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Hilbert schemes, polygraphs and the Macdonald positivity conjecture |
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| Authors: | Mark Haiman |
| Affiliation: | Department of Mathematics, University of California at San Diego, La Jolla, California 92093-0112 |
| Abstract: | We study the isospectral Hilbert scheme (1) We prove the strong form of the (2) We show that the Hilbert scheme The proofs rely on a study of certain subspace arrangements |
| Keywords: | Macdonald polynomials Hilbert schemes Cohen-Macaulay Gorenstein sheaf cohomology |
| 点击此处可从《Journal of the American Mathematical Society》浏览原始摘要信息 | |
| 点击此处可从《Journal of the American Mathematical Society》下载全文 | |
, defined as the reduced fiber product of 
with the Hilbert scheme 
of points in the plane 
, over the symmetric power 
. By a theorem of Fogarty, 
is smooth. We prove that 
is normal, Cohen-Macaulay and Gorenstein, and hence flat over 
. We derive two important consequences. 
conjecture of Garsia and the author, giving a representation-theoretic interpretation of the Kostka-Macdonald coefficients 
. This establishes the Macdonald positivity conjecture, namely that ![$K_{lambda mu }(q,t)in {mathbb N} [q,t]$](http://www.ams.org/jams/2001-14-04/S0894-0347-01-00373-3/gif-abstract0/img11.gif){kind=small}
. 
is isomorphic to the 
-Hilbert scheme 
of Nakamura, in such a way that 
is identified with the universal family over 
. From this point of view, 
describes the fiber of a character sheaf 
at a torus-fixed point of 
corresponding to 
. 
, called polygraphs, whose coordinate rings 
carry geometric information about 
. The key result is that 
is a free module over the polynomial ring in one set of coordinates on 
. This is proven by an intricate inductive argument based on elementary commutative algebra.