{
  "id": "1609.09538",
  "version": "v1",
  "published": "2016-09-29T21:45:21.000Z",
  "updated": "2016-09-29T21:45:21.000Z",
  "title": "Levi Subgroup Actions on Schubert Varieties, Induced Decompositions of their Coordinate Rings, Sphericity and Singularity Consequences",
  "authors": [
    "Reuven Hodges",
    "Venkatramani Lakshmibai"
  ],
  "comment": "49 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $L_w$ be the Levi part of the stabilizer $Q_w$ in $GL_N(\\mathbb{C})$ (for left multiplication) of a Schubert variety $X(w)$ in the Grassmannian $G_{d,N}$. For the natural action of $L_w$ on $\\mathbb{C}[X(w)]$, the homogeneous coordinate ring of $X(w)$ (for the Pl\\\"ucker embedding), we give a combinatorial description of the decomposition of $\\mathbb{C}[X(w)]$ into irreducible $L_w$-modules; in fact, our description holds more generally for the action of the Levi part $L$ of any parabolic group $Q$ that is a subgroup of $Q_w$. Using this combinatorial description, we give a classification of all Schubert varieties $X(w)$ in the Grassmannian $G_{d,N}$ for which $\\mathbb{C}[X(w)]$ has a decomposition into irreducible $L_w$-modules that is multiplicity free. This classification is then used to show that certain classes of Schubert varieties are spherical $L_w$-varieties. These classes include all smooth Schubert varieties, all determinantal Schubert varieties, as well as all Schubert varieties in $G_{2,N}$ and $G_{3,N}$. Also, as an important consequence, we get interesting results related to the singular locus of $X(w)$ and multiplicities at $T$-fixed points in $X(w)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-29T21:45:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20G05",
      "20G20",
      "14B05",
      "14M27"
    ],
    "keywords": [
      "schubert variety",
      "levi subgroup actions",
      "coordinate ring",
      "singularity consequences",
      "induced decompositions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}