{
  "id": "2204.04924",
  "version": "v1",
  "published": "2022-04-11T07:52:38.000Z",
  "updated": "2022-04-11T07:52:38.000Z",
  "title": "Calculating the $p$-canonical basis of Hecke algebras",
  "authors": [
    "Joel Gibson",
    "Lars Thorge Jensen",
    "Geordie Williamson"
  ],
  "comment": "21 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "We describe an algorithm for computing the $p$-canonical basis of the Hecke algebra, or one of its antispherical modules. The algorithm does not operate in the Hecke category directly, but rather uses a faithful embedding of the Hecke category inside a semisimple category to build a \"model\" for indecomposable objects and bases of their morphism spaces. Inside this semisimple category, objects are sequences of Coxeter group elements, and morphisms are (sparse) matrices over a fraction field, making it quite amenable to computations. This strategy works for the full Hecke category over any base field, but in the antispherical case we must instead work over $\\mathbb{Z}_{(p)}$ and use an idempotent lifting argument to deduce the result for a field of characteristic $p > 0$. We also describe a less sophisticated algorithm which is much more suited to the case of finite groups. We provide complete implementations of both algorithms in the MAGMA computer algebra system.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-11T07:52:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hecke algebra",
      "canonical basis",
      "semisimple category",
      "magma computer algebra system",
      "hecke category inside"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}