{
  "id": "2005.08329",
  "version": "v1",
  "published": "2020-05-17T18:08:55.000Z",
  "updated": "2020-05-17T18:08:55.000Z",
  "title": "Differential operators on Schur and Schubert polynomials",
  "authors": [
    "Gleb Nenashev"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "This paper is about decreasing operators on back stable Schubert polynomials. We study two operators $\\xi$ and $\\nabla$ of degree $-1$, which satisfy Leibniz rule. Furthermore, we show that all other such operators are linear combinations of $\\xi$ and $\\nabla$. For the case of Schur functions, these two operators fully determine the product of Schur functions, i.e., it is possible to define Littlewood-Richardson coefficients only from $\\xi$ and $\\nabla$. This new point of view on Schur functions gives us an elementary proof of The Giambelli identity and Jacobi-Trudi identities. For the case of Schubert polynomials, we construct a bigger class of decreasing operators as expressions in terms of $\\xi$ and $\\nabla$, which are indexed by Young diagrams. Surprisingly, these operators are related to Stanley symmetric functions. In particular, we extend bosonic operators from Schur to Schubert polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-17T18:08:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "differential operators",
      "schur functions",
      "satisfy leibniz rule",
      "extend bosonic operators",
      "define littlewood-richardson coefficients"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}