Hilbert schemes, polygraphs, and the Macdonald positivity conjecture
摘要
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.
引用
@article{arxiv.math/0010246,
title = {Hilbert schemes, polygraphs, and the Macdonald positivity conjecture},
author = {Mark Haiman},
journal= {arXiv preprint arXiv:math/0010246},
year = {2007}
}
备注
62 pages. Submitted to Journal of the A.M.S. New version contains expository improvements and corrects minor errors. Also available at http://math.ucsd.edu/~mhaiman