{
  "id": "1409.0174",
  "version": "v1",
  "published": "2014-08-31T01:36:50.000Z",
  "updated": "2014-08-31T01:36:50.000Z",
  "title": "The boundary of the irreducible components for invariant subspace varieties",
  "authors": [
    "Justyna Kosakowska",
    "Markus Schmidmeier"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "Given partitions $\\alpha$, $\\beta$, $\\gamma$, the short exact sequences $0\\to N_\\alpha \\to N_\\beta \\to N_\\gamma \\to 0$ of nilpotent linear operators of Jordan types $\\alpha$, $\\beta$, $\\gamma$, respectively, define a constructible subset $\\mathbb V_{\\alpha,\\gamma}^\\beta$ of an affine variety. Geometrically, the varieties $\\mathbb V_{\\alpha,\\gamma}^\\beta$ are of particular interest as they occur naturally and since they typically consist of several irreducible components. In fact, each Littlewood-Richardson (LR-) tableau $\\Gamma$ of shape $(\\alpha,\\beta,\\gamma)$ contributes one irreducible component $\\overline{\\mathbb V}_\\Gamma$. We consider the partial order $\\Gamma\\leq_{\\sf bound}^*\\widetilde{\\Gamma}$ on LR-tableaux which is the transitive closure of the relation given by $\\mathbb V_{\\widetilde{\\Gamma}}\\cap \\overline{\\mathbb V}_\\Gamma\\neq \\emptyset$. In this paper we compare the boundary relation with partial orders given by algebraic, combinatorial and geometric conditions. It is known that in the case where the parts of $\\alpha$ are at most two, all those partial orders are equivalent. We prove that those partial orders are also equivalent in the case where $\\beta\\setminus\\gamma$ is a horizontal and vertical strip. Moreover, we discuss how the orders differ in general.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-31T01:36:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14L30",
      "16G20",
      "47A15"
    ],
    "keywords": [
      "invariant subspace varieties",
      "irreducible component",
      "partial order",
      "nilpotent linear operators",
      "short exact sequences"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1409.0174K"
    }
  }
}