{
  "id": "2207.03683",
  "version": "v1",
  "published": "2022-07-08T04:31:58.000Z",
  "updated": "2022-07-08T04:31:58.000Z",
  "title": "Grassmannians in the Lattice points of Dilations of the Standard Simplex",
  "authors": [
    "Praise Adeyemo"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A remarkable connection between the cohomology ring ${\\rm H^{\\ast}(Gr}(d, d+r),\\Z)$ of the Grasssmannian ${\\rm Gr}(d,d+r)$ and the lattice points of the dilation $r\\Delta_{d}$ of the standard d-simplex is investigated. The natural grading on the cohomology induces different gradings of the lattice points of $r\\Delta_{d}$. This leads to different refinements of the Ehrhart polynomial $L_{\\Delta_{d}}(r)$ of the standard $d$-simplex. We study two of these refinements which are defined by the weights $(1,1,\\dots,1)$ and $(1,2,\\dots, d)$. One of the refinements interprets the Poincar\\'e polynomial ${\\rm P(Gr}(d,d+r),z)$ as the counting of the lattice points which lie on the slicing hyperplanes of the dilation $r\\Delta_d$. Therefore, on the combinatorial level the Poincar\\'e polynomial of the Grassmannian Gr$(d,d+r)$ is a refinement of the Ehrhart polynomial $L_{\\Delta_d}(r)$ of the standard $d$-simplex $\\Delta_{d}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-08T04:31:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lattice points",
      "standard simplex",
      "poincare polynomial",
      "ehrhart polynomial",
      "cohomology induces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}