{
  "id": "2403.04101",
  "version": "v1",
  "published": "2024-03-06T23:10:51.000Z",
  "updated": "2024-03-06T23:10:51.000Z",
  "title": "Schur positivity of difference of products of derived Schur polynomials",
  "authors": [
    "Julius Ross",
    "Kuang-Yu Wu"
  ],
  "comment": "20 pages, 18 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "To any Schur polynomial $s_{\\lambda}$ one can associated its derived polynomials $s_{\\lambda}{(i)}$ $i=0,\\ldots,|\\lambda|$ by the rule $$s_{\\lambda}(x_1+t,\\ldots,x_n+t) = \\sum_i s_{\\lambda}^{(i)}(x_1,\\ldots,x_n) t^i.$$ We conjecture that $$(s_{\\lambda}^{(i)})^2 - s_{\\lambda}^{(i-1)} s_{\\lambda}^{(i+1)}$$ is always Schur positive and prove this when $i=1$ for rectangles $\\lambda = (k^\\ell)$, for hooks $\\lambda = (k, 1^{\\ell -1})$, and when $\\lambda = (k,k,1)$ or $\\lambda = (3,2^{k-1})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-06T23:10:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E05"
    ],
    "keywords": [
      "derived schur polynomials",
      "schur positivity",
      "difference"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}