Hilbert schemes, polygraphs and the Macdonald positivity conjecture

摘要: We study the isospectral Hilbert scheme X_n, defined as reduced fiber product of C^2n with H_n points in plane, over symmetric power S^n C^2. prove that X_n is normal, Cohen-Macaulay, and Gorenstein, hence flat H_n. derive two important consequences. (1) strong form "n! conjecture" Garsia author, giving a representation-theoretic interpretation Kostka-Macdonald coefficients K_{lambda,mu}(q,t). This establishes Macdonald positivity conjecture, K_{lambda,mu}(q,t) always polynomial non-negative integer coefficients. (2) show isomorphic to orbits C^2n//S_n, such way identified universal family C^2n//S_n.