{
  "id": "2407.12546",
  "version": "v1",
  "published": "2024-07-17T13:28:32.000Z",
  "updated": "2024-07-17T13:28:32.000Z",
  "title": "Minimal equivariant embeddings of the Grassmannian and flag manifold",
  "authors": [
    "Lek-Heng Lim",
    "Ke Ye"
  ],
  "comment": "11 pages",
  "categories": [
    "math.RT",
    "math.DG"
  ],
  "abstract": "We show that the flag manifold $\\operatorname{Flag}(k_1,\\dots, k_p, \\mathbb{R}^n)$, with Grassmannian the special case $p=1$, has an $\\operatorname{SO}_n(\\mathbb{R})$-equivariant embedding in an Euclidean space of dimension $(n-1)(n+2)/2$, two orders of magnitude below the current best known result. We will show that the value $(n-1)(n+2)/2$ is the smallest possible and that any $\\operatorname{SO}_n(\\mathbb{R})$-equivariant embedding of $\\operatorname{Flag}(k_1,\\dots, k_p, \\mathbb{R}^n)$ in an ambient space of minimal dimension is equivariantly equivalent to the aforementioned one.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-17T13:28:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14M15",
      "57R40",
      "57S25",
      "14R20",
      "22E46",
      "22E70"
    ],
    "keywords": [
      "minimal equivariant embeddings",
      "flag manifold",
      "grassmannian",
      "ambient space",
      "minimal dimension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}