Hilbert schemes, polygraphs, and the Macdonald positivity conjecture
| dc.creator | Haiman, Mark | |
| dc.date | 2000-10-25 | |
| dc.date | 2000-12-08 | |
| dc.date.accessioned | 2026-07-07T04:38:15Z | |
| dc.date.available | 2026-07-07T04:38:15Z | |
| dc.description | 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. | |
| dc.description | 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 | |
| dc.identifier | https://arxiv.org/abs/math/0010246 | |
| dc.identifier | http://arxiv.org/abs/math/0010246 | |
| dc.identifier | J. Amer. Math. Soc. 14 (2001), 941-1006 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/60203 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Combinatorics | |
| dc.subject | Quantum Algebra | |
| dc.subject | 14C05 (primary), 05E05, 14M05 (secondary) | |
| dc.title | Hilbert schemes, polygraphs, and the Macdonald positivity conjecture | |
| dc.type | text |