{
  "id": "2107.03310",
  "version": "v1",
  "published": "2021-07-07T15:38:40.000Z",
  "updated": "2021-07-07T15:38:40.000Z",
  "title": "Coxeter combinatorics for sum formulas in the representation theory of algebraic groups",
  "authors": [
    "Jonathan Gruber"
  ],
  "comment": "22 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $G$ be a simple algebraic group over an algebraically closed field $\\mathbb{F}$ of characteristic $p\\geq h$, the Coxeter number of $G$. We observe an easy `recursion formula' for computing the Jantzen sum formula of a Weyl module with $p$-regular highest weight. We also discuss a `duality formula' that relates the Jantzen sum formula to Andersen's sum formula for tilting filtrations and we give two different representation theoretic explanations of the recursion formula. As a corollary, we also obtain an upper bound on the length of the Jantzen filtration of a Weyl module with $p$-regular highest weight in terms of the length of the Jantzen filtration of a Weyl module with highest weight in an adjacent alcove.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-07T15:38:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20G05"
    ],
    "keywords": [
      "algebraic group",
      "coxeter combinatorics",
      "representation theory",
      "regular highest weight",
      "jantzen sum formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}