{
  "id": "2408.08251",
  "version": "v1",
  "published": "2024-08-15T16:44:18.000Z",
  "updated": "2024-08-15T16:44:18.000Z",
  "title": "Combinatorics of the irreducible components of $\\mathcal{H}_n^Γ$ in type $D$ and $E$",
  "authors": [
    "Raphaël Paegelow"
  ],
  "categories": [
    "math.CO",
    "math.RT"
  ],
  "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 $\\mathbb{C}^2$, when $\\Gamma$ 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 $0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-15T16:44:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "irreducible components",
      "fixed point",
      "combinatorics",
      "binary dihedral group",
      "binary tetrahedral group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}