# On inductive construction of Procesi bundles

## TL;DR

This paper provides a geometric and representation-theoretic proof of an inductive formula for Procesi bundles on Hilbert schemes, leading to new partial results related to Haiman's $n!$ theorem.

## Contribution

It introduces a new proof of the inductive formula for Procesi bundles and establishes weaker versions of the $n!$ theorem regarding the properties of the isospectral Hilbert scheme.

## Key findings

- Proved the inductive formula for Procesi bundles using geometric methods.
- Showed the normalization of Haiman's isospectral Hilbert scheme is Cohen-Macaulay and Gorenstein.
- Established the normalization morphism is bijective, improving previous results.

## 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.