Hilbert schemes, polygraphs and the Macdonald positivity conjecture
Hilbert schemes, polygraphs and the Macdonald positivity conjecture
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DOI：
10.1090/s0894-0347-01-00373-3
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发表时间：
2000-10
影响因子：
3.9
通讯作者：
M. Haiman
中科院分区：
文献类型：
--
作者：
M. Haiman
本文研究了对称幂空间S^nC^2上的等谱Hilbert格式X_n，它定义为C^2n与平面上点的Hilbert格式H_n的约化纤维积。我们证明了X_n是正规的，Cohen-Macaulay和Gorenstein，因而是H_n上平坦的.我们得出两个重要的结果。(1)我们证明了“n！猜想”，给出了Kostka-Macdonald系数K_{lambda，mu}（q，t）的表示论解释。这就建立了Macdonald正性猜想，即K_{lambda，mu}（q，t）总是一个非负整数系数的多项式. (2)证明了Hilbert方案H_n与轨道C^2n//S_n的Hilbert方案同构，使得X_n与C^2n//S_n上的泛族相同.
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.