{
  "id": "1906.06494",
  "version": "v1",
  "published": "2019-06-15T08:18:32.000Z",
  "updated": "2019-06-15T08:18:32.000Z",
  "title": "Finite Reflection Groups: Invariant functions and functions of the Invariants in finite class of differentiability",
  "authors": [
    "Gerard Barbançon"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $W$ be a finite reflection group. A $W$-invariant function of class~$C^{\\infty}$ may be expressed as a functions of class $C^{\\infty}$ of the basic invariants. In finite class of differentiability, the situation is not this simple. Let~$h$ be the greatest Coxeter number of the irreducible components of $W$ and $P$ be~the Chevalley mapping, if $f$ is an invariant function of class $C^{hr}$, and $F$ is the function of invariants associated by $f=F\\circ P$, then $F$ is of class $C^r$. However if~$F$ is of class $C^r$, in general $f=F\\circ P$ is of class $C^r$ and not of class $C^{hr}$. Here we determine the space of $W$-invariant functions that may be written as functions of class $C^r$ of the polynomial invariants and the subspace of functions $F$ of class $C^r$ of the invariants such that the invariant function $f=F\\circ P$ is of class $C^{hr}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-15T08:18:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F55",
      "58B10"
    ],
    "keywords": [
      "invariant function",
      "finite reflection group",
      "finite class",
      "differentiability",
      "greatest coxeter number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}