{
  "id": "1505.04294",
  "version": "v1",
  "published": "2015-05-16T17:11:20.000Z",
  "updated": "2015-05-16T17:11:20.000Z",
  "title": "FI-modules and the cohomology of modular representations of symmetric groups",
  "authors": [
    "Rohit Nagpal"
  ],
  "comment": "59 pages",
  "categories": [
    "math.RT",
    "math.GR",
    "math.GT"
  ],
  "abstract": "An FI-module $V$ over a commutative ring $\\bf{k}$ encodes a sequence $(V_n)_{n \\geq 0}$ of representations of the symmetric groups $(\\mathfrak{S}_n)_{n \\geq 0}$ over $\\bf{k}$. In this paper, we show that for a \"finitely generated\" FI-module $V$ over a field of characteristic $p$, the cohomology groups $H^t(\\mathfrak{S}_n, V_n)$ are eventually periodic in $n$. We describe a recursive way to calculate the period and the periodicity range and show that the period is always a power of $p$. As an application, we show that if $\\mathcal{M}$ is a compact, connected, oriented manifold of dimension $\\geq 2$ and $\\mathit{conf}_n(\\mathcal{M})$ is the configuration space of unordered $n$-tuples of distinct points in $\\mathcal{M}$ then the mod-$p$ cohomology groups $H^{t}(\\mathit{conf}_n(\\mathcal{M}),\\bf{k})$ are eventually periodic in $n$ with period a power of $p$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-05-16T17:11:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "symmetric groups",
      "modular representations",
      "cohomology groups",
      "eventually periodic",
      "periodicity range"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 59,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150504294N"
    }
  }
}