{
  "abstract": "We generalize known results on normalized symmetric coordinates and their dual differential operators, established for the symmetric group, to the complex monomial reflection group $G(r,n):=μ_r\\wr S_n$ (with $G(1,n)=S_n$). The central tool, proved in detail, is a \\emph{transfer principle}: the substitution $y_i=x_i^r$ identifies the invariant ring of $G(r,n)$ with that of $S_n$ and transports, term by term, the corresponding operators and coordinates into explicit rational objects in the original variables $x_i$. From this we deduce the $G(r,n)$-analogues of the known results for $S_n$ existence and uniqueness of the dual coordinates $U_k$, and a Weyl algebra structure localized at the discriminant of $G(r,n)$ with complete and self-contained proofs for the points that do not follow directly from the transfer (Leibniz rule, main theorem). We then treat the \\emph{total diagonal}: its preimage splits into $r^{n-1}$ lines permuted transitively by the group, and there, unlike the transferred operators $Δ_i$, the \\emph{raw} derivatives $\\partial_{x_i}$ exhibit a phenomenon specific to $r\\ge2$ that we describe completely via a Fa di Bruno-type structure formula. Finally, we give a closed formula for the constants of this structure formula (via Bell polynomials), completely resolve the degeneracy at an isolated point $x_i=0$ (the operator $Δ_i$ extends holomorphically there, with $Δ_iϕ=\\partial_i^rϕ/r!$), and deduce from this a partial analogue of the description of the tangent space to the GIT quotient $\\CC^n/G(r,n)$; the case of several coordinates vanishing simultaneously remains open and is precisely delineated. Full proofs of all new results are given in detail.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.18139",
  "author_total": 3,
  "authors": [
    {
      "name": "Ferdinand Kafando"
    },
    {
      "name": "Jean Kaboré"
    },
    {
      "name": "Ibrahim Nonkané"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9tbfx64ptqc19.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.18139",
  "fulltext_url": null,
  "id": "9tbfx64ptqc19",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.18139",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AC"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7203895439",
  "pdf_url": "https://arxiv.org/pdf/2608.18139",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9tbfx64ptqc19.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2608.18139",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.18139"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Differential Operators on $G(r,n)$-Invariant Functions",
  "title_text": "Differential Operators on G(r,n)-Invariant Functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/9tbfx64ptqc19",
  "venue": "arXiv",
  "versions": []
}