论文标题
有限组和维度功能的DADE组
Dade Groups for Finite Groups and Dimension Functions
论文作者
论文摘要
令$ g $为有限的组，$ k $是特征性$ p> 0 $的代数关闭的字段。我们将DADE $ kg $模块的概念定义为$ p $ groups的内部渗透模块的概括。我们表明，在适当的等价关系下，达德$ kg $ - 模块的等效类别在张量产品下形成了一个组，并且以这种方式获得的小组对lassueur定义的Dade组$ d（g）$是同构。我们还考虑相对syzygies $ω_x$生成的$ d（g）的子组$ d^ω（g）$，其中$ x $是有限的$ g $ set。如果$ c（g，p）$表示在$ g $的$ p $ -subgroup上定义的一组超级类功能，则有天然生成器$ω_x$ $ c（g，p）$，我们证明存在一个定义明确的群体iMormorphism unormorphism $ cok $ c（g，p）$ $ $ d $ $ $ $ $ $ $ g）$（$ g）$ g（$ g）。本文的主要定理是验证$ c（g，p）$的子组由$ k $的尺寸函数组成的$ g $的尺寸函数在$ψ_g$的内核中。
Let $G$ be a finite group and $k$ an algebraically closed field of characteristic $p>0$. We define the notion of a Dade $kG$-module as a generalization of endo-permutation modules for $p$-groups. We show that under a suitable equivalence relation, the set of equivalence classes of Dade $kG$-modules forms a group under tensor product, and the group obtained this way is isomorphic to the Dade group $D(G)$ defined by Lassueur. We also consider the subgroup $D^Ω (G)$ of $D(G)$ generated by relative syzygies $Ω_X$, where $X$ is a finite $G$-set. If $C(G,p)$ denotes the group of superclass functions defined on the $p$-subgroups of $G$, there are natural generators $ω_X$ of $C(G,p)$, and we prove the existence of a well-defined group homomorphism $Ψ_G:C(G,p)\to D^Ω(G)$ that sends $ω_X$ to $Ω_X$. The main theorem of the paper is the verification that the subgroup of $C(G,p)$ consisting of the dimension functions of $k$-orientable real representations of $G$ lies in the kernel of $Ψ_G$.
