. CRM Workshop on Colegero-Sutherland-Moser models, to appear.
D. Bressoud and D. Zeilberger, A proof of Andrews'q-Dyson conjecture, Discrete Math.54 (1985) 201–224.
I. Cherednik, A unification of Kniznik-Zamolodchikov and Dunkl operators via affine Hecke algebras, Inv. Math.106 (1991) 411–432.
I. Cherednik, Integration of quantum many-body problems by affine Knizhnik-Zamolodchikov equations, Commum. Math. Phys.106 (1994) 65–95.
I. Cherednik, Nonsymmetric Macdonald polynomials, Int. Math. Res. Not.10 (1995) 483–515.
G. Duchamp, D. Krob, A. Lascoux, B. Leclerc, T. Scharf, and J. Thibon, Euler-Poincaré characteristic and polynomial representations of Iwahori-Hecke algebras, RIMS31 (1995) 179–201.
C.F. Dunkl, Intertwining operators and polynomials associated with the symmetric group, Monat. Math., to appear.
L. Habsieger, Uneq-intégrale de Selberg-Askey, SIAM J. Math. Analysis19 (1988) 1475–1489.
K.W.J. Kadell, A proof of Askey's conjecturedq-analogue of Selberg's integral and a conjecture of Morris, SIAM J. Math. Analysis19 (1988) 969–986.
J. Kaneko,q-Selberg integrals and Macdonald polynomials, Ann. Sci. Éc. Norm. Sup. 4e série29 (1996) 1086–1110.
J. Kaneko, Constant term identities of Forrester-Zeilberger-Cooper, Discrete Math.173 (1997) 79–90.
A.N. Kirillov and M. Noumi, Affine Hecke algebras and raising operators for the Macdonald polynomials, Duke Math. J.93 (1998) 1–39.
F. Knop and S. Sahi, Difference equations and symmetric polynomials defined by their zeros, Int. Math. Res. Not.10 (1996) 473–486.
F. Knop and S. Sahi, A recursion and combinatorial formula for the Jack polynomials, Inv. Math.128 (1996) 9–22.
H. Konno, Relativistic Calogero-Sutherland model: Spin generalization, quantum affine symmetry and dynamical correlation functions, Nucl. Phys. B473 (1996) 579–601.
I.G. Macdonald, Notes on Shubert Polynomials, Lacim, Montreal 1991.
I.G. Macdonald, Hall Polynomials and Symmetric Functions, Oxford University Press, Oxford, 2nd Ed., 1995.
I.G. Macdonald, Affine Hecke algebras and orthogonal polynomials, In: Séminaire Bourbaki, 47ème année:797, 1994–5.
K. Mimachi and M. Noumi, A reproducing kernel for nonsymmetric Macdonald polynomials, Duke Math. J.91 (1998) 621–634.
W.G. Morris, Constant term identities for finite and affine root systems, conjectures and theorems, Ph.D. thesis, University of Wisconsin, 1982.
A. Okounkov, Binomial formula for Macdonald polynomials, Math. Res. Lett.4 (1997) 533–553.
S. Sahi, A new scalar product for the nonsymmetric Jack polynomials, Int. Math. Res. Not.20 (1996) 997–1004.
J.R. Stembridge, A short proof of Macdonald's conjecture for the root systems of type A, Proc. Amer. Math. Soc.102 (1988) 777–785.
Author information
Authors and Affiliations
Additional information
Supported by an APA scholarship.
Rights and permissions
About this article
Cite this article
Marshall, D. Symmetric and nonsymmetric Macdonald polynomials. Annals of Combinatorics 3, 385–415 (1999). https://doi.org/10.1007/BF01608794
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01608794