{
  "id": "2110.15504",
  "version": "v2",
  "published": "2021-10-29T02:57:06.000Z",
  "updated": "2022-06-27T23:44:57.000Z",
  "title": "A Remark on Random Vectors and Irreducible Representations",
  "authors": [
    "Alexander Kushkuley"
  ],
  "categories": [
    "math.PR",
    "cs.NA",
    "math.NA",
    "math.RT"
  ],
  "abstract": "It was observed in \\cite{Al2} that the expectation of a squared scalar product of two random independent unit vectors that are uniformly distributed on the unit sphere in $\\mathbb{R}^n $ is equal to $1/n$. It is shown below that this is a characteristic property of random unit vectors defined on invariant probability subspaces of irreducible real representations of compact Lie groups.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-06-27T23:44:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20Cxx",
      "37L40",
      "60B99",
      "65C50",
      "65C20"
    ],
    "keywords": [
      "random vectors",
      "irreducible representations",
      "random independent unit vectors",
      "random unit vectors",
      "invariant probability subspaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}