The Procesi bundle over the $Γ$-fixed points of the Hilbert scheme of points in $\mathbb{C}^2$
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866908639271321600 |
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| author | Bellamy, Gwyn Paegelow, Raphaël |
| author_facet | Bellamy, Gwyn Paegelow, Raphaël |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_17819 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Procesi bundle over the $Γ$-fixed points of the Hilbert scheme of points in $\mathbb{C}^2$ Bellamy, Gwyn Paegelow, Raphaël Algebraic Geometry Representation Theory 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. |
| title | The Procesi bundle over the $Γ$-fixed points of the Hilbert scheme of points in $\mathbb{C}^2$ |
| topic | Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/2404.17819 |