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Hilbert schemes, polygraphs and the Macdonald positivity conjecture

Authors:	Mark Haiman

Institution:	Department of Mathematics, University of California at San Diego, La Jolla, California 92093-0112

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 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 -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 carry geometric information about $X_{n}$ . The key result is that 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.

Keywords:	Macdonald polynomials Hilbert schemes Cohen-Macaulay Gorenstein sheaf cohomology

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