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arXiv · 1901.05862

On inductive construction of Procesi bundles

Ivan Losev
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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-theoretic 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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Ivan Losev. 2019-01-17. On inductive construction of Procesi bundles. https://arxiv.org/abs/1901.05862

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