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Vector invariant ideals of abelian group algebras under the actions of the unitary groups and orthogonal groups
Czechoslovak Mathematical Journal, Tome 66 (2016) no. 4, pp. 1059-1078
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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.
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.
DOI : 10.1007/s10587-016-0310-x
Classification : 16S34, 20G40
Keywords: vector invariant ideal; group algebra; unitary group; orthogonal group 
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Zeng, Lingli; Nan, Jizhu. Vector invariant ideals of abelian group algebras under the actions of the unitary groups and orthogonal groups. Czechoslovak Mathematical Journal, Tome 66 (2016) no. 4, pp. 1059-1078. doi: 10.1007/s10587-016-0310-x

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