<%@ Page Language="C#" MasterPageFile="~/Main.master" AutoEventWireup="true" Title="Volume 5 Issue 3" %>
Yang-Baxter Graphs, Jack and Macdonald Polynomials
Alain Lascoux
CNRS, Institut Gaspard Monge, Université de Marne-la-Vallée 77454 Marne-la-Vallée Cedex, France
alain.lascoux@univ-mlv.fr
Annals of Combinatorics 5 (3) p.397-424 September, 2001
AMS Subject Classification: 05E05
Abstract:
We describe properties of the affine graph underlying the recursions between the different varieties of nonsymmetric Macdonald and Jack polynomials. We use an arbitrary function of one variable in the definition of affine edges, and of Cherednik's elements, to unify the different theories. We describe the symmetrizing operators furnishing the symmetric polynomials from the nonsymmetric ones.
Keywords: Macdonald polynomials, Jack polynomials, Yang-Baxter

References:

1.  T.H. Baker and P.J. Forrester, Symmetric Jack polynomials from non-symmetric theory, Ann. Combin. 3 (1999) 159–170.

2.  T.H. Baker and P.J. Forrester, A q-analogue of the type A Dunkl operator and integral kernel, Internat. Math. Res. Notices 14 (1997) 667–686.

3.  T.H. Baker and P.J. Forrester, Isomorphisms of type A affine Hecke algebras and multivariate orthogonal polynomials, Pacific J. Math. 194 (2000) 19–41.

4.  I.N. Bernstein, I.M. Gelfand, and S. Gelfand, Schubert cells and cohomology of the spaces G=P, Russian Math. Surveys 28 (1973) 1–26.

5.  I. Cherednik, A unification of the Knizhnik-Zamolodchikov and Dunkl operators via affine Hecke algebras, Inv. Math. 106 (1991) 411–432.

6.  I. Cherednik, Integration of quantum many-body problems by affine Knizhnik- Zamolodchikov equations, Commun. Math. Phys. 106 (1994) 65–95.

7.  I. Cherednik, Non-symmetric Macdonald polynomials, Internat. Math. Res. Notices 10 (1995) 483–515.

8.  M. Demazure, Une formule des caractères, Bull. Sci. Math. 98 (1974) 163–172.

9.  G. Duchamp, D. Krob, A. Lascoux, B. Leclerc, T. Scharf, and J.Y. Thibon, Euler-Poincaré characteristics and polynomial representations of Iwahori-Hecke algebras, Pub. RIMS (Kyoto) 31 (1995) 179–201.

10.  C.F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989) 167–183.

11.  S. Fomin and A. Kirillov, The Yang-Baxter equation, symmetric functions and Schubert polynomials, Discrete Math. 153 (1996) 123–143.

12.  F. Hirzebruch, Gesammelte Abhandlungen, Springer, 1987.

13.  J. Kaneko, q-Selberg integrals and Macdonald polynomials, Ann. Sci. Éc. Norm. Sup. 4e série, 29 (1996) 1086–1110.

14.  Y. Kajihara and M. Noumi, Raising operators of row type for Macdonald polynomials, Compositio Math.

15.  A. Kirillov and M. Noumi, Affine Hecke algebra and raising operators for Macdonald polynomials, Duke Math. J. 93 (1998) 1–39.

16.  A. Kirillov and M. Noumi, q-difference raising operators for Macdonald polynomials and the integrality of transition coefficients, CRM Proceedings and Lecture Notes 22 (1999) 227–243.

17.  F. Knop, Integrality of two variable Kostka functions. J. Reine Ang. Math. 482 (1997) 177– 189.

18.  F. Knop, Symmetric and non-symmetric quantum Capelli polynomials, Comment. Math. Helvet. 72 (1997) 84–100.

19.  F. Knop and S. Sahi, Difference equations and symmetric polynomials defined by their zeros, Internat. Math. Res. Notices 10 (1996) 473–486.

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

21.  L. Lapointe, A. Lascoux, and J. Morse, Determinantal expressions for Macdonald polynomials, Internat. Math. Res. Notices 18 (1998) 957–978.

22.  L. Lapointe, A. Lascoux, and J. Morse, Determinantal expressions for Jack polynomials, Elect. J. Combin. 7 (1) (2000).

23.  L. Lapointe and L. Vinet, A Rodrigues formula for the Jack polynomials and Stanley conjecture, Internat. Math. Res. Notices 9 (1995) 419–424.

24.  L. Lapointe and L. Vinet, Creation operators for the Macdonald and Jack polynomials, Lett. Math. Phys. 40 (1997) 269–286.

25.  A. Lascoux, Polynômes symétriques, Foncteurs de Schur et Grassmanniennes, Thèse, Universit é Paris 7 (1977).

26.  A. Lascoux, About the “y” in the cy-characteristic of Hirzebruch, In: Algebraic Geometry: Hirzebruch 70 (Warsaw, 1998), Contemp. Math. 241, 1999, pp. 285–296.

27.  A. Lascoux, B. Leclerc, and J.Y. Thibon, Twisted action of the symmetric group on the cohomology of a flag manifold, Banach Center Publ. 36 (1996) 111–124.

28.  A. Lascoux, B. Leclerc, and J.Y. Thibon, Flag varieties and the Yang-Baxter equation, Lett. Math. Phys. 40 (1997) 75–90.

29.  A. Lascoux and M.P. Schützenberger, Symmetry and Flag manifolds, Invariant Theory, Springer L.N. 996 (1983) 118–144.

30.  A. Lascoux and M.P. Schützenberger, Symmetrization operators on polynomial rings, Funct. Anal. 21 (1987) 77–78.

31.  A. Lascoux and M.P. Schützenberger, Fonctorialité des polynômes de Schubert, In: Invariant Theory, Contemp. Math. 88, 1989, pp. 585–598.

32.  A. Lascoux and M.P. Schützenberger, Algèbre des différences divisées, Discrete Math. 99 (1992) 165–179.

33.  G. Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989) 599-685.

34.  I.G. Macdonald, Notes on Schubert polynomials, LACIM, Publ. Université Montréal, 1991.

35.  I.G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, Oxford, 2nd Edition, 1995.

36.  I.G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Séminaire Bourbaki, 47ème année 797 (1995).

37.  K. Mimachi and M. Noumi, A reproducing kernel for nonsymmetric Macdonald polynomials, q-alg/9610014.

38.  A. Okounkov and A. Vershik, A new approach to representation theory of symmetric groups, Selecta Math. 2 (1996) 581–605.

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

40.  S. Sahi, Interpolation, integrality and a generalization of Macdonald’s polynomials, Internat. Math. Res. Notices 10 (1996) 457–471.

41.  S. Sahi, A new scalar product for nonsymmetric Jack polynomials, Internat. Math. Res. Notices 20 (1996) 997–1004.

42.  S. Sahi, The binomial formula for nonsymmetric Macdonald polynomials, q-alg/9703024.

43.  A. Vershik, Local algebras and a new version of Young’s orthonormal form, Topics in Algebra, Banach Center Publ. 26 (1990) 467–473.

44.  C.N. Yang, Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett. 19 (1967) 1312–1315.

45.  A. Young, The Collected Papers of Alfred Young, University of Toronto Press, 1977.