{
  "id": "2109.01067",
  "version": "v1",
  "published": "2021-09-02T16:36:17.000Z",
  "updated": "2021-09-02T16:36:17.000Z",
  "title": "Join operation for the Bruhat order and Verma modules",
  "authors": [
    "Hankyung Ko",
    "Volodymyr Mazorchuk",
    "Rafael Mrđen"
  ],
  "comment": "46 pages, several figures",
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "We observe that the join operation for the Bruhat order on a Weyl group agrees with the intersections of Verma modules in type $A$. The statement is not true in other types, and we propose a conjectural statement of a weaker correspondence. Namely, we introduce distinguished subsets of the Weyl group on which the join operation conjecturally agrees with the intersections of Verma modules. We also relate our conjecture with a statement about the socles of the cokernels of inclusions between Verma modules. The latter determines the first Ext spaces between a simple module and a Verma module. We give a conjectural complete description of such socles, which we verify in a number of cases. Along the way, we determine the poset structure of the join-irreducible elements in Weyl groups and obtain closed formulae for certain families of Kazhdan-Lusztig polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-02T16:36:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "verma module",
      "join operation",
      "bruhat order",
      "weyl group",
      "conjectural complete description"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}