八元数、Albert向量与群 ${}^2 \mathrm{E}_6(F)$
Octonions, Albert vectors and the group ${}^2 \mathrm{E}_6(F)$
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中文总结 AI 辅助
本文用显式八元数方法研究扭群${}^2\mathrm{E}_6(F)$，在范数满射时给出白点三轨道定理和稳定子的坐标证明，并证明中心商单性；有限域上恢复经典轨道长度和群阶，非满射时各向异性轨道由范数余类参数化。
中文摘要 AI 辅助
我们将类型为 $\mathrm{E}_6$ 的群的显式八元数方法推广到与二次伽罗瓦扩张 $K/F$ 相关的扭群 ${}^2\mathrm{SE}_{6,K}(F)$。这些群作用于 $K$ 上的Albert空间，保持Dickson–Freudenthal行列式和一个Hermite形式。当域范数 $K^{\times}\to F^{\times}$ 是满射时，我们给出白点上的三轨道定理和代表性白向量的稳定子的显式生成元和坐标证明。在此假设下，我们还证明了中心商的单性。在有限域上，我们从稳定子阶和白点总数恢复了经典的轨道长度和群阶。最后，在不假设范数满射的情况下，我们证明两个各向同性白点轨道保持不变，而各向异性轨道由 $F^{\times}/\mathrm{N}_{K/F}(K^{\times})$ 参数化。
英文摘要
We extend the explicit octonionic approach to groups of type $\mathrm{E}_6$ to the twisted groups ${}^2\mathrm{SE}_{6,K}(F)$ associated with a quadratic Galois extension $K/F$. These groups act on the Albert space over $K$, preserving the Dickson--Freudenthal determinant and a Hermitean form. When the field norm $K^{\times}\to F^{\times}$ is surjective, we give explicit generators and coördinate proofs of the three-orbit theorem on white points and of the stabilisers of representative white vectors. Under this hypothesis we also prove simplicity of the central quotient. Over finite fields we recover the classical orbit lengths and group orders from the stabiliser orders and the total number of white points. Finally, without assuming norm surjectivity, we show that the two isotropic white-point orbits remain unchanged, while the non-isotropic orbits are parametrised by $F^{\times}/\mathrm{N}_{K/F}(K^{\times})$.
