{
  "id": "1505.05910",
  "version": "v1",
  "published": "2015-05-21T21:32:42.000Z",
  "updated": "2015-05-21T21:32:42.000Z",
  "title": "A crystal to rigged configuration bijection and the filling map for type $D_4^{(3)}$",
  "authors": [
    "Travis Scrimshaw"
  ],
  "comment": "39 pages, 6 figures",
  "categories": [
    "math.CO",
    "math.QA",
    "math.RT"
  ],
  "abstract": "We give a bijection $\\Phi$ from rigged configurations to a tensor product of Kirillov--Reshetikhin crystals of the form $B^{r,1}$ and $B^{1,s}$ in type $D_4^{(3)}$. We show that the cocharge statistic is sent to the energy statistic for tensor products $\\bigotimes_{i=1}^N B^{r_i,1}$ and $\\bigotimes_{i=1}^N B^{1,s_i}$. We extend this bijection to a single $B^{r,s}$, show that it preserves statistics, and obtain the so-called Kirillov--Reshetikhin tableaux model for $B^{r,s}$. Additionally, we show $\\Phi$ commutes with the virtualization map and that $B^{1,s}$ is naturally a virtual crystal in type $D_4^{(1)}$, thus defining the affine crystal structure on rigged configurations corresponding to $B^{1,s}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-05-21T21:32:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10",
      "17B37"
    ],
    "keywords": [
      "rigged configuration bijection",
      "filling map",
      "tensor product",
      "kirillov-reshetikhin tableaux model",
      "affine crystal structure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150505910S"
    }
  }
}