具有伽罗瓦作用量的Clifford理论
On Clifford theory with Galois action
摘要：
设$\widehat{G}$是有限群，$N $是$\widehat{G}$和$θ\in \operatorname{Irr}N$的正规子群.设$\mathbb{F}$是复数的一个子域，并假定$θ$在$\mathbb{F}$上的伽罗瓦轨道在$\widehat{G}$中是不变的。我们证明了存在另一个具有相同形式的三元组$(\widehat{G}_1,N_1,θ_1)$，使得$\widehat{G}$ over $θ$和$\widehat{G}_1$ over $θ_1$的特征标理论在域$\mathbb{F}$上基本上是“相同的”，并且使得以下成立：$\widehat{G}_1$具有包含在$N_1$中的循环正规子群$C$，使得对于$C$的某些线性特征符$λ$，并且使得$N_1/C$同构于域扩张$\mathbb{F}(λ)/\mathbb{F}(θ_1)$的（阿贝尔）伽罗瓦群。更确切地说，“相同”意味着两个三元组产生由A定义的Brauer-Clifford群$\operatorname{BrCliff}(G,\mathbb{F}(θ))$的相同元素。图鲁尔
英文摘要：
Let $\widehat{G}$ be a finite group, $N $ a normal subgroup of $\widehat{G}$ and $θ\in \operatorname{Irr}N$. Let $\mathbb{F}$ be a subfield of the complex numbers and assume that the Galois orbit of $θ$ over $\mathbb{F}$ is invariant in $\widehat{G}$. We show that there is another triple $(\widehat{G}_1,N_1,θ_1)$ of the same form, such that the character theories of $\widehat{G}$ over $θ$ and of $\widehat{G}_1$ over $θ_1$ are essentially "the same" over the field $\mathbb{F}$ and such that the following holds: $\widehat{G}_1$ has a cyclic normal subgroup $C$ contained in $N_1$, such that $θ_1=λ^{N_1}$ for some linear character $λ$ of $C$, and such that $N_1/C$ is isomorphic to the (abelian) Galois group of the field extension $\mathbb{F}(λ)/\mathbb{F}(θ_1)$. More precisely, "the same" means that both triples yield the same element of the Brauer-Clifford group $\operatorname{BrCliff}(G,\mathbb{F}(θ))$ defined by A. Turull.
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