论文标题
Schur索引和Witt-Berman定理的扩展
Schur Index and Extensions of Witt-Berman's Theorems
论文作者
论文摘要
让$ g $是有限的组，而特征$ 0 $或$ g $的$ f $ $ f $。伯曼以1952美元的价格，witt和1956美元的$中的$ 1956 $，证明了$ g $的不相等的$ f $ - 代表的数量等于$ g $的$ f $ - 结合元素的数量，其中“ $ f $ -conconjugacy”在某种程度上定义了。在本文中，我们以自然的方式定义了$ g $上的$ f $ - 结合性，并证明了上述Witt-Berman定理。此外，我们还提供了一个明确的公式，用于计算对应于$ g $的不可约$ f $ - 代表对应的组代数$ f [g] $的原始中央dempotent（pci），可以从$ g $ of $ g $的“ $ f $ - character table”中获得。 令$ g $为有限的集团，其中一个普通亚组$ h $的索引$ p $，a Prime。在$ 1955 $中，如果$ f $是代数关闭的，则伯曼计算了与$ f [h] $的pci的原始中央式（PCI）的$ f [g] $相对应的$ f [g] $。在本文中，我们给出了该伯曼定理的完整证明，并在$ F $不一定在代数上关闭时扩展了此结果。此外，使用古典Schur的理论和Wedderburn的理论，我们将$ f $ a $ h $的不可还原$ f $ ementation的诱导代表分解为不可约组件。
Let $G$ be a finite group, and $F$ a field of characteristic $0$ or prime to the order of $G$. In $1952$, Witt and in $1956$, Berman independently proved that the number of inequivalent irreducible $F$-representations of $G$ is equal to the number of $F$-conjugacy classes of the elements of $G$, where "$F$-conjugacy" was defined in a certain way. In this paper, we define $F$-conjugacy on $G$ in a natural way and give a proof of the above Witt-Berman theorem. In addition, we give an explicit formula for computing a primitive central idempotent (pci) of the group algebra $F[G]$ corresponding to an irreducible $F$-representation of $G$, which can be obtained from the "$F$-character table" of $G$. Let $G$ be a finite group with a normal subgroup $H$ of index $p$, a prime. In $1955$, in case $F$ is algebraically closed, Berman computed the primitive central idempotent (pci) of $F[G]$ corresponding to an irreducible $F$-representation of $G$, in terms of pci's of $F[H]$. In this paper, we give a complete proof of this Berman's theorem, and extend this result when $F$ is not necessarily algebraically closed. Also, using classical Schur's theory and Wedderburn's theory, we work out decomposition of induced representation of an irreducible $F$-representation of $H$, into irreducible components.
