{
  "id": "2101.06851",
  "version": "v1",
  "published": "2021-01-18T02:32:41.000Z",
  "updated": "2021-01-18T02:32:41.000Z",
  "title": "Subregular $J$-rings of Coxeter systems via quiver path algebras",
  "authors": [
    "Ivan Dimitrov",
    "Charles Paquette",
    "David Wehlau",
    "Tianyuan Xu"
  ],
  "comment": "49 pages, 7 figures",
  "categories": [
    "math.RT",
    "math.GR",
    "math.RA"
  ],
  "abstract": "We study the subregular $J$-ring $J_C$ of a Coxeter system $(W,S)$, a subring of Lusztig's $J$-ring. We prove that $J_C$ is isomorphic to a quotient of the path algebra of the double quiver of $(W,S)$ by a suitable ideal that we associate to a family of Chebyshev polynomials. As applications, we use quiver representations to study the category mod-$A_K$ of finite dimensional right modules of the algebra $A_K=K\\otimes_\\Z J_C$ over an algebraically closed field $K$ of characteristic zero. Our results include classifications of Coxeter systems for which mod-$A_K$ is semisimple, has finitely many simple modules up to isomorphism, or has a bound on the dimensions of simple modules. Incidentally, we show that every group algebra of a free product of finite cyclic groups is Morita equivalent to the algebra $A_K$ for a suitable Coxeter system; this allows us to specialize the classifications to the module categories of such group algebras.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-18T02:32:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20C08",
      "16G20",
      "16D60",
      "20C07",
      "20E06"
    ],
    "keywords": [
      "quiver path algebras",
      "subregular",
      "group algebra",
      "simple modules",
      "finite dimensional right modules"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}