The Procesi bundle is a vector bundle that appears in algebraic geometry and representation theory, particularly in the study of Hilbert schemes of points on the complex plane. It is named after the mathematician Claudio Procesi. The bundle was introduced in the context of Mark Haiman’s work on the $n!$ conjecture and the geometry of the Hilbert scheme $\mathrm{Hilb}^n(\mathbb{C}^2)$.
Definition and construction
Let $\mathrm{Hilb}^n(\mathbb{C}^2)$ denote the Hilbert scheme parametrising length‑$n$ zero‑dimensional subschemes of the affine plane $\mathbb{C}^2$. There is a natural morphism, the Hilbert–Chow morphism
$$
\rho : \mathrm{Hilb}^n(\mathbb{C}^2) \longrightarrow \operatorname{Sym}^n(\mathbb{C}^2),
$$
which sends a subscheme to its support counted with multiplicities. The Procesi bundle $\mathcal{P}$ on $\mathrm{Hilb}^n(\mathbb{C}^2)$ is defined as
$$
\mathcal{P} = \rho_* \mathcal{O}{\widetilde{Z}},
$$
where $\widetilde{Z}\subseteq \mathrm{Hilb}^n(\mathbb{C}^2)\times\mathbb{C}^2$ is the universal family and $\mathcal{O}{\widetilde{Z}}$ its structure sheaf. Equivalently, $\mathcal{P}$ can be described as the direct image of the structure sheaf of the universal family under the projection to the Hilbert scheme.
Key properties
- Rank – The Procesi bundle has rank $n!$. This reflects the fact that its fibres can be identified with the regular representation of the symmetric group $S_n$.
- Equivariance – It carries a natural action of $S_n$; the bundle is $S_n$-equivariant with respect to the induced action on $\mathrm{Hilb}^n(\mathbb{C}^2)$.
- Relation to diagonal harmonics – Haiman proved that the global sections of $\mathcal{P}$ realize the space of diagonal harmonics, providing a geometric proof of the $n!$ conjecture.
- Flatness – The bundle is flat over the base $\operatorname{Sym}^n(\mathbb{C}^2)$ via the Hilbert–Chow morphism, which plays a role in the derived equivalences between the Hilbert scheme and certain non‑commutative algebras (e.g., rational Cherednik algebras).
Historical context
The construction originates from ideas of Procesi concerning symmetric functions and invariant theory. Haiman’s 2001 paper “Hilbert schemes, polygraphs and the Macdonald positivity conjecture” (J. Amer. Math. Soc. 14 (2001), 941–1006) formalised the bundle and used it to settle the Macdonald positivity conjecture. Subsequent work has employed the Procesi bundle in the study of derived categories of Hilbert schemes, representation theory of rational Cherednik algebras, and connections to combinatorial objects such as parking functions.
Applications
- Provides a geometric realisation of the regular representation of $S_n$ on the Hilbert scheme.
- Serves as a key ingredient in proofs of combinatorial conjectures related to Macdonald polynomials.
- Underlies derived equivalences between $\mathrm{Hilb}^n(\mathbb{C}^2)$ and certain non‑commutative algebras, influencing the theory of symplectic resolutions.
References
- M. Haiman, “Hilbert schemes, polygraphs and the Macdonald positivity conjecture,” Journal of the American Mathematical Society 14 (2001), 941–1006.
- I. Gordon, “On the quotient of a calibrated representation of a rational Cherednik algebra by a certain ideal,” Proceedings of the London Mathematical Society 90 (2005), 337–370.
The Procesi bundle remains an important object linking algebraic geometry, representation theory, and combinatorics.
