{
  "id": "1806.10852",
  "version": "v1",
  "published": "2018-06-28T09:44:15.000Z",
  "updated": "2018-06-28T09:44:15.000Z",
  "title": "The Kazhdan-Lusztig polynomials of uniform matroids",
  "authors": [
    "Alice L. L. Gao",
    "Linyuan Lu",
    "Matthew H. Y. Xie",
    "Arthur L. B. Yang",
    "Philip B. Zhang"
  ],
  "comment": "23 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "The Kazhdan-Lusztig polynomial of a matroid was introduced by Elias, Proudfoot, and Wakefield [{\\it Adv. Math. 2016}]. Let $U_{m,d}$ denote the uniform matroid of rank $d$ on a set of $m+d$ elements. Gedeon, Proudfoot, and Young [{\\it J. Combin. Theory Ser. A, 2017}] pointed out that they can derive an explicit formula of the Kazhdan-Lusztig polynomials of $U_{m,d}$ using equivariant Kazhdan-Lusztig polynomials. In this paper we give two alternative explicit formulas, which allow us to prove the real-rootedness of the Kazhdan-Lusztig polynomials of $U_{m,d}$ for $2\\leq m\\leq 15$ and all $d$'s. The case $m=1$ was previously proved by Gedeon, Proudfoot, and Young [{\\it S\\'{e}m. Lothar. Combin. 2017}]. We further determine the $Z$-polynomials of all $U_{m,d}$'s and prove the real-rootedness of the $Z$-polynomials of $U_{m,d}$ for $2\\leq m\\leq 15$ and all $d$'s. Our formula also enables us to give an alternative proof of Gedeon, Proudfoot, and Young's formula for the Kazhdan-Lusztig polynomials of $U_{m,d}$'s without using the equivariant Kazhdan-Lusztig polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-28T09:44:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "26C10",
      "33F10"
    ],
    "keywords": [
      "uniform matroid",
      "equivariant kazhdan-lusztig polynomials",
      "youngs formula",
      "alternative explicit formulas",
      "real-rootedness"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}