{
  "id": "2101.05711",
  "version": "v1",
  "published": "2021-01-14T16:36:43.000Z",
  "updated": "2021-01-14T16:36:43.000Z",
  "title": "Norton algebras of the Hamming Graphs via linear characters",
  "authors": [
    "Jia Huang"
  ],
  "comment": "27 pages, comments welcome",
  "categories": [
    "math.CO"
  ],
  "abstract": "The Norton product is defined on each eigenspace of a distance regular graph by the orthogonal projection of the entry-wise product. The resulting algebra, known as the Norton algebra, is a commutative nonassociative algebra that is useful in group theory due to its interesting automorphism group. We provide a formula for the Norton product on each eigenspace of a Hamming graph using linear characters. We construct a large subgroup of automorphisms of the Norton algebra of a Hamming graph and completely describe the automorphism group in some cases. We also show that the Norton product on each eigenspace of a Hamming graph is as nonassociative as possible, except for some special cases in which it is either associative or equally as nonassociative as the so-called double minus operation previously studied by the author, Mickey, and Xu. Our results restrict to the hypercubes and extend to the halved and/or folded cubes, the bilinear forms graphs, and more generally, all Cayley graphs of finite abelian groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-14T16:36:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E30",
      "17A36",
      "20B25"
    ],
    "keywords": [
      "hamming graph",
      "norton algebra",
      "linear characters",
      "norton product",
      "automorphism group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}