{
  "id": "2302.03643",
  "version": "v1",
  "published": "2023-02-07T17:49:02.000Z",
  "updated": "2023-02-07T17:49:02.000Z",
  "title": "Top-degree components of Grothendieck and Lascoux polynomials",
  "authors": [
    "Jianping Pan",
    "Tianyi Yu"
  ],
  "comment": "27 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We define a filtered algebra $\\widehat{V_1} \\subset \\widehat{V_2} \\subset \\cdots \\subset \\widehat{V} \\subset \\mathbb{Q}[x_1, x_2, \\dots]$ which gives an algebraic interpretation of a classical $q$-analogue of Bell numbers. The space $\\widehat{V_n}$ is the span of the Castelnuovo-Mumford polynomials $\\widehat{\\mathfrak{G}}_w$ with $w \\in S_n$. Pechenik, Speyer and Weigandt define $\\widehat{\\mathfrak{G}}_w$ as the top-degree components of the Grothendieck polynomials and extract a basis of $\\widehat{V_n}$. We describe another basis consisting of $\\widehat{\\mathfrak{L}}_\\alpha$, the top-degree components of Lascoux polynomials. Our basis connects the Hilbert series of $\\widehat{V_n}$ and $\\widehat{V}$ to rook-theoretic results of Garsia and Remmel. To understand $\\widehat{\\mathfrak{L}}_\\alpha$, we introduce a combinatorial construction called a ``snow diagram'' that augments and decorates any diagram $D$. When $D$ is the key diagram of $\\alpha$, its snow diagram yields the leading monomial of $\\widehat{\\mathfrak{L}}_\\alpha$. When $D$ is the Rothe diagram of $w$, its snow diagram yields the leading monomial of $\\widehat{\\mathfrak{G}}_w$, agreeing with the work of Pechenik, Speyer and Weigandt.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-07T17:49:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E05"
    ],
    "keywords": [
      "top-degree components",
      "lascoux polynomials",
      "snow diagram yields",
      "leading monomial",
      "hilbert series"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}