{
  "id": "2405.05128",
  "version": "v2",
  "published": "2024-05-08T15:25:05.000Z",
  "updated": "2024-07-19T10:59:12.000Z",
  "title": "Degree of the Grassmannian as an affine variety",
  "authors": [
    "Lek-Heng Lim",
    "Ke Ye"
  ],
  "comment": "16 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "The degree of the Grassmannian with respect to the Pl\\\"ucker embedding is well-known. However, the Pl\\\"ucker embedding, while ubiquitous in pure mathematics, is almost never used in applied mathematics. In applied mathematics, the Grassmannian is usually embedded as projection matrices $\\operatorname{Gr}(k,\\mathbb{R}^n) \\cong \\{P \\in \\mathbb{R}^{n \\times n} : P^{\\scriptscriptstyle\\mathsf{T}} = P = P^2,\\; \\operatorname{tr}(P) = k\\}$ or as involution matrices $\\operatorname{Gr}(k,\\mathbb{R}^n) \\cong \\{X \\in \\mathbb{R}^{n \\times n} : X^{\\scriptscriptstyle\\mathsf{T}} = X,\\; X^2 = I,\\; \\operatorname{tr}(X)=2k - n\\}$. We will determine an explicit expression for the degree of the Grassmannian with respect to these embeddings. In so doing, we resolved a conjecture of Devriendt--Friedman--Sturmfels about the degree $\\operatorname{Gr}(2, \\mathbb{R}^n)$ and in fact generalized it to $\\operatorname{Gr}(k, \\mathbb{R}^n)$. We also proved a set theoretic variant of another conjecture of Devriendt--Friedman--Sturmfels about the limit of $\\operatorname{Gr}(k,\\mathbb{R}^n)$ in the sense of Gr\\\"obner degneration.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-07-19T10:59:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14E25",
      "14F45"
    ],
    "keywords": [
      "affine variety",
      "grassmannian",
      "set theoretic variant",
      "applied mathematics",
      "pure mathematics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}