G(r,n)-不变函数上的微分算子

中文摘要 AI 辅助

我们将对称群上已建立的关于归一化对称坐标及其对偶微分算子的已知结果，推广到复单项式反射群$G(r,n):=\mu_r\wr S_n$（其中$G(1,n)=S_n$）。经详细证明的核心工具是一个\emph{转移原理}：代换$y_i=x_i^r$将$G(r,n)$的不变环与$S_n$的不变环等同，并逐项将对应的算子和坐标转化为原变量$x_i$中的显式有理对象。由此我们推导出$S_n$已知结果的$G(r,n)$类似结论：对偶坐标$U_k$的存在性与唯一性，以及在$G(r,n)$判别式处局部化的Weyl代数结构，对无法直接由转移得到的要点（莱布尼茨法则、主定理）给出了完整自包含的证明。随后我们处理\emph{全对角线}：其原像分裂为$r^{n-1}$条被群可迁置换的直线，在此处，与转移算子$\Delta_i$不同，\emph{原始}导数$\partial_{x_i}$呈现出仅在$r\geq2$时特有的现象，我们通过法阿·迪·布鲁诺型结构公式对其进行了完整描述。最后，我们给出该结构公式常数的闭式（通过贝尔多项式），完全解决了孤立点$x_i=0$处的退化问题（算子$\Delta_i$在该处全纯延拓，满足$\Delta_i\phi=\partial_i^r\phi/r!$），并由此推导出GIT商空间$\mathbb{C}^n/G(r,n)$的切空间描述的部分类似结论；多个坐标同时为零的情况仍未解决，且被明确界定。所有新结果均给出了完整证明。

英文摘要

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.

发表机构

• Université Thomas SANKARA（托马斯·桑卡拉大学）
• Laboratoire de Sciences et Technologies (LaST)（科学与技术实验室）
• IUFIC

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