On inductive construction of Procesi bundles

DOI10.1007/S40879-021-00500-WarXiv1901.05862MaRDI QIDQ6312626

Ivan Losev

Publication date: 17 January 2019

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.

Mathematics Subject Classification ID

Deformation quantization, star products (53D55) Representation theory of associative rings and algebras (16G99) McKay correspondence (14E16)

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