{
  "id": "1910.02251",
  "version": "v1",
  "published": "2019-10-05T11:47:11.000Z",
  "updated": "2019-10-05T11:47:11.000Z",
  "title": "$τ$-Tilting Finiteness of Non-distributive Algebras and their Module Varieties",
  "authors": [
    "Kaveh Mousavand"
  ],
  "comment": "17 pages",
  "categories": [
    "math.RT",
    "math.AG",
    "math.CO"
  ],
  "abstract": "We treat the $\\tau$-tilting finiteness of those minimal representation-infinite (min-rep-infinite) algebras which are non-distributive. Building upon the new results of Bongartz, we fully determine which algebras in this family are $\\tau$-tilting finite and which ones are not. This complements our previous work in which we carried out a similar analysis for the min-rep-infinite biserial algebras. Consequently, we obtain nontrivial explicit sufficient conditions for $\\tau$-tilting infiniteness of a large family of algebras. This also produces concrete families of \"minimal $\\tau$-tilting infinite algebras\"-- the modern counterpart of min-rep-infinite algebras, independently introduced by the author and Wang. We further use our results on the family of non-distributive algebras to establish a conjectural connection between the $\\tau$-tilting theory and two geometric notions in the study of module varieties introduced by Chindris, Kinser and Weyman. We verify the conjectures for the algebras studied in this note: For the min-rep-infinite algebras which are non-distributive or biserial, we show that if $\\Lambda$ has the dense orbit property, then it must be $\\tau$-tilting finite. Moreover, we prove that such an algebra is Schur-representation-finite if and only if it is $\\tau$-tilting finite. The latter result gives a categorical interpretation of Schur-representation-finiteness over this family of min-rep-infinite algebras.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-05T11:47:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G20",
      "16G60",
      "05E10",
      "14K10"
    ],
    "keywords": [
      "module varieties",
      "non-distributive algebras",
      "tilting finiteness",
      "min-rep-infinite algebras",
      "nontrivial explicit sufficient conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}