有限群在特征为零的任意域上的表示的一些注记

Remarks on Representations of Finite Groups over an Arbitrary Field of Characteristic Zero

中文关键词:  $\Gamma_K$-作用  $\Gamma_K$-类  正交关系.

英文关键词:$\Gamma_K$-action  $\Gamma_K$-classes  orthogonality relations.

中文摘要:

      设$G$是一个有限群,$K$是特征为零的域.众所周知,如果$K$是$G$的分裂域,则$G$是交换群当且仅当它的任意$K$-不可约表示是一次的.在本文中,我们将上面这个结果推广到特征为零的任意域$K$的情形(即$K$不必是$G$的分裂域),并且我们也得到了在这种情形下$G$的不可约$K$-特征标的正交关系. 我们的结果推广了一些著名定理.

英文摘要:

      Let $G$ be a finite group and $K$ a field of characteristic zero. It is well-known that if $K$ is a splitting field for $G$, then $G$ is abelian if and only if any irreducible representation of $G$ has degree 1. In this paper, we generalize this result to the case that $K$ is an arbitrary field of characteristic zero (that is, $K$ need not be a splitting field for $G$), and we also obtain the orthogonality relations of irreducible $K$-characters of $G$ in this case. Our results generalize some well-known theorems.