{
  "id": "1712.06737",
  "version": "v1",
  "published": "2017-12-19T01:14:50.000Z",
  "updated": "2017-12-19T01:14:50.000Z",
  "title": "Conormal Varieties on the Cominuscule Grassmannian",
  "authors": [
    "Rahul Singh",
    "Venkatraman Lakshmibai"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $G$ be a simply connected, almost simple group over an algebraically closed field $\\mathbf k$, and $P$ a maximal parabolic subgroup corresponding to omitting a cominuscule root. We construct a compactification $\\phi:T^*G/P\\rightarrow X(u)$, where $X(u)$ is a Schubert variety corresponding to the loop group $LG$. Let $N^*X(w)\\subset T^*G/P$ be the conormal variety of some Schubert variety $X(w)$ in $G/P$; hence we obtain that the closure of $\\phi(N^*X(w))$ in $X(u)$ is a $B$-stable compactification of $N^*X(w)$. We further show that this compactification is a Schubert subvariety of $X(u)$ if and only if $X(w_0w)\\subset G/P$ is smooth, where $w_0$ is the longest element in the Weyl group of $G$. This result is applied to compute the conormal fibre at the $0$ matrix in any determinantal variety.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-19T01:14:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14M15"
    ],
    "keywords": [
      "conormal variety",
      "cominuscule grassmannian",
      "compactification",
      "determinantal variety",
      "maximal parabolic subgroup corresponding"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}