{
  "id": "2405.10726",
  "version": "v1",
  "published": "2024-05-17T12:17:45.000Z",
  "updated": "2024-05-17T12:17:45.000Z",
  "title": "$τ$-Tilting finiteness of group algebras over generalized symmetric groups",
  "authors": [
    "Naoya Hiramae"
  ],
  "comment": "15 pages. arXiv admin note: text overlap with arXiv:2405.10021",
  "categories": [
    "math.RT",
    "math.GR",
    "math.RA"
  ],
  "abstract": "In this paper, we show that weakly symmetric $\\tau$-tilting finite algebras have positive definite Cartan matrices, which implies that we can prove $\\tau$-tilting infiniteness of weakly symmetric algebras by calculating their Cartan matrices. Similarly, we obtain the condition on Cartan matrices that selfinjective algebras are $\\tau$-tilting infinite. By applying this result, we show that a group algebra of $(\\mathbb{Z}/p^l\\mathbb{Z})^n\\rtimes H$ is $\\tau$-tilting infinite when $p^l\\geq n$ and $\\#\\mathrm{IBr}\\,H\\geq\\min\\{p,3\\}$, where $p>0$ is the characteristic of the ground field, $H$ is a subgroup of the symmetric group $\\mathfrak{S}_n$ of degree $n$, the action of $H$ permutes the entries of $(\\mathbb{Z}/p^l\\mathbb{Z})^n$, and $\\mathrm{IBr}\\,H$ denotes the set of irreducible $p$-Brauer characters of $H$. Moreover, we show that under the assumption that $p^l\\geq n$ and $H$ is a $p'$-subgroup of $\\mathfrak{S}_n$, $\\tau$-tilting finiteness of a group algebra of a group $(\\mathbb{Z}/p^l\\mathbb{Z})^n\\rtimes H$ is determined by its $p$-hyperfocal subgroup.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-17T12:17:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G10",
      "20C20"
    ],
    "keywords": [
      "group algebra",
      "generalized symmetric groups",
      "tilting finiteness",
      "weakly symmetric",
      "tilting infinite"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}