Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Procesi bundle over the $Γ$-fixed points of the Hilbert scheme of points in $\mathbb{C}^2$

Gwyn Bellamy, Raphaël Paegelow

TL;DR

This work analyzes the fibers of the Procesi bundle over the Γ-fixed points of the Hilbert scheme of n points in the plane, proving a Reduction Theorem that expresses each fiber as an induced image from a fiber over a zero-dimensional component in a smaller Hilbert scheme. The main tool is an explicit isomorphism $S_p\cong S_{g_Γ}\times Γ$ that reduces the $(\mathfrak{S}_n\times Γ)$-module structure to a smaller setting, enabling concrete decompositions and combinatorial consequences. In type A, this yields explicit decompositions of fibers over monomial ideals in terms of ℓ-core data and induced representations, with two edge-case proofs using symmetric function theory and representation theory. In type D, the paper describes how fibers decompose under the binary dihedral group, relating the $BD_{ℓ}$-modules to smaller cores and showing symmetry phenomena among characters, thereby linking Hilbert schemes, McKay correspondence, and symmetric function methods.

Abstract

For $Γ$ a finite subgroup of $\mathrm{SL}_2(\mathbb{C})$ and $n \geq 1$, we study the fibers of the Procesi bundle over the $Γ$-fixed points of the Hilbert scheme of $n$ points in the plane. For each irreducible component of this fixed point locus, our approach reduces the study of the fibers of the Procesi bundle, as an $(\mathfrak{S}_n \times Γ)$-module, to the study of the fibers of the Procesi bundle over an irreducible component of dimension zero in a smaller Hilbert scheme. When $Γ$ is of type $A$, our main result shows, as a corollary, that the fiber of the Procesi bundle over the monomial ideal associated with a partition $λ$ is induced, as an $(\mathfrak{S}_n \times Γ)$-module, from the fiber of the Procesi bundle over the monomial ideal associated with the core of $λ$. We give different proofs of this corollary in two edge cases, using only representation theory and symmetric functions.

The Procesi bundle over the $Γ$-fixed points of the Hilbert scheme of points in $\mathbb{C}^2$

TL;DR

This work analyzes the fibers of the Procesi bundle over the Γ-fixed points of the Hilbert scheme of n points in the plane, proving a Reduction Theorem that expresses each fiber as an induced image from a fiber over a zero-dimensional component in a smaller Hilbert scheme. The main tool is an explicit isomorphism that reduces the -module structure to a smaller setting, enabling concrete decompositions and combinatorial consequences. In type A, this yields explicit decompositions of fibers over monomial ideals in terms of ℓ-core data and induced representations, with two edge-case proofs using symmetric function theory and representation theory. In type D, the paper describes how fibers decompose under the binary dihedral group, relating the -modules to smaller cores and showing symmetry phenomena among characters, thereby linking Hilbert schemes, McKay correspondence, and symmetric function methods.

Abstract

For a finite subgroup of and , we study the fibers of the Procesi bundle over the -fixed points of the Hilbert scheme of points in the plane. For each irreducible component of this fixed point locus, our approach reduces the study of the fibers of the Procesi bundle, as an -module, to the study of the fibers of the Procesi bundle over an irreducible component of dimension zero in a smaller Hilbert scheme. When is of type , our main result shows, as a corollary, that the fiber of the Procesi bundle over the monomial ideal associated with a partition is induced, as an -module, from the fiber of the Procesi bundle over the monomial ideal associated with the core of . We give different proofs of this corollary in two edge cases, using only representation theory and symmetric functions.
Paper Structure (14 sections, 34 theorems, 100 equations)

This paper contains 14 sections, 34 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. The Hilbert scheme of points in the plane
  3. The Procesi bundle
  4. Combinatorial consequences in type $A$
  5. Notation
  6. Acknowledgements
  7. Root systems and irreducible components of $\mathcal{H}_n^{\Gamma}$
  8. The Reduction Theorem
  9. Combinatorial consequences in type $A$
  10. Corollary of the reduction theorem
  11. Independant proofs in two edge cases
  12. When $\gamma_{\ell}$ is very small
  13. When $\gamma_{\ell}$ is small and $\ell$ is prime
  14. Combinatorial consequences in type $D$

Key Result

Theorem 1

For each irreducible component $\mathcal{C}$ of $\mathcal{H}_n^{\Gamma}$, there exists an isomorphism of groups ${\textphnc{\As} \colon S_p \xrightarrow{\,{\raisebox{-0.65ex}{$\sim$}}\,} \mathfrak{S}_{\mathrm{g}_{\Gamma}} \times \Gamma}$, making $\mathscr{P}^{\mathrm{g}_{\Gamma}}_{|I_{0}}$ into a $S

Theorems & Definitions (81)

  • Theorem
  • Corollary
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 71 more