{
  "id": "2106.15486",
  "version": "v1",
  "published": "2021-06-29T15:11:44.000Z",
  "updated": "2021-06-29T15:11:44.000Z",
  "title": "Positive Jantzen sum formulas for cyclotomic Hecke algebras",
  "authors": [
    "Andrew Mathas"
  ],
  "comment": "LaTeX, 32 pages, TikZ diagrams and LaTeX3 tables",
  "categories": [
    "math.RT",
    "math.CO",
    "math.QA"
  ],
  "abstract": "We prove a ``positive'' Jantzen sum formula for the Specht modules of the cyclotomic Hecke algebras of type~$A$. That is, in the Grothendieck group, we show that the sum of the pieces of the Jantzen filtration is equal to an explicit non-negative linear combination of modules $E^\\nu_{f,e}$, which are modular reductions of simple modules for closely connected Hecke algebras in characteristic zero. The coefficient of $E^\\nu_{f,e}$ in the sum formula is determined by the graded decomposition numbers in characteristic zero, which are known, and the characteristic of the field. As a consequence we see that the decomposition numbers of a cyclotomic Hecke algebra at an $e$th root of unity in characteristic $p$ depend on the decomposition numbers of related cyclotomic Hecke algebras at $ep^r$th roots of unity in characteristic zero, for $r\\ge0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-29T15:11:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20G43",
      "20C08",
      "20C30",
      "05E10"
    ],
    "keywords": [
      "positive jantzen sum formulas",
      "decomposition numbers",
      "characteristic zero",
      "th root",
      "related cyclotomic hecke algebras"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}