{
  "id": "1703.08945",
  "version": "v1",
  "published": "2017-03-27T06:24:32.000Z",
  "updated": "2017-03-27T06:24:32.000Z",
  "title": "Uniform description of the rigged configuration bijection",
  "authors": [
    "Travis Scrimshaw"
  ],
  "comment": "51 pages, 1 figure, 2 tables",
  "categories": [
    "math.CO",
    "math.RT"
  ],
  "abstract": "We give a uniform description of the bijection $\\Phi$ from rigged configurations to tensor products of Kirillov--Reshetikhin crystals of the form $\\bigotimes_{i=1}^N B^{r_i,1}$ in dual untwisted types: simply-laced types and types $A_{2n-1}^{(2)}$, $D_{n+1}^{(2)}$, $E_6^{(2)}$, and $D_4^{(3)}$. We give a uniform proof that $\\Phi$ is a bijection and preserves statistics. We describe $\\Phi$ uniformly using virtual crystals for all remaining types, but our proofs are type-specific. We also give a uniform proof that $\\Phi$ is a bijection for $\\bigotimes_{i=1}^N B^{r_i,s_i}$ when $r_i$, for all $i$, map to $0$ under an automorphism of the Dynkin diagram. Furthermore, we give a description of the Kirillov--Reshetikhin crystals $B^{r,1}$ using tableaux of a fixed height $k_r$ depending on $r$ in all affine types. Additionally, we are able to describe crystals $B^{r,s}$ using $k_r \\times s$ shaped tableaux that are conjecturally the crystal basis for Kirillov--Reshetikhin modules for various nodes $r$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-27T06:24:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10",
      "17B37",
      "05A19",
      "81R50",
      "82B23"
    ],
    "keywords": [
      "rigged configuration bijection",
      "uniform description",
      "kirillov-reshetikhin crystals",
      "uniform proof",
      "affine types"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 51,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}