summary:Let $F$ be a finite field of characteristic $p$ and $K$ a field which contains a primitive $p$th root of unity and ${\rm char} K\neq p$. Suppose that a classical group $G$ acts on the $F$-vector space $V$. Then it can induce the actions on the vector space $V\oplus V$ and on the group algebra $K[V\oplus V]$, respectively. In this paper we determine the structure of $G$-invariant ideals of the group algebra $K[V\oplus V]$, and establish the relationship between the invariant ideals of $K[V]$ and the vector invariant ideals of $K[V\oplus V]$, if $G$ is a unitary group or orthogonal group. Combining the results obtained by Nan and Zeng (2013), we solve the problem of vector invariant ideals for all classical groups over finite fields.
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机译：摘要：让$ F $为特征$ p $的有限域，而$ K $为包含原始$ p $ th统一根和$ {\ rm char} K \ neq p $的域。假设经典群$ G $作用于$ F $-向量空间$ V $。然后，它可以在向量空间$ V \ oplus V $和组代数$ K [V \ oplus V] $上引发作用。在本文中，我们确定群代数$ K [V \ oplus V] $的$ G $不变理想的结构，并建立$ K [V] $不变理想与$向量不变理想之间的关系。如果$ G $是a群或正交群，则K [V \ oplus V] $。结合Nan和Zeng（2013）的结果，我们解决了有限域上所有经典群的向量不变理想的问题。
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