{
  "id": "2112.11148",
  "version": "v2",
  "published": "2021-12-21T12:29:31.000Z",
  "updated": "2022-03-28T11:50:41.000Z",
  "title": "Schurian-finiteness of blocks of type $A$ Hecke algebras",
  "authors": [
    "Susumu Ariki",
    "Liron Speyer"
  ],
  "comment": "34 pages, comments welcome. v2 is extended to 44 pages. The main theorems have been strengthened, in particular to include small characteristics. Many improvements and minor corrections throughout",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "For any algebra $A$ over an algebraically closed field $\\mathbb{F}$, we say that an $A$-module $M$ is Schurian if $\\mathrm{End}_A(M) \\cong \\mathbb{F}$. We say that $A$ is Schurian-finite if there are only finitely many isomorphism classes of Schurian $A$-modules, and Schurian-infinite otherwise. By work of Demonet, Iyama and Jasso it is known that Schurian-finiteness is equivalent to $\\tau$-tilting-finiteness, so that we may draw on a wealth of known results in the subject. We prove that if $e\\geq 3$, then principal blocks of type $A$ Hecke algebras are Schurian-finite if and only if they have weight $0$ or $1$. By results of Erdmann and Nakano, this means that these principal blocks are Schurian-finite if and only if they have finite representation type under our assumption on $e$, or equivalently that they are Schurian-infinite if and only if they have wild representation type. Along the way, we also prove a graded version of the Scopes equivalence, which may be of independent interest. If $e=2$ and $p\\ne 2$, then blocks of weight $0$, $1$ or $2$ are Schurian-finite. Again by Erdmann and Nakano, these weight $2$ blocks have infinite tame representation type. When $e=2$, blocks of weight at least $3$ have wild representation type, but our methods cannot determine whether these blocks are Schurian-infinite. Our methods do, however, apply to a large number of other blocks, and we discuss these in the final section.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-03-28T11:50:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20C08",
      "16G10"
    ],
    "keywords": [
      "hecke algebras",
      "wild representation type",
      "schurian-finiteness",
      "principal blocks",
      "infinite tame representation type"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}