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

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 $0$.