{
  "id": "2201.11715",
  "version": "v1",
  "published": "2022-01-27T18:15:58.000Z",
  "updated": "2022-01-27T18:15:58.000Z",
  "title": "Structure of blocks with normal defect and abelian $p'$ inertial quotient",
  "authors": [
    "David Benson",
    "Radha Kessar",
    "Markus Linckelmann"
  ],
  "comment": "21 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $k$ be an algebraically closed field of prime characteristic $p$. Let $kGe$ be a block of a group algebra of a finite group $G$, with normal defect group $P$ and abelian $p'$ inertial quotient $L$. Then we show that $kGe$ is a matrix algebra over a quantised version of the group algebra of a semidirect product of $P$ with a certain subgroup of $L$. To do this, we first examine the associated graded algebra, using a Jennings--Quillen style theorem. As an example, we calculate the associated graded of the basic algebra of the non-principal block in the case of a semidirect product of an extraspecial $p$-group $P$ of exponent $p$ and order $p^3$ with a quaternion group of order eight with the centre acting trivially. In the case $p=3$ we give explicit generators and relations for the basic algebra as a quantised version of $kP$. As a second example, we give explicit generators and relations in the case of a group of shape $2^{1+4}:3^{1+2}$ in characteristic two.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-01-27T18:15:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20C20"
    ],
    "keywords": [
      "inertial quotient",
      "basic algebra",
      "group algebra",
      "semidirect product",
      "explicit generators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}