{
  "id": "2304.07796",
  "version": "v1",
  "published": "2023-04-16T14:26:44.000Z",
  "updated": "2023-04-16T14:26:44.000Z",
  "title": "Linkage and translation for tensor products of representations of simple algebraic groups and quantum groups",
  "authors": [
    "Jonathan Gruber"
  ],
  "comment": "22 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\mathbf{G}$ be either a simple linear algebraic group over an algebraically closed field of characteristic $\\ell>0$ or a quantum group at an $\\ell$-th root of unity. We define a tensor ideal of singular $\\mathbf{G}$-modules in the category $\\mathrm{Rep}(\\mathbf{G})$ of finite-dimensional $\\mathbf{G}$-modules and study the associated quotient category $\\mathrm{\\underline{Re}p}(\\mathbf{G})$, called the regular quotient. Our main results are a 'linkage principle' and a 'translation principle' for tensor products: Let $\\mathrm{\\underline{Re}p}_0(\\mathbf{G})$ be the essential image in $\\mathrm{\\underline{Re}p}(\\mathbf{G})$ of the principal block of $\\mathrm{Rep}(\\mathbf{G})$. We first show that $\\mathrm{\\underline{Re}p}_0(\\mathbf{G})$ is closed under tensor products in $\\mathrm{\\underline{Re}p}(\\mathbf{G})$. Then we prove that the monoidal structure of $\\mathrm{\\underline{Re}p}(\\mathbf{G})$ is governed to a large extent by the monoidal structure of $\\mathrm{\\underline{Re}p}_0(\\mathbf{G})$. These results can be combined to give an external tensor product decomposition $\\mathrm{\\underline{Re}p}(\\mathbf{G}) \\cong \\mathrm{Ver}(\\mathbf{G}) \\boxtimes \\mathrm{\\underline{Re}p}_0(\\mathbf{G})$, where $\\mathrm{Ver}(\\mathbf{G})$ denotes the Verlinde category of $\\mathbf{G}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-16T14:26:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20G05",
      "20G42",
      "17B10",
      "17B55"
    ],
    "keywords": [
      "simple algebraic groups",
      "quantum group",
      "translation",
      "simple linear algebraic group",
      "external tensor product decomposition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}