{
  "id": "2106.05198",
  "version": "v1",
  "published": "2021-06-09T16:42:19.000Z",
  "updated": "2021-06-09T16:42:19.000Z",
  "title": "On homological properties of strict polynomial functors of degree p",
  "authors": [
    "Patryk Jaśniewski"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "We study the homological algebra in the category $\\mathcal{P}_p$ of strict polynomial functors of degree $p$ over a field of positive characteristic $p$. We determine the decomposition matrix of our category and we calculate the Ext-groups between functors important from the point of view of representation theory. Our results include computations of the Ext-algebras for simple functors and Schur functors and the Ext-groups between Schur and Weyl functors. We observe that the category $\\mathcal{P}_p$ has a Kazhdan-Lusztig theory and we show that the dg algebras computing the Ext-algebras for simple functors and Schur functors are formal. These last results allow one to describe the bounded derived category of $\\mathcal{P}_p$ as derived categories of certain explicitly described graded algebras. We also generalize our results to all blocks of $p$-weight 1 in $\\mathcal{P}_e$ for $e > p$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-09T16:42:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "strict polynomial functors",
      "homological properties",
      "schur functors",
      "simple functors",
      "derived category"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}