{
  "id": "2105.08823",
  "version": "v1",
  "published": "2021-05-18T20:49:10.000Z",
  "updated": "2021-05-18T20:49:10.000Z",
  "title": "Euler obstructions for the Lagrangian Grassmannian",
  "authors": [
    "Paul LeVan",
    "Claudiu Raicu"
  ],
  "categories": [
    "math.AG",
    "math.CO"
  ],
  "abstract": "We prove a case of a positivity conjecture of Mihalcea-Singh, concerned with the local Euler obstructions associated to the Schubert stratification of the Lagrangian Grassmannian LG(n,2n). Combined with work of Aluffi-Mihalcea-Sch\\\"urmann-Su, this further implies the positivity of the Mather classes for Schubert varieties in LG(n,2n), which Mihalcea-Singh had verified for the other cominuscule spaces of classical Lie type. Building on the work of Boe and Fu, we give a positive recursion for the local Euler obstructions, and use it to show that they provide a positive count of admissible labelings of certain trees, analogous to the ones describing Kazhdan-Lusztig polynomials. Unlike in the case of the Grassmannians in types A and D, for LG(n,2n) the Euler obstructions e_{y,w} may vanish for certain pairs (y,w) with y <= w in the Bruhat order. Our combinatorial description allows us to classify all the pairs (y,w) for which e_{y,w}=0. Restricting to the big opposite cell in LG(n,2n), which is naturally identified with the space of n x n symmetric matrices, we recover the formulas for the local Euler obstructions associated with the matrix rank stratification.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-18T20:49:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14M15",
      "14M12",
      "05C05",
      "32S05",
      "32S60"
    ],
    "keywords": [
      "local euler obstructions",
      "matrix rank stratification",
      "lagrangian grassmannian lg",
      "big opposite cell",
      "schubert stratification"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}