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

TL;DR

This paper provides a combinatorial model for the irreducible components of the fixed points of the Hilbert scheme under certain finite subgroups of SL(2,C), revealing their structure and fixed points in type D and E.

Contribution

It introduces a symmetric core combinatorial model for these components and characterizes fixed points under specific binary polyhedral groups.

Findings

Irreducible components are indexed by symmetric cores.

Fixed points under certain groups are actually fixed by SL(2,C).

Components containing a T1-fixed point are zero-dimensional.

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 $0$.