{
  "id": "1110.1699",
  "version": "v5",
  "published": "2011-10-08T05:20:28.000Z",
  "updated": "2016-02-23T07:14:08.000Z",
  "title": "Quiver Schur algebras for linear quivers",
  "authors": [
    "Jun Hu",
    "Andrew Mathas"
  ],
  "comment": "Major revision to improve readability. We have added a proof that our quiver Schur algebras are graded Morita equivalent to those of Stroppel-Webster. This result is then used to match up the KLR and category O gradings in the degenerate case. Explicit formulas for the inverse parabolic Kazhdan-Lusztig polynomials are also given",
  "journal": "Proc. Lond. Math. Soc. (3) 110 (2015), no. 6, 1315-1386",
  "doi": "10.1112/plms/pdv007",
  "categories": [
    "math.RT",
    "math.CO",
    "math.QA",
    "math.RA"
  ],
  "abstract": "We define a graded quasi-hereditary covering for the cyclotomic quiver Hecke algebras $\\mathcal{R}^\\Lambda_n$ of type $A$ when $e=0$ (the linear quiver) or $e\\ge n$. We show that these algebras are quasi-hereditary graded cellular algebras by giving explicit homogeneous bases for them. When $e=0$ we show that the KLR grading on the quiver Hecke algebras is compatible with the gradings on parabolic category $\\mathcal{O}$ previously introduced in the works of Beilinson, Ginzburg and Soergel and Backelin. As a consequence, we show that when $e=0$ our graded Schur algebras are Koszul over field of characteristic zero. Finally, we give an LLT-like algorithm for computing the graded decomposition numbers of the quiver Schur algebras in characteristic zero when $e=0$.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2014-06-14T01:54:33.000Z",
      "title": "Quiver Schur algebras I: linear quivers",
      "abstract": "We define a graded quasi-hereditary covering for the cyclotomic quiver Hecke algebras $\\R$ of type $A$ when $e=0$ (the linear quiver) or $e\\ge n$. We show that these algebras are quasi-hereditary graded cellular algebras by giving explicit homogeneous bases for them. When $e=0$ we show that the KLR grading on the quiver Hecke algebras is compatible with the gradings on parabolic category $\\O$ previously introduced in the works of Beilinson, Ginzburg and Soergel and Backelin. As a consequence, we show that when $e=0$ our graded Schur algebras are Koszul over field of characteristic zero. Finally, we give an LLT-like algorithm for computing the graded decomposition numbers of the quiver Schur algebras in characteristic zero when $e=0$.",
      "journal": null,
      "doi": null
    },
    {
      "version": "v5",
      "updated": "2016-02-23T07:14:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20C08",
      "20C30",
      "05E10"
    ],
    "keywords": [
      "quiver schur algebras",
      "linear quiver",
      "characteristic zero",
      "cyclotomic quiver hecke algebras",
      "quasi-hereditary graded cellular algebras"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1110.1699H"
    }
  }
}