{
  "id": "1309.3762",
  "version": "v2",
  "published": "2013-09-15T13:53:03.000Z",
  "updated": "2015-06-17T23:23:54.000Z",
  "title": "Schubert decompositions for quiver Grassmannians of tree modules",
  "authors": [
    "Oliver Lorscheid"
  ],
  "comment": "22 pages",
  "categories": [
    "math.RT",
    "math.AG"
  ],
  "abstract": "Let $Q$ be a quiver, $M$ a representation of $Q$ with an ordered basis $\\cB$ and $\\ue$ a dimension vector for $Q$. In this note we extend the methods of \\cite{L12} to establish Schubert decompositions of quiver Grassmannians $\\Gr_\\ue(M)$ into affine spaces to the ramified case, i.e.\\ the canonical morphism $F:T\\to Q$ from the coefficient quiver $T$ of $M$ w.r.t.\\ $\\cB$ is not necessarily unramified. In particular, we determine the Euler characteristic of $\\Gr_\\ue(M)$ as the number of \\emph{extremal successor closed subsets of $T_0$}, which extends the results of Cerulli Irelli (\\cite{Cerulli11}) and Haupt (\\cite{Haupt12}) (under certain additional assumptions on $\\cB$).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-09-15T13:53:03.000Z",
      "title": "Schubert dcompositions for quiver Grassmannians of tree modules",
      "abstract": "Let $Q$ be a quiver, $M$ a representation of $Q$ with an ordered basis $B$ and $e$ a dimension vector for $Q$. In this note we extend the methods of the author's previous paper \\cite{Lorscheid12} to establish Schubert decompositions of quiver Grassmannians $\\Gr_e(M)$ into affine spaces to the ramified case, i.e. the canonical morphism $F:T\\to Q$ from the coefficient quiver $T$ of $M$ w.r.t. $B$ is not necessarily unramified. In particular, we determine the Euler characteristic of $\\Gr_e(M)$ as the number of extremal successor closed subsets of $T_0$, which extends results of Cerulli Irelli and Haupt (under certain additional assumptions on $B$).",
      "comment": "16 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-06-17T23:23:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14M15",
      "16G20"
    ],
    "keywords": [
      "quiver grassmannians",
      "tree modules",
      "schubert dcompositions",
      "extremal successor closed subsets",
      "establish schubert decompositions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1309.3762L"
    }
  }
}