{
  "id": "1506.07678",
  "version": "v1",
  "published": "2015-06-25T09:43:37.000Z",
  "updated": "2015-06-25T09:43:37.000Z",
  "title": "Asymptotic of Number of Similarity Classes of Commuting Tuples",
  "authors": [
    "Uday Bhaskar Sharma"
  ],
  "comment": "13 Pages, 1 Table",
  "categories": [
    "math.CO"
  ],
  "abstract": "We have for positive integers $n$, $k$ and finite field $\\mathbb{F}_q$, $c(n,k,q)$, as the number of simultaneous similarity classes of $k$-tuples of commuting $n\\times n$ matrices over the $\\mathbb{F}_q$. In this paper, it has been shown that $c(n,k,q)$ as a function of $k$ for fixed $n$ and $q$ is asymptotically $q^{m(n)k}$, where $m(n) = \\left[\\frac{n^2}{4}\\right] + 1$, which is the dimension of the maximal commutative subalgebra of $M_n(\\mathbb{F}_q)$ (the algebra of $n\\times n$ matrices over $\\mathbb{F}_q$).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-25T09:43:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A16"
    ],
    "keywords": [
      "commuting tuples",
      "asymptotic",
      "finite field",
      "simultaneous similarity classes",
      "maximal commutative subalgebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150607678B"
    }
  }
}