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

Raphaël Paegelow

TL;DR

The paper develops a combinatorial model for the irreducible components of the $\Gamma$-fixed Hilbert scheme $\mathcal{H}_n^{\Gamma}$ containing a monomial ideal, focusing on type $D$ (binary dihedral) and type $E$ (exceptional binary polyhedral) groups. It introduces a $BD_{2\ell}$-Residue and a folding from type $D$ to type $C$, enabling a parametrization of components by symmetric $2\ell$-cores with $|\lambda| \equiv n \pmod{2\ell}$ and $|\lambda| \le n$, and proves a complete combinatorial description in type $D$ (including a core-based equivalence for monomial-ideal components). In contrast, type $E$ exhibits an absence of such combinatorics: the $\Gamma$-fixed points that are also $\mathbb{T}_1$-fixed coincide with $SL_2(\mathbb{C})$-fixed points, and all relevant components are zero-dimensional, a fact extended to $\tilde{E}_7$ and $\tilde{E}_8$ via subgroup containment. The results connect the geometry of $\mathcal{H}_n^{\Gamma}$ with representation theory through quiver varieties and explicit residue maps (notably $\mathrm{Res}_{\tilde{E}_6}$) that encode the $\mathrm{BT}$-decomposition of $\mathbb{C}[x,y]/I_{\lambda}$, offering a precise, dimension-controlled description of fixed-point components.

Abstract

In this article, we give a combinatorial model in terms of symmetric cores of the indexing set of the irreducible components of $\mathcal{H}_n^Γ$ (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 $0$.

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

TL;DR

The paper develops a combinatorial model for the irreducible components of the -fixed Hilbert scheme containing a monomial ideal, focusing on type (binary dihedral) and type (exceptional binary polyhedral) groups. It introduces a -Residue and a folding from type to type , enabling a parametrization of components by symmetric -cores with and , and proves a complete combinatorial description in type (including a core-based equivalence for monomial-ideal components). In contrast, type exhibits an absence of such combinatorics: the -fixed points that are also -fixed coincide with -fixed points, and all relevant components are zero-dimensional, a fact extended to and via subgroup containment. The results connect the geometry of with representation theory through quiver varieties and explicit residue maps (notably ) that encode the -decomposition of , offering a precise, dimension-controlled description of fixed-point components.

Abstract

In this article, we give a combinatorial model in terms of symmetric cores of the indexing set of the irreducible components of (the -fixed points of the Hilbert scheme of points in ) containing a monomial ideal, whenever is a finite subgroup of isomorphic to the binary dihedral group. Moreover, we show that if is a subgroup of isomorphic to the binary tetrahedral group, to the binary octahedral group or to the binary icosahedral group, then the -fixed points of which are also fixed under , the maximal diagonal torus of , are in fact -fixed points. Finally, we prove that in that case, the irreducible components of containing a -fixed point are of dimension .
Paper Structure (10 sections, 17 theorems, 29 equations, 4 figures)

This paper contains 10 sections, 17 theorems, 29 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Acknowledgement
  3. Starting point
  4. From type $D$ to type $C$
  5. $\mathrm{BD}_{2\ell}$-Residue
  6. Combinatorial description in type $D$
  7. Absence of combinatorics in type $E$
  8. Dimension of the irreducible components containing a $\mathbb{T}_1$-fixed point
  9. $\tilde{E}_6$-Residue
  10. Zero-dimensionnal irreducible components

Key Result

Proposition 3.2

The set $\Phi(\tilde{D}_{\ell+2}^{\sigma})$ is a crystallographic root system of type $\tilde{C}_{\ell}$.

Figures (4)

  • Figure :
  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (57)

  • Definition 2.1
  • Definition 3.1
  • Proposition 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Remark 3.6
  • Definition 3.7
  • Proposition 3.8
  • Remark 3.9
  • ...and 47 more