{
  "id": "1609.09042",
  "version": "v1",
  "published": "2016-09-28T19:06:36.000Z",
  "updated": "2016-09-28T19:06:36.000Z",
  "title": "Operations on Arc Diagrams and Degenerations for Invariant Subspaces of Linear Operators. Part II",
  "authors": [
    "Mariusz Kaniecki",
    "Justyna Kosakowska",
    "Markus Schmidmeier"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "For a partition $\\beta$, denote by $N_\\beta$ the nilpotent linear operator of Jordan type $\\beta$. Given partitions $\\beta$, $\\gamma$, we investigate the representation space ${}_2{\\mathbb V}_{\\gamma}^\\beta$ of all short exact sequences $$ \\mathcal E: 0\\to N_\\alpha \\to N_\\beta \\to N_\\gamma \\to 0$$ where $\\alpha$ is any partition with each part at most 2. Due to the condition on $\\alpha$, the isomorphism type of a sequence $\\mathcal E$ is given by an arc diagram $\\Delta$; denote by ${\\mathbb V}_\\Delta$ the subset of ${}_2{\\mathbb V}_{\\gamma}^\\beta$ of all sequences isomorphic to $\\mathcal E$. Thus, the space ${}_2{\\mathbb V}_{\\gamma}^\\beta$ carries a stratification given by the subsets of type ${\\mathbb V}_\\Delta$. We compute the dimension of each stratum and show that the boundary of a stratum ${\\mathbb V}_\\Delta$ consists exactly of those ${\\mathbb V}_{\\Delta'}$ where $\\Delta'$ is obtained from $\\Delta$ by a non-empty sequence of arc moves of five possible types {\\bf (A) -- (E)}. The case where all three partitions are fixed has been studied in [3] and [4]. There, arc moves of types {\\bf (A) -- (D)} suffice to describe the boundary of a ${\\mathbb V}_\\Delta$ in ${\\mathbb V}_{\\alpha,\\gamma}^\\beta$. Our fifth move {\\bf (E)}, \"explosion\", is needed to break up an arc into two poles to allow for changes in the partition $\\alpha$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-28T19:06:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "arc diagram",
      "invariant subspaces",
      "arc moves",
      "degenerations",
      "operations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}