{
  "id": "2505.07097",
  "version": "v1",
  "published": "2025-05-11T19:06:30.000Z",
  "updated": "2025-05-11T19:06:30.000Z",
  "title": "Compatibility of Higher Specht Polynomials and Decompositions of Representations",
  "authors": [
    "Shaul Zemel"
  ],
  "comment": "54 pages",
  "categories": [
    "math.CO",
    "math.RT"
  ],
  "abstract": "%We show how to normalize the higher Specht polynomials of Ariki, Terasoma, and Yamada in a compatible way in order to define a stable version of these polynomials. We also decompose the non-transitive actions of Haglund, Rhoades, and Shimozono into orbits, and show how the associated basis of higher Specht polynomials of Gillespie and Rhoades respects that decomposition. For a given $n$, the orbits of the action of $S_{n}$ are associated with subsets of the set of positive integers that are smaller than $n$, and we relate the representation associated with a set $I$ to the ones of $S_{n+1}$ associated with $I$ and with its union with $n$, the latter being a lifting of the Branching Rule.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-11T19:06:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20C30",
      "05E05",
      "05E10"
    ],
    "keywords": [
      "higher specht polynomials",
      "representation",
      "decomposition",
      "compatibility",
      "rhoades respects"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 54,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}