{
  "id": "2408.14727",
  "version": "v1",
  "published": "2024-08-27T01:36:31.000Z",
  "updated": "2024-08-27T01:36:31.000Z",
  "title": "Projective (or spin) representations of finite groups. III",
  "authors": [
    "Satoe Yamanaka",
    "Tatsuya Tsurii",
    "Itsumi Mikami",
    "Takeshi Hirai"
  ],
  "comment": "6 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "In previous papers I and II under the same title, we proposed a practical method called Efficient stairway up to the Sky, and apply it to some typical finite groups $G$, with Schur multiplier $M(G)$ containing prime number 3, to construct explicitly their representation groups $R(G)$, and then, to construct a complete set of representatives of linear IRs of $R(G)$, which gives naturally, through sectional restrictions, a complete set of representatives of spin IRs of $G$. In the present paper, we are concerned mainly with group $G=G_{39}$ of order 27 in a list of Tahara's paper, with $M(G)={\\mathbb Z}_3\\times {\\mathbb Z}_3$. In this case, to arrive up to the Sky, we have two steps of one-step efficient central extensions. By the 1st step, we obtain a covering group of order 81, and by the 2nd step we arrive to $R(G)$ of order 243. At the 1st step, to construct explicitly a complete set of representatives of IRs of the group, we apply Mackey's induced representations, and at the 2nd step, with a help of this result, we apply so-called classical method for semidirect product groups given by Hirai and arrive to a complete list of IRs of $R(G)$. Then, using explicit realization of these IRs, we can compute their characters (called spin characters).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-27T01:36:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite groups",
      "complete set",
      "1st step",
      "2nd step",
      "one-step efficient central extensions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}