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Hilbert schemes, polygraphs and the Macdonald positivity conjecture

Author: Mark Haiman
Journal: J. Amer. Math. Soc. 14 (2001), 941-1006
MSC (2000): Primary 14C05; Secondary 05E05, 14M05
Published electronically: May 29, 2001
MathSciNet review: 1839919
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We study the isospectral Hilbert scheme $X_{n}$, defined as the reduced fiber product of $(\mathbb{C}^{2})^{n}$ with the Hilbert scheme $H_{n}$ of points in the plane $\mathbb{C}^{2}$, over the symmetric power $S^{n}\mathbb{C}^{2} = (\mathbb{C}^{2})^{n}/S_{n}$. By a theorem of Fogarty, $H_{n}$ is smooth. We prove that $X_{n}$ is normal, Cohen-Macaulay and Gorenstein, and hence flat over $H_{n}$. We derive two important consequences.

(1) We prove the strong form of the $n!$ conjecture of Garsia and the author, giving a representation-theoretic interpretation of the Kostka-Macdonald coefficients $K_{\lambda \mu }(q,t)$. This establishes the Macdonald positivity conjecture, namely that $K_{\lambda \mu }(q,t)\in {\mathbb N} [q,t]$.

(2) We show that the Hilbert scheme $H_{n}$ is isomorphic to the $G$-Hilbert scheme $(\mathbb{C}^{2})^{n}{//}S_n$ of Nakamura, in such a way that $X_{n}$ is identified with the universal family over $({\mathbb C}^2)^n{//}S_n$. From this point of view, $K_{\lambda \mu }(q,t)$ describes the fiber of a character sheaf $C_{\lambda }$ at a torus-fixed point of $({\mathbb C}^2)^n{//}S_n$corresponding to $\mu $.

The proofs rely on a study of certain subspace arrangements $Z(n,l)\subseteq (\mathbb{C}^{2})^{n+l}$, called polygraphs, whose coordinate rings $R(n,l)$ carry geometric information about $X_{n}$. The key result is that $R(n,l)$ is a free module over the polynomial ring in one set of coordinates on $(\mathbb{C}^{2})^{n}$. This is proven by an intricate inductive argument based on elementary commutative algebra.

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  • 1. Victor V. Batyrev, Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs, J. Eur. Math. Soc. (JEMS) 1 (1999), no. 1, 5-33, arXiv:math.AG/9803071. CMP 99:09
  • 2. David Bayer and Michael Stillman, Macaulay: A computer algebra system for algebraic geometry, Version 3.0, Software distributed via ftp:, 1994.
  • 3. F. Bergeron, A. M. Garsia, M. Haiman, and G. Tesler, Identities and positivity conjectures for some remarkable operators in the theory of symmetric functions, Methods Appl. Anal. 6 (1999), no. 3, 363-420. CMP 2001:07
  • 4. N. Bergeron and A. M. Garsia, On certain spaces of harmonic polynomials, Hypergeometric functions on domains of positivity, Jack polynomials, and applications (Tampa, FL, 1991), Amer. Math. Soc., Providence, RI, 1992, pp. 51-86. MR 94a:05212
  • 5. Joël Briançon, Description de ${\rm Hilb}^{n}{C}\{x,y\}$, Invent. Math. 41 (1977), no. 1, 45-89. MR 56:15637
  • 6. Tom Bridgeland, Alastair King, and Miles Reid, The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. 14 (2001), no. 3, 555-578.
  • 7. William Brockman and Mark Haiman, Nilpotent orbit varieties and the atomic decomposition of the $q$-Kostka polynomials, Canad. J. Math. 50 (1998), no. 3, 525-537. MR 2000c:05153
  • 8. Jan Cheah, Cellular decompositions for nested Hilbert schemes of points, Pacific J. Math. 183 (1998), no. 1, 39-90. MR 99d:14002
  • 9. Ivan Cherednik, Double affine Hecke algebras and Macdonald's conjectures, Ann. of Math. (2) 141 (1995), no. 1, 191-216. MR 96m:33010
  • 10. Corrado De Concini and Claudio Procesi, Symmetric functions, conjugacy classes and the flag variety, Invent. Math. 64 (1981), no. 2, 203-219. MR 82m:14030
  • 11. Geir Ellingsrud and Stein Arild Strømme, On the homology of the Hilbert scheme of points in the plane, Invent. Math. 87 (1987), no. 2, 343-352. MR 88c:14008
  • 12. Jacques Emsalem, Géométrie des points épais, Bull. Soc. Math. France 106 (1978), no. 4, 399-416. MR 80j:14008
  • 13. Pavel I. Etingof and Alexander A. Kirillov, Jr., Macdonald's polynomials and representations of quantum groups, Math. Res. Lett. 1 (1994), no. 3, 279-296, hep-th/9312103. MR 96m:17025
  • 14. John Fogarty, Algebraic families on an algebraic surface, Amer. J. Math 90 (1968), 511-521. MR 38:5778
  • 15. A. M. Garsia and M. Haiman, A graded representation model for Macdonald's polynomials, Proc. Nat. Acad. Sci. U.S.A. 90 (1993), no. 8, 3607-3610. MR 94b:05206
  • 16. -, A remarkable $q,t$-Catalan sequence and $q$-Lagrange inversion, J. Algebraic Combin. 5 (1996), no. 3, 191-244. MR 97k:05208
  • 17. A. M. Garsia and C. Procesi, On certain graded ${S}_ n$-modules and the $q$-Kostka polynomials, Adv. Math. 94 (1992), no. 1, 82-138. MR 93j:20030
  • 18. A. M. Garsia and J. Remmel, Plethystic formulas and positivity for $q,t$-Kostka coefficients, Mathematical essays in honor of Gian-Carlo Rota (Cambridge, MA, 1996), Birkhäuser Boston, Boston, MA, 1998, pp. 245-262. MR 99j:05189d
  • 19. A. M. Garsia and G. Tesler, Plethystic formulas for Macdonald $q,t$-Kostka coefficients, Adv. Math. 123 (1996), no. 2, 144-222. MR 99j:05189e
  • 20. Mark L. Green, Generic initial ideals, Six lectures on commutative algebra (Bellaterra, 1996), Birkhäuser, Basel, 1998, pp. 119-186. MR 99m:13040
  • 21. Alexander Grothendieck, Techniques de construction et théorèmes d'existence en géométrie algébrique. IV. Les schémas de Hilbert (Exp. no. 221), Séminaire Bourbaki, Vol. 6, Année 1960/61, Soc. Math. France, Paris, 1995, pp. 249-276. MR 99f:00038
  • 22. Mark Haiman, Conjectures on the quotient ring by diagonal invariants, J. Algebraic Combin. 3 (1994), no. 1, 17-76. MR 95a:20014
  • 23. -, $t,q$-Catalan numbers and the Hilbert scheme, Discrete Math. 193 (1998), no. 1-3, 201-224, Selected papers in honor of Adriano Garsia (Taormina, 1994). MR 2000k:05264
  • 24. -, Macdonald polynomials and geometry, New perspectives in geometric combinatorics (Billera, Björner, Greene, Simion, and Stanley, eds.), MSRI Publications, vol. 38, Cambridge University Press, 1999, pp. 207-254. CMP 2000:07
  • 25. Robin Hartshorne, Residues and duality, Springer-Verlag, Berlin, 1966, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, Vol. 20. MR 36:5145
  • 26. -, Local cohomology, Springer-Verlag, Berlin, 1967, A seminar given by A. Grothendieck, Harvard University, Fall, 1961. Lecture Notes in Mathematics, Vol. 41. MR 37:219
  • 27. R. Hotta and T. A. Springer, A specialization theorem for certain Weyl group representations and an application to the Green polynomials of unitary groups, Invent. Math. 41 (1977), no. 2, 113-127. MR 58:5945
  • 28. Y. Ito and I. Nakamura, McKay correspondence and Hilbert schemes, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), no. 7, 135-138. MR 97k:14003
  • 29. -, Hilbert schemes and simple singularities, New trends in algebraic geometry (Warwick, 1996), Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. MR 2000i:14004
  • 30. D. Kaledin, McKay correspondence for symplectic quotient singularities, Electronic preprint, arXiv:math.AG/9907087, 1999.
  • 31. Shin-ichi Kato, Spherical functions and a $q$-analogue of Kostant's weight multiplicity formula, Invent. Math. 66 (1982), no. 3, 461-468. MR 84b:22030
  • 32. Anatol N. Kirillov and Masatoshi Noumi, Affine Hecke algebras and raising operators for Macdonald polynomials, Duke Math. J. 93 (1998), no. 1, 1-39, arXiv:q-alg/9605004. MR 99j:05189a
  • 33. Friedrich Knop, Integrality of two variable Kostka functions, J. Reine Angew. Math. 482 (1997), 177-189, arXiv:q-alg/9603027. MR 99j:05189c
  • 34. Hanspeter Kraft, Conjugacy classes and Weyl group representations, Young tableaux and Schur functions in algebra and geometry (Torun, 1980), Soc. Math. France, Paris, 1981, pp. 191-205. MR 83h:20017
  • 35. Luc Lapointe and Luc Vinet, Operator construction of the Jack and Macdonald symmetric polynomials, Special functions and differential equations (Madras, 1997), Allied Publ., New Delhi, 1998, pp. 271-279. MR 2000a:05210
  • 36. Alain Lascoux and Marcel-Paul Schützenberger, Sur une conjecture de H. O. Foulkes, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 7, A323-A324. MR 57:12672
  • 37. G. Lusztig, Green polynomials and singularities of unipotent classes, Adv. in Math. 42 (1981), no. 2, 169-178. MR 83c:20059
  • 38. -, Singularities, character formulas, and a $q$-analog of weight multiplicities, Analysis and topology on singular spaces, II, III (Luminy, 1981), Soc. Math. France, Paris, 1983, pp. 208-229. MR 85m:17005
  • 39. I. G. Macdonald, A new class of symmetric functions, Actes du 20e Séminaire Lotharingien, vol. 372/S-20, Publications I.R.M.A., Strasbourg, 1988, pp. 131-171.
  • 40. -, Symmetric functions and Hall polynomials, second ed., The Clarendon Press, Oxford University Press, New York, 1995, With contributions by A. Zelevinsky, Oxford Science Publications. MR 96h:05207
  • 41. -, Affine Hecke algebras and orthogonal polynomials, Astérisque 237 (1996), 189-207, Séminaire Bourbaki 1994/95, Exp. no. 797. MR 99f:33024
  • 42. I. Nakamura, Hilbert schemes of abelian group orbits, Journal of Algebraic Geometry, to appear.
  • 43. M. Reid, McKay correspondence, Electronic preprint, arXiv:alg-geom/9702016, 1997.
  • 44. Siddhartha Sahi, Interpolation, integrality, and a generalization of Macdonald's polynomials, Internat. Math. Res. Notices 10 (1996), 457-471. MR 99j:05189b
  • 45. T. A. Springer, A construction of representations of Weyl groups, Invent. Math. 44 (1978), no. 3, 279-293. MR 58:11154
  • 46. Glenn Tesler, Isotypic decompositions of lattice determinants, J. Combin. Theory Ser. A 85 (1999), no. 2, 208-227. MR 2000i:05184
  • 47. Misha Verbitsky, Holomorphic symplectic geometry and orbifold singularities, Asian J. Math. 4 (2000), no. 3, 553-564, arXiv:math.AG/9903175. CMP 2001:05
  • 48. Weiqiang Wang, Hilbert schemes, wreath products, and the McKay correspondence, Electronic preprint, arXiv:math.AG/9912104, 1999.

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Additional Information

Mark Haiman
Affiliation: Department of Mathematics, University of California at San Diego, La Jolla, California 92093-0112

Keywords: Macdonald polynomials, Hilbert schemes, Cohen-Macaulay, Gorenstein, sheaf cohomology
Received by editor(s): August 15, 2000
Received by editor(s) in revised form: January 29, 2001
Published electronically: May 29, 2001
Additional Notes: This research was supported in part by N.S.F. Mathematical Sciences grants DMS-9701218 and DMS-0070772.
Article copyright: © Copyright 2001 American Mathematical Society

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