Vanishing theorems and character formulas for the Hilbert scheme of points in the plane

2002/01/16 by Mark Haiman · 1 citation
Mathematics · #math.AG #math.CO #msc:14C05 #msc:05E05 #msc:14F17

paper · pdf · doi:10.1007/s002220200219

published as Invent. Math. 149, no. 2 (2002) 371-407 · 33 pages

arxiv created 2002/01/16 · arxiv updated 2009/11/30

Abstract

Earlier we showed that the Hilbert scheme of n points in the plane can be identified with the Hilbert scheme of regular S_n orbits on C²ⁿ. Using this result, together with a recent theorem of Bridgeland, King and Reid on the generalized McKay correspondence, we prove vanishing theorems for tensor powers of tautological bundles on the Hilbert scheme. We apply the vanishing theorems to establish (among other things) the character formula for diagonal harmonics conjectured by Garsia and the author. In particular we prove that the dimension of the space of diagonal harmonics is equal to (n+1)^n-1.

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