{
  "id": "2010.07060",
  "version": "v1",
  "published": "2020-10-14T13:08:35.000Z",
  "updated": "2020-10-14T13:08:35.000Z",
  "title": "Dual canonical basis for unipotent group and base affine space",
  "authors": [
    "Jian-rong Li"
  ],
  "comment": "19 pages, 2 figures",
  "categories": [
    "math.RT",
    "math.QA"
  ],
  "abstract": "Denote by $N \\subset SL_k$ the subgroup of unipotent upper triangular matrices. In this paper, we show that the dual canonical basis of $\\mathbb{C}[N]$ can be parameterized by semi-standard Young tableaux. Moreover, we give an explicit formula for every element in the the dual canonical basis. Let $N^- \\subset SL_k$ be the subgroup of unipotent lower-triangular matrices and let $\\mathbb{C}[SL_k]^{N^-}$ be the coordinate ring of the base affine space $SL_k/N^{-}$. Denote by $\\widetilde{\\mathbb{C}[SL_k]^{N^-}}$ the quotient of $\\mathbb{C}[SL_k]^{N^-}$ by identifying the leading principal minors with $1$. We also give an explicit description of the dual canonical basis of $\\widetilde{\\mathbb{C}[SL_k]^{N^-}}$ and give a conjectural description of the dual canonical basis of $\\mathbb{C}[SL_k]^{N^-}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-14T13:08:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13F60",
      "17B37"
    ],
    "keywords": [
      "dual canonical basis",
      "base affine space",
      "unipotent group",
      "unipotent upper triangular matrices",
      "semi-standard young tableaux"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}