Remarks on Representations of Finite Groups over an Arbitrary Field of Characteristic Zero |
Received:April 17, 2009 Revised:September 15, 2009 |
Key Words:
$\Gamma_K$-action $\Gamma_K$-classes orthogonality relations.
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Fund Project:Supported by the National Natural Science Foundation of China (Grant No.10771132) and the Natural Science Foundation of Shandong Province (Grant No.Y2008A03). |
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Abstract: |
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. |
Citation: |
DOI:10.3770/j.issn:1000-341X.2011.03.007 |
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