共轭不变量仅在重标号范围内确定元交换置换

中文摘要 AI 辅助

设$\boldsymbol{\textit{H}}$为Hurwitz四元数，$p$为奇素数，$Q \boldsymbol{\textit{H}}$为范数$q \neq p$的素元。元交换$PQ = Q'P'$诱导范数$p$的素元的$p+1$个左相伴类上的置换$\boldsymbol{\textit{\textbackslash pi}}_Q$。Cohn与Kumar通过仅依赖$Q$的共轭不变量数据（即$q$和$\text{tr}\boldsymbol{\textit{\textbackslash,}}Q$）的公式计算其符号和不动点数量，Leite与Machiavelo则计算其完整循环结构。本文证明这恰好是此类不变量所能承载的边界：任何在单位共轭下不变的量$I(Q)$都无法将$\boldsymbol{\textit{\textbackslash pi}}_Q$确定为内在类集的带标号置换，甚至无法确定单个指定类的像。证明结合了等变恒等式$\boldsymbol{\textit{\textbackslash pi}}_{uQu^{-1}} = \boldsymbol{\textit{\textbackslash rho}}_u \boldsymbol{\textit{\textbackslash pi}}_Q \boldsymbol{\textit{\textbackslash rho}}_u^{-1}$，以及$(p,q)=(3,5)$处的极小、完全显式的见证：四个素元$2+i$、$2+j$、$2+k$、$2-i$构成单一单位共轭轨道，因此在所有共轭不变函数下均一致，但诱导出四个两两不同的包含相同四个类的4-循环。本文进一步观察到同构$\boldsymbol{\textit{\textbackslash mathcal}}H/p\boldsymbol{\textit{\textbackslash mathcal}}H \to M_2(\boldsymbol{\textit{\textbackslash mathbb}}F_p)$在$\text{PGL}_2(\boldsymbol{\textit{\textbackslash mathbb}}F_p)$下形成挠子，因此类的任何射影标号都不是典范的；且通过轨道-稳定子定理，单个目标$\boldsymbol{\textit{\textbackslash pi}}_Q(C)$的数据恰好是$\text{PGL}_2(\boldsymbol{\textit{\textbackslash mathbb}}F_p)/G_C$中的一个陪集$g_Q G_C$。见证中的全部16个重构均列于附录，并已通过两条独立途径经机器验证。

英文摘要

Let $\mathcal{H}$ be the Hurwitz quaternions, $p$ an odd prime, and $Q \in \mathcal{H}$ a prime of norm $q \neq p$. Metacommutation $PQ = Q'P'$ induces a permutation $π_Q$ of the $p+1$ left-associate classes of primes of norm $p$. Cohn and Kumar compute its sign and fixed-point count, and Leite and Machiavelo its full cycle structure, by formulas depending only on conjugation-invariant data of $Q$ (namely $q$ and $\mathrm{tr}\,Q$). We prove this is exactly the boundary of what such invariants can carry: no quantity $I(Q)$ invariant under unit conjugation determines $π_Q$ as a labelled permutation of the intrinsic class set, or even the image of a single specified class. The proof combines an equivariance identity $π_{uQu^{-1}} = ρ_u π_Q ρ_u^{-1}$ with a minimal, fully explicit witness at $(p,q) = (3,5)$: the four primes $2+i$, $2+j$, $2+k$, $2-i$ form a single unit-conjugacy orbit, hence agree under every conjugation-invariant function, yet induce four pairwise distinct $4$-cycles of the same four classes. We further observe that isomorphisms $\mathcal{H}/p\mathcal{H} \to M_2(\mathbb{F}_p)$ form a torsor under $\mathrm{PGL}_2(\mathbb{F}_p)$, so no projective labelling of the classes is canonical, and that by orbit-stabilizer the datum of one destination $π_Q(C)$ is exactly a coset $g_Q G_C$ in $\mathrm{PGL}_2(\mathbb{F}_p)/G_C$. All sixteen refactorizations in the witness are listed in the appendix and have been verified by machine along two independent routes.

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7 pages