{
  "id": "0809.0988",
  "version": "v1",
  "published": "2008-09-05T09:36:40.000Z",
  "updated": "2008-09-05T09:36:40.000Z",
  "title": "Homotopy, homology, and $GL_2$",
  "authors": [
    "Vanessa Miemietz",
    "Will Turner"
  ],
  "comment": "28 pages",
  "doi": "10.1112/plms/pdp040",
  "categories": [
    "math.RT",
    "math.CT"
  ],
  "abstract": "We define weak 2-categories of finite dimensional algebras with bimodules, along with collections of operators $\\mathbb{O}_{(c,x)}$ on these 2-categories. We prove that special examples $\\mathbb{O}_p$ of these operators control all homological aspects of the rational representation theory of the algebraic group $GL_2$, over a field of positive characteristic. We prove that when $x$ is a Rickard tilting complex, the operators $\\mathbb{O}_{(c,x)}$ honour derived equivalences, in a differential graded setting. We give a number of representation theoretic corollaries, such as the existence of tight $\\mathbb{Z}_+$-gradings on Schur algebras $S(2,r)$, and the existence of braid group actions on the derived categories of blocks of these Schur algebras.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-09-05T09:36:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20G05",
      "18D05"
    ],
    "keywords": [
      "schur algebras",
      "finite dimensional algebras",
      "rational representation theory",
      "representation theoretic corollaries",
      "braid group actions"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0809.0988M"
    }
  }
}