Hilbert schemes, polygraphs, and the Macdonald positivity conjecture

Details Hilbert schemes, polygraphs, and the Macdonald positivity conjecture Hilbert schemes, polygraphs, and the Macdonald positivity conjecture

2000-10-25

arxiv.org/abs/math/0010246v2

J. Amer. Math. Soc. 14 (2001), 941-1006

Mathematics

Algebraic Geometry

62 pages. Submitted to Journal of the A.M.S. New version contains expository improvements and corrects minor errors. Also avai

Scientific paper

We study the isospectral Hilbert scheme X_n, defined as the reduced fiber product of C^2n with the Hilbert scheme H_n of points in the plane, over the symmetric power S^n C^2. 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, that K_{lambda,mu}(q,t) is always a polynomial with non-negative integer coefficients. (2) We show that the Hilbert scheme H_n is isomorphic to the Hilbert scheme of orbits C^2n//S_n, in such a way that X_n is identified with the universal family over C^2n//S_n.

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