{
  "id": "2308.00790",
  "version": "v1",
  "published": "2023-08-01T19:00:46.000Z",
  "updated": "2023-08-01T19:00:46.000Z",
  "title": "Conjugate weight enumerators and invariant theory",
  "authors": [
    "Gabriele Nebe",
    "Leonie Scheeren"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "The Galois group of a finite field extension $K/F$ defines a grading on the symmetric algebra of the $F$-space $K^v$ which we use to introduce the notion of homogeneous conjugate invariants for subgroups $G\\leq \\GL_v(K)$. If the Weight Enumerator Conjecture holds for a finite representation $\\rho $ then the genus-$m$ conjugate complete weight enumerators of self-dual codes generate the corresponding space of conjugate invariants of the associated genus-$m$ Clifford-Weil group ${\\mathcal C}_m(\\rho ) \\leq \\GL_{v^m}(K)$. This generalisation of a paper by Bannai, Oura and Da Zhao provides new examples of Clifford-Weil orbits that form projective designs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-01T19:00:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13A50",
      "94B60",
      "11S20"
    ],
    "keywords": [
      "conjugate weight enumerators",
      "invariant theory",
      "weight enumerator conjecture holds",
      "conjugate complete weight enumerators",
      "finite field extension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}