{
  "id": "2307.06678",
  "version": "v1",
  "published": "2023-07-13T10:55:34.000Z",
  "updated": "2023-07-13T10:55:34.000Z",
  "title": "The Frobenius transform of a symmetric function",
  "authors": [
    "Mitchell Lee"
  ],
  "comment": "27 pages",
  "categories": [
    "math.CO",
    "math.RT"
  ],
  "abstract": "We define an abelian group homomorphism $\\mathscr{F}$, which we call the Frobenius transform, from the ring of symmetric functions to the ring of the symmetric power series. The matrix entries of $\\mathscr{F}$ in the Schur basis are the restriction coefficients $r_\\lambda^\\mu = \\dim \\operatorname{Hom}_{\\mathfrak{S}_n}(V_\\mu, \\mathbb{S}^\\lambda \\mathbb{C}^n)$, which are known to be nonnegative integers but have no known combinatorial interpretation. The Frobenius transform satisfies the identity $\\mathscr{F}\\{fg\\} = \\mathscr{F}\\{f\\} \\ast \\mathscr{F}\\{g\\}$, where $\\ast$ is the Kronecker product. We prove for all symmetric functions $f$ that $\\mathscr{F}\\{f\\} = \\mathscr{F}_{\\mathrm{Sur}}\\{f\\} \\cdot (1 + h_1 + h_2 + \\cdots)$, where $\\mathscr{F}_{\\mathrm{Sur}}\\{f\\}$ is a symmetric function with the same degree and leading term as $f$. Then, we compute the matrix entries of $\\mathscr{F}_{\\mathrm{Sur}}\\{f\\}$ and $\\mathscr{F}^{-1}_{\\mathrm{Sur}}\\{f\\}$ in the complete homogeneous, elementary, and power sum symmetric function bases, giving combinatorial interpretations of the coefficients where possible. In particular, the matrix entries of $\\mathscr{F}^{-1}_{\\mathrm{Sur}}\\{f\\}$ in the elementary basis count words with a constraint on their Lyndon factorization. As an example application of our main results, we prove that $r_\\lambda^\\mu = 0$ if the Young diagram of $\\mu$ contains a square of side length greater than $2^{\\lambda_1 - 1}$, and this inequality is tight.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-13T10:55:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E05"
    ],
    "keywords": [
      "matrix entries",
      "power sum symmetric function bases",
      "combinatorial interpretation",
      "elementary basis count words",
      "abelian group homomorphism"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}