{
  "id": "2207.06184",
  "version": "v1",
  "published": "2022-07-13T13:29:44.000Z",
  "updated": "2022-07-13T13:29:44.000Z",
  "title": "Block decomposition via the geometric Satake equivalence",
  "authors": [
    "Emilien Zabeth"
  ],
  "comment": "50 pages, preliminary version",
  "categories": [
    "math.RT"
  ],
  "abstract": "We give a new proof for the description of the blocks in the category of representations of a reductive algebraic group $\\mathbf{G}$ over a field of positive characteristic $\\ell$ (originally due to Donkin), by working in the Satake category of the Langlands dual group and applying Smith-Treumann theory as developed by Riche and Williamson. On the representation theoretic side, our methods enable us to give a bound for the length of a minimum chain linking two weights in the same block, and to give a new proof for the block decomposition of a quantum group at an $\\ell$-th root of unity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-13T13:29:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "geometric satake equivalence",
      "block decomposition",
      "langlands dual group",
      "representation theoretic side",
      "th root"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 50,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}