{
  "id": "2204.00066",
  "version": "v1",
  "published": "2022-03-31T20:15:10.000Z",
  "updated": "2022-03-31T20:15:10.000Z",
  "title": "Jordan types of triangular matrices over a finite field",
  "authors": [
    "Dmitry Fuchs",
    "Alexandre Kirillov"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\lambda$ be a partition of an integer $n$ and ${\\mathbb F}_q$ be a finite field of order $q$. Let $P_\\lambda(q)$ be the number of strictly upper triangular $n\\times n$ matrices of the Jordan type $\\lambda$. It is known that the polynomial $P_\\lambda$ has a tendency to be divisible by high powers of $q$ and $Q=q-1$, and we put $P_\\lambda(q)=q^{d(\\lambda)}Q^{e(\\lambda)}R_\\lambda(q)$, where $R_\\lambda(0)\\neq0$ and $R_\\lambda(1)\\neq0$. In this article, we study the polynomials $P_\\lambda(q)$ and $R_\\lambda(q)$. Our main results: an explicit formula for $d(\\lambda)$ (an explicit formula for $e(\\lambda)$ is known, see Proposition 3.3 below), a recursive formula for $R_\\lambda(q)$ (a similar formula for $P_\\lambda(q)$ is known, see Proposition 3.1 below), the stabilization of $R_\\lambda$ with respect to extending $\\lambda$ by adding strings of 1's, and an explicit formula for the limit series $R_{\\lambda1^\\infty}$. Our studies are motivated by projected applications to the orbit method in the representation theory of nilpotent algebraic groups over finite fields.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-31T20:15:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite field",
      "jordan type",
      "triangular matrices",
      "explicit formula",
      "nilpotent algebraic groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}