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arXiv · 2408.08251

Combinatorics of the irreducible components of $\mathcal{H}_n^{\Gamma}$ in type $D$ and $E$

Raphaël Paegelow
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Abstract

In this article, we give a combinatorial model in terms of symmetric cores of the indexing set of the irreducible components of $\mathcal{H}_n^{\Gamma}$ (the $\Gamma$-fixed points of the Hilbert scheme of $n$ points in $\mathbb{C}^2$) containing a monomial ideal, whenever $\Gamma$ is a finite subgroup of $\mathrm{SL}_2(\mathbb{C})$ isomorphic to the binary dihedral group. Moreover, we show that if $\Gamma$ 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 $\Gamma$-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 $\mathcal{H}_n^{\Gamma}$ containing a $\mathbb{T}_1$-fixed point are of dimension $0$.

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Raphaël Paegelow. 2024-08-15. Combinatorics of the irreducible components of $\mathcal{H}_n^{\Gamma}$ in type $D$ and $E$. https://arxiv.org/abs/2408.08251

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