Differential Operators on G(r,n)-Invariant Functions
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):=μrwr Sn (with G(1,n)=Sn). The central tool, proved in detail, is a \emph{transfer principle}: the substitution yi=xir identifies the invariant ring of G(r,n) with that of Sn and transports, term by term, the corresponding operators and coordinates into explicit rational objects in the original variables xi. From this we deduce the G(r,n)-analogues of the known results for Sn existence and uniqueness of the dual coordinates Uk, 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 rn−1 lines permuted transitively by the group, and there, unlike the transferred operators Δi, the \emph{raw} derivatives ∂xi exhibit a phenomenon specific to r≥2 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 xi=0 (the operator Δi extends holomorphically there, with Δiϕ=∂irϕ/r!), and deduce from this a partial analogue of the description of the tangent space to the GIT quotient CCn/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.
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BibTeX
@misc{kafando2026differential,
title = {{Differential Operators on $G(r,n)$-Invariant Functions}},
author = {Ferdinand Kafando and Jean Kaboré and Ibrahim Nonkané},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.18139},
eprint = {2608.18139},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.18139},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Differential Operators on G(r,n)-Invariant Functions
AU - Ferdinand Kafando
AU - Jean Kaboré
AU - Ibrahim Nonkané
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2608.18139
UR - https://arxiv.org/abs/2608.18139
AB - 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_rwr 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 ∂_x_i exhibit a phenomenon specific to r≥2 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=∂_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.
ER -