{
  "id": "2206.04509",
  "version": "v1",
  "published": "2022-06-09T13:48:03.000Z",
  "updated": "2022-06-09T13:48:03.000Z",
  "title": "Weak faces and a formula for weights of highest weight modules, via parabolic partial sum property for roots",
  "authors": [
    "G. Krishna Teja"
  ],
  "comment": "12 pages, final version. This is an extended abstract of arXiv:2012.07775 and arXiv:2106.14929, accepted in FPSAC 2022",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\mathfrak{g}$ be a finite or an affine type Lie algebra over $\\mathbb{C}$ with root system $\\Delta$. We show a parabolic generalization of the partial sum property for $\\Delta$, which we term the parabolic partial sum property. It allows any root $\\beta$ involving (any) fixed subset $S$ of simple roots, to be written as an ordered sum of roots, each involving exactly one simple root from $S$, with each partial sum also being a root. We show three applications of this property to weights of highest weight $\\mathfrak{g}$-modules: (1)~We provide a minimal description for the weights of all non-integrable simple highest weight $\\mathfrak{g}$-modules, refining the weight formulas shown by Khare [J. Algebra} 2016] and Dhillon-Khare [Adv. Math. 2017]. (2)~We provide a Minkowski difference formula for the weights of an arbitrary highest weight $\\mathfrak{g}$-module. (3)~We completely classify and show the equivalence of two combinatorial subsets - weak faces and 212-closed subsets - of the weights of all highest weight $\\mathfrak{g}$-modules. These two subsets were introduced and studied by Chari-Greenstein [Adv. Math. 2009], with applications to Lie theory including character formulas. We also show ($3'$) a similar equivalence for root systems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-09T13:48:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "parabolic partial sum property",
      "highest weight modules",
      "weak faces",
      "root system",
      "affine type lie algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}