Abstract
A Procesi bundle, a rank n! vector bundle on the Hilbert scheme \(H_n\) of n points in \(\mathbb {C}^2\), was first constructed by Mark Haiman in his proof of the n! theorem by using a complicated combinatorial argument. Since then alternative constructions of this bundle were given by Bezrukavnikov–Kaledin and by Ginzburg. In this paper we give a Geometric Representation Theory proof of the inductive formula for the Procesi bundle that plays an important role in Haiman’s construction. Then we use the inductive formula to prove a weaker version of the n! theorem: the normalization of Haiman’s isospectral Hilbert scheme is Cohen–Macaulay and Gorenstein, and the normalization morphism is bijective. This improves an earlier result of Ginzburg.
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