{
  "id": "1901.06945",
  "version": "v1",
  "published": "2019-01-21T15:01:06.000Z",
  "updated": "2019-01-21T15:01:06.000Z",
  "title": "The action of the Weyl group on the $E_8$ root system",
  "authors": [
    "Rosa Winter",
    "Ronald van Luijk"
  ],
  "comment": "55 pages and 19 pages appendix",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $\\Gamma$ be the graph on the roots of the $E_8$ root system, where any two distinct vertices $e$ and $f$ are connected by an edge with color equal to the inner product of $e$ and $f$. For any set $c$ of colors, let $\\Gamma_c$ be the subgraph of $\\Gamma$ consisting of all the $240$ vertices, and all the edges whose color lies in $c$. We consider cliques, i.e., complete subgraphs, of $\\Gamma$ that are either monochromatic, or of size at most $3$, or a maximal clique in $\\Gamma_c$ for some color set $c$, or whose vertices are the vertices of a face of the $E_8$ root polytope. We prove that, apart from two exceptions, two such cliques are conjugate under the automorphism group of $\\Gamma$ if and only if they are isomorphic as colored graphs. Moreover, for an isomorphism $f$ from one such clique $K$ to another, we give necessary and sufficient conditions for $f$ to extend to an automorphism of $\\Gamma$, in terms of the restrictions of $f$ to certain special subgraphs of $K$ of size at most 7.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-21T15:01:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "root system",
      "weyl group",
      "color equal",
      "automorphism group",
      "inner product"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 55,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}