Combinatorics of the irreducible components of $\mathcal{H}_n^Γ$ in type $D$ and $E$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910971800322048 |
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| author | Paegelow, Raphaël |
| author_facet | Paegelow, Raphaël |
| contents | In this article, we give a combinatorial model in terms of symmetric cores of the indexing set of the irreducible components of $\mathcal{H}_n^Γ$ (the $Γ$-fixed points of the Hilbert scheme of $n$ points in $\mathbb{C}^2$) containing a monomial ideal, whenever $Γ$ is a finite subgroup of $\mathrm{SL}_2(\mathbb{C})$ isomorphic to the binary dihedral group. Moreover, we show that if $Γ$ is a subgroup of $\mathrm{SL}_2(\mathbb{C})$ isomorphic to the binary tetrahedral group, to the binary octahedral group or to the binary icosahedral group, then the $Γ$-fixed points of $\mathcal{H}_n$ which are also fixed under $\mathbb{T}_1$, the maximal diagonal torus of $\mathrm{SL}_2(\mathbb{C})$, are in fact $\mathrm{SL}_2(\mathbb{C})$-fixed points. Finally, we prove that in that case, the irreducible components of $\mathcal{H}_n^Γ$ containing a $\mathbb{T}_1$-fixed point are of dimension $0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_08251 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Combinatorics of the irreducible components of $\mathcal{H}_n^Γ$ in type $D$ and $E$ Paegelow, Raphaël Combinatorics Representation Theory In this article, we give a combinatorial model in terms of symmetric cores of the indexing set of the irreducible components of $\mathcal{H}_n^Γ$ (the $Γ$-fixed points of the Hilbert scheme of $n$ points in $\mathbb{C}^2$) containing a monomial ideal, whenever $Γ$ is a finite subgroup of $\mathrm{SL}_2(\mathbb{C})$ isomorphic to the binary dihedral group. Moreover, we show that if $Γ$ is a subgroup of $\mathrm{SL}_2(\mathbb{C})$ isomorphic to the binary tetrahedral group, to the binary octahedral group or to the binary icosahedral group, then the $Γ$-fixed points of $\mathcal{H}_n$ which are also fixed under $\mathbb{T}_1$, the maximal diagonal torus of $\mathrm{SL}_2(\mathbb{C})$, are in fact $\mathrm{SL}_2(\mathbb{C})$-fixed points. Finally, we prove that in that case, the irreducible components of $\mathcal{H}_n^Γ$ containing a $\mathbb{T}_1$-fixed point are of dimension $0$. |
| title | Combinatorics of the irreducible components of $\mathcal{H}_n^Γ$ in type $D$ and $E$ |
| topic | Combinatorics Representation Theory |
| url | https://arxiv.org/abs/2408.08251 |