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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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Abstract: 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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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