On the Connection Between Macdonald Polynomials and Demazure Characters
Yasmine B. Sanderson
Rutgers University Department of Mathematics New Brunswick N.J. 08901
DOI: 10.1023/A:1008786420650
Abstract
We show that the specialization of nonsymmetric Macdonald polynomials at t = 0 are, up to multiplication by a simple factor, characters of Demazure modules for [^( sl( n))] \widehat{sl(n)} . This connection furnishes Lie-theoretic proofs of the nonnegativity and monotonicity of Kostka polynomials.
Pages: 269–275
Keywords: affine Lie algebras; Macdonald polynomials; Demazure character
Full Text: PDF
References
1. L.M. Butler, “Subgroup lattices and symmetric functions,” Memoirs of the A.M.S 112(539), (1994).
2. I. Cherednik, “Double affine Hecke algebras and Macdonald's conjectures,” Ann. Math. 141 (1995), 191-216.
3. M. Demazure, “Désingularisation des variétés de Schubert généralisées,” Ann. Scient. Ec. Norm. Sup. 6 (1974), 53-88.
4. A. Garsia and C. Procesi, “On certain graded Sn-modules and the q-Kostka polynomials,” Adv. Math 94(1) (1992), 82-138.
5. A.N. Kirillov, “Dilogarithm identities,” hep-th/9408113 v2, 25, Aug 1994.
6. F. Knop, “Integrality of two variable Kostka functions,” J. reine angew. Math. 482 (1997), 177-189.
7. S. Kumar, “Demazure character formula in arbitrary Kac-Moody setting,” Invent. Math. 89 (1987), 395-423.
8. A. Kuniba, K.C. Misra, M. Okado, T. Takagi, and J. Uchiyama, “Characters of Demazure modules and solvable lattice models,” q-alg 9707004, 3 July 1997.
9. A. Lascoux and M.-P. Sch\ddot utzenberger, “Sur une conjecture de H. O. Foulkes,” C.R. Acad. Sci. Paris 286A (1978), no. 7, 323-324.
10. G. Lusztig, “Green polynomials and singularities of unipotent classes,” Advances in Math. 42 (1981), 169-178.
11. I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, Oxford Math. Mon. Clarendon Press, Oxford, 1995.
12. I.G. Macdonald, “Affine Hecke algebras and orthogonal polynomials,” Seminaire Bourbaki, 1996, Vol. 1994/1995, Astérisque No. 237, Exp. No. 797, 4, 189-207.
13. O. Mathieu, “Formule de Demazure-Weyl et généralisation du théor`eme de Borel-Weil-Bott,” C.R. Acad. Sci. Paris, Sér. I Math. 303(9) (1986), 391-394.
14. E. Opdam, “Harmonic analysis for certain representations of graded Hecke algebras,” Acta Math. 175 (1995), 75-121.
15. Y. Sanderson, “Real characters of Demazure modules for rank two affine Lie algebras,” J. Algebra 184 (1996), 985-1000.
2. I. Cherednik, “Double affine Hecke algebras and Macdonald's conjectures,” Ann. Math. 141 (1995), 191-216.
3. M. Demazure, “Désingularisation des variétés de Schubert généralisées,” Ann. Scient. Ec. Norm. Sup. 6 (1974), 53-88.
4. A. Garsia and C. Procesi, “On certain graded Sn-modules and the q-Kostka polynomials,” Adv. Math 94(1) (1992), 82-138.
5. A.N. Kirillov, “Dilogarithm identities,” hep-th/9408113 v2, 25, Aug 1994.
6. F. Knop, “Integrality of two variable Kostka functions,” J. reine angew. Math. 482 (1997), 177-189.
7. S. Kumar, “Demazure character formula in arbitrary Kac-Moody setting,” Invent. Math. 89 (1987), 395-423.
8. A. Kuniba, K.C. Misra, M. Okado, T. Takagi, and J. Uchiyama, “Characters of Demazure modules and solvable lattice models,” q-alg 9707004, 3 July 1997.
9. A. Lascoux and M.-P. Sch\ddot utzenberger, “Sur une conjecture de H. O. Foulkes,” C.R. Acad. Sci. Paris 286A (1978), no. 7, 323-324.
10. G. Lusztig, “Green polynomials and singularities of unipotent classes,” Advances in Math. 42 (1981), 169-178.
11. I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, Oxford Math. Mon. Clarendon Press, Oxford, 1995.
12. I.G. Macdonald, “Affine Hecke algebras and orthogonal polynomials,” Seminaire Bourbaki, 1996, Vol. 1994/1995, Astérisque No. 237, Exp. No. 797, 4, 189-207.
13. O. Mathieu, “Formule de Demazure-Weyl et généralisation du théor`eme de Borel-Weil-Bott,” C.R. Acad. Sci. Paris, Sér. I Math. 303(9) (1986), 391-394.
14. E. Opdam, “Harmonic analysis for certain representations of graded Hecke algebras,” Acta Math. 175 (1995), 75-121.
15. Y. Sanderson, “Real characters of Demazure modules for rank two affine Lie algebras,” J. Algebra 184 (1996), 985-1000.