{
  "id": "1201.4549",
  "version": "v3",
  "published": "2012-01-22T11:04:27.000Z",
  "updated": "2012-08-15T20:05:24.000Z",
  "title": "On the combinatorial structure of crystals of types A,B,C",
  "authors": [
    "Vladimir Danilov",
    "Alexander Karzanov",
    "Gleb Koshevoy"
  ],
  "comment": "54 pages. Compared with the previous version, some illustrations and references are added",
  "categories": [
    "math.CO"
  ],
  "abstract": "Regular $A_n$-, $B_n$- and $C_n$-crystals are edge-colored directed graphs, with ordered colors $1,2,...,n$, which are related to representations of quantized algebras $U_q(\\mathfrak{sl}_{n+1})$, $U_q(\\mathfrak{sp}_{2n})$ and $U_q(\\mathfrak{so}_{2n+1})$, respectively. We develop combinatorial methods to reveal refined structural properties of such objects. Firstly, we study subcrystals of a regular $A_n$-crystal $K$ and characterize pairwise intersections of maximal subcrystals with colors $1,...,n-1$ and colors $2,...,n$. This leads to a recursive description of the structure of $K$ and provides an efficient procedure of assembling $K$. Secondly, using merely combinatorial means, we demonstrate a relationship between regular $B_n$-crystals (resp. $C_n$-crystals) and regular symmetric $A_{2n-1}$-crystals (resp. $A_{2n}$-crystals).",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-08-15T20:05:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B37",
      "05C75",
      "05E99"
    ],
    "keywords": [
      "combinatorial structure",
      "reveal refined structural properties",
      "combinatorial methods",
      "regular symmetric",
      "maximal subcrystals"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 54,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1201.4549D"
    }
  }
}