{
  "id": "1301.1140",
  "version": "v4",
  "published": "2013-01-07T09:48:58.000Z",
  "updated": "2014-09-11T22:12:42.000Z",
  "title": "Faces and maximizer subsets of highest weight modules",
  "authors": [
    "Apoorva Khare"
  ],
  "comment": "Minor changes; and Theorem 4 (as well as its proof, i.e., Section 7) from the previous version removed. The result has been significantly strengthened, in forthcoming work titled \"Standard parabolic subsets of highest weight modules\"",
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "In this paper we study general highest weight modules $\\mathbb{V}^\\lambda$ over a complex semisimple Lie algebra $\\mathfrak{g}$. We present three formulas for the support of a large family of modules $\\mathbb{V}^\\lambda$, which include but are not restricted to all simple modules and all parabolic Verma modules. These formulas are direct and do not involve cancellations, and were not previously known in the literature. Our results extend the notion of the Weyl polytope to highest weight $\\mathfrak{g}$-modules $\\mathbb{V}^\\lambda$. We also show that for all simple modules, the convex hull of the weights is a $W_J$-invariant polyhedron for some parabolic subgroup $W_J$. We compute its vertices, faces, and symmetries - more generally, we do this for all parabolic Verma modules, and for all modules $\\mathbb{V}^\\lambda$ with highest weight $\\lambda$ not on a simple root hyperplane. To show our results, we extend the notion of convexity to arbitrary additive subgroups $\\mathbb{A} \\subset (\\mathbb{R},+)$ of coefficients. Our techniques enable us to completely classify \"weak $\\mathbb{A}$-faces\" of the support sets ${\\rm wt}(\\mathbb{V}^\\lambda)$ for arbitrary $\\mathbb{V}^\\lambda$, in the process extending results of Vinberg, Chari-Dolbin-Ridenour, and Cellini-Marietti to all highest weight modules.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-04-23T01:42:15.000Z",
      "abstract": "In this paper we compute, in three ways, the set of weights of all simple highest weight modules (and others) over a complex semisimple Lie algebra $\\lie{g}$. This extends the notion of the Weyl polytope to a large class of highest weight $\\lie{g}$-modules $\\V$. Our methods involve computing the convex hull of the weights; this is precisely the Weyl polytope when $\\V$ is finite-dimensional. We also show that for all simple modules, the convex hull of the weights is a $W_J$-invariant polyhedron for some parabolic subgroup $W_J$. We compute its vertices, (weak) faces, and symmetries - more generally, we do this for all parabolic Verma modules, and for all modules $\\V$ with $\\lambda$ not on a simple root hyperplane. Our techniques also enable us to completely classify inclusion relations between \"weak faces\" of the set $\\wt(\\V)$ of weights of arbitrary $\\V$, in the process extending results of Vinberg, Chari-Dolbin-Ridenour, and Cellini-Marietti to all highest weight modules.",
      "comment": "See Announcement titled \"Weights of simple highest weight modules over a complex semisimple Lie algebra\" (which will be uploaded to arXiv in April 2013) for list of new results. Most important are: (1) Weights of all simple highest weight modules computed - Theorems 5 and 6 are new; (2) Pages 1-2, and Sections 6 and 9 are new; (3) Assumptions (b),(d) of Theorems 2 and 3 are new. (32 pages; LaTeX)",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2014-09-11T22:12:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B20",
      "17B10",
      "52B20"
    ],
    "keywords": [
      "maximizer subsets",
      "convex hull",
      "weyl polytope",
      "complex semisimple lie algebra",
      "simple highest weight modules"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1301.1140K"
    }
  }
}