Hilbert schemes, polygraphs and the Macdonald positivity conjecture

@article{Haiman2000HilbertSP,
  title={Hilbert schemes, polygraphs and the Macdonald positivity conjecture},
  author={Mark D. Haiman},
  journal={Journal of the American Mathematical Society},
  year={2000},
  volume={14},
  pages={941-1006}
}
  • M. Haiman
  • Published 25 October 2000
  • Mathematics
  • Journal of the American Mathematical Society
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… 
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