{
  "id": "1511.02387",
  "version": "v1",
  "published": "2015-11-07T19:05:48.000Z",
  "updated": "2015-11-07T19:05:48.000Z",
  "title": "The Saxl Conjecture for Fourth Powers via the Semigroup Property",
  "authors": [
    "Sammy Luo",
    "Mark Sellke"
  ],
  "comment": "50 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "The tensor square conjecture states that for $n \\geq 10$, there is an irreducible representation $V$ of the symmetric group $S_n$ such that $V \\otimes V$ contains every irreducible representation of $S_n$. Our main result is that for large enough $n$, there exists an irreducible representation $V$ such that $V^{\\otimes 4}$ contains every irreducible representation. We also show that tensor squares of certain irreducible representations contain $(1-o(1))$-fraction of irreducible representations with respect to two natural probability distributions. Our main tool is the semigroup property, which allows us to break partitions down into smaller ones.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-07T19:05:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "irreducible representation",
      "semigroup property",
      "saxl conjecture",
      "fourth powers",
      "tensor square conjecture states"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 50,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151102387L"
    }
  }
}