{
  "id": "2004.00788",
  "version": "v1",
  "published": "2020-04-02T03:07:38.000Z",
  "updated": "2020-04-02T03:07:38.000Z",
  "title": "Ordered set partitions, Garsia-Procesi modules, and rank varieties",
  "authors": [
    "Sean T. Griffin"
  ],
  "comment": "49 pages. Full version of extended abstract accepted to FPSAC 2020",
  "categories": [
    "math.CO",
    "math.AC"
  ],
  "abstract": "We introduce a family of ideals $I_{n,\\lambda,s}$ in $\\mathbb{Q}[x_1,\\dots,x_n]$ for $\\lambda$ a partition of $k\\leq n$ and an integer $s \\geq \\ell(\\lambda)$. This family contains both the Tanisaki ideals $I_\\lambda$ and the ideals $I_{n,k}$ of Haglund-Rhoades-Shimozono as special cases. We study the corresponding quotient rings $R_{n,\\lambda,s}$ as symmetric group modules. When $n=k$ and $s$ is arbitrary, we recover the Garsia-Procesi modules, and when $\\lambda=(1^k)$ and $s=k$, we recover the generalized coinvariant algebras of Haglund-Rhoades-Shimozono. We give a monomial basis for $R_{n,\\lambda,s}$, unifying the monomial bases studied by Garsia-Procesi and Haglund-Rhoades-Shimozono, and realize the $S_n$-module structure of $R_{n,\\lambda,s}$ in terms of an action on $(n,\\lambda,s)$-ordered set partitions. We also prove formulas for the Hilbert series and graded Frobenius characteristic of $R_{n,\\lambda,s}$. We then connect our work with Eisenbud-Saltman rank varieties using results of Weyman. As an application of our work, we give a monomial basis, Hilbert series formula, and graded Frobenius characteristic formula for the coordinate ring of the scheme-theoretic intersection of a rank variety with diagonal matrices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-02T03:07:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ordered set partitions",
      "rank variety",
      "garsia-procesi modules",
      "monomial basis",
      "hilbert series formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}