{
  "id": "2004.05283",
  "version": "v1",
  "published": "2020-04-11T02:25:37.000Z",
  "updated": "2020-04-11T02:25:37.000Z",
  "title": "Covering $Irrep(S_n)$ With Tensor Products and Powers",
  "authors": [
    "Mark Sellke"
  ],
  "categories": [
    "math.CO",
    "math.RT"
  ],
  "abstract": "We study when a tensor product of irreducible representations of the symmetric group $S_n$ contains all irreducibles as subrepresentations - we say such a tensor product covers $Irrep(S_n)$. Our results show that this behavior is typical. We first give a general criterion for such a tensor product to have this property. Using this criterion we show that the tensor product of a constant number of random irreducibles covers $Irrep(S_n)$ asymptotically almost surely. We also consider, for a fixed irreducible representation, the degree of tensor power needed to cover $Irrep(S_n)$. We show that the trivial lower bound based on dimension is tight up to a universal constant factor for every irreducible representation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-11T02:25:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "irreducible representation",
      "random irreducibles covers",
      "universal constant factor",
      "tensor product covers",
      "trivial lower bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}