{
  "id": "math/0410357",
  "version": "v3",
  "published": "2004-10-15T20:40:39.000Z",
  "updated": "2005-10-04T02:43:23.000Z",
  "title": "Extending $π$-systems to bases of root systems",
  "authors": [
    "Helmer Aslaksen",
    "Mong Lung Lang"
  ],
  "comment": "6 pages, LaTeX. Corrected typo in statement of theorem and clarified proof",
  "journal": "Journal of Algebra 287 (2005), 496-500",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $R$ be an indecomposable root system. It is well known that any root is part of a basis $B$ of $R$. But when can you extend a set of two or more roots to a basis $B$ of $R$? A $\\pi$-system is a linearly independent set of roots, $C$, such that if $\\alpha$ and $\\beta$ are in $C$, then $\\alpha - \\beta$ is not a root. We will use results of Dynkin and Bourbaki to show that with two exceptions, $A_3 \\subset B_n$ and $A_7 \\subset E_8$, an indecomposable $\\pi$-system whose Dynkin diagram is a subdiagram of the Dynkin diagram of $R$ can always be extended to a basis of $R$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2005-10-04T02:43:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B05"
    ],
    "keywords": [
      "dynkin diagram",
      "linearly independent set",
      "indecomposable root system",
      "exceptions",
      "subdiagram"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.....10357A"
    }
  }
}