{
  "id": "1611.01251",
  "version": "v1",
  "published": "2016-11-04T02:51:43.000Z",
  "updated": "2016-11-04T02:51:43.000Z",
  "title": "Ordered set partitions and the 0-Hecke algebra",
  "authors": [
    "Jia Huang",
    "Brendon Rhoades"
  ],
  "comment": "30 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let the symmetric group $\\mathfrak{S}_n$ act on the polynomial ring $\\mathbb{Q}[\\mathbf{x}_n] = \\mathbb{Q}[x_1, \\dots, x_n]$ by variable permutation. The coinvariant algebra is the graded $\\mathfrak{S}_n$-module $R_n := {\\mathbb{Q}[\\mathbf{x}_n]} / {I_n}$, where $I_n$ is the ideal in $\\mathbb{Q}[\\mathbf{x}_n]$ generated by invariant polynomials with vanishing constant term. Haglund, Rhoades, and Shimozono introduced a new quotient $R_{n,k}$ of the polynomial ring $\\mathbb{Q}[\\mathbf{x}_n]$ depending on two positive integers $k \\leq n$ which reduces to the classical coinvariant algebra of the symmetric group $\\mathfrak{S}_n$ when $k = n$. The quotient $R_{n,k}$ carries the structure of a graded $\\mathfrak{S}_n$-module; Haglund et. al. determine its graded isomorphism type and relate it to the Delta Conjecture in the theory of Macdonald polynomials. We introduce and study a related quotient $S_{n,k}$ of $\\mathbb{F}[\\mathbf{x}_n]$ which carries a graded action of the 0-Hecke algebra $H_n(0)$, where $\\mathbb{F}$ is an arbitrary field. We prove 0-Hecke analogs of the results of Haglund, Rhoades, and Shimozono. In the classical case $k = n$, we recover earlier results of Huang concerning the 0-Hecke action on the coinvariant algebra.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-04T02:51:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ordered set partitions",
      "symmetric group",
      "invariant polynomials",
      "vanishing constant term",
      "polynomial ring"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}