{
  "id": "2412.14091",
  "version": "v1",
  "published": "2024-12-18T17:38:17.000Z",
  "updated": "2024-12-18T17:38:17.000Z",
  "title": "The positive orthogonal Grassmannian",
  "authors": [
    "Yassine El Maazouz",
    "Yelena Mandelshtam"
  ],
  "comment": "25 pages, 7 figures. Comments welcome",
  "categories": [
    "math.CO",
    "math.AG"
  ],
  "abstract": "The Pl\\\"ucker positive region $\\mathrm{OGr}_+(k,2k)$ of the orthogonal Grassmannian emerged as the positive geometry behind the ABJM scattering amplitudes. In this paper we initiate the study of the positive orthogonal Grassmannian $\\mathrm{OGr}_+(k,n)$ for general values of $k$ and $n$. We determine the boundary structure of the quadric $\\mathrm{OGr}_+(1,n)$ in $\\mathbb{P}_+^{n-1}$ and show that it is a positive geometry. We show that $\\mathrm{OGr}_+(k,2k+1)$ is isomorphic to $\\mathrm{OGr}_+(k+1,2k+2)$ and connect its combinatorial structure to matchings on $[2k+2]$. Finally, we show that in the case $n > 2k+1$, the positroid cells of $\\mathrm{Gr}_+(k,n)$ do not induce a CW cell decomposition of $\\mathrm{OGr}_+(k,n)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-18T17:38:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E14",
      "14M15",
      "15B48"
    ],
    "keywords": [
      "positive orthogonal grassmannian",
      "positive geometry",
      "cw cell decomposition",
      "combinatorial structure",
      "boundary structure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}