Vector invariant ideals of abelian group algebras under the actions of the unitary groups and orthogonal groups

摘要

Let $F$ be a finite field of characteristic $p$ and $K$ a field which contains a primitive $p$th root of unity and $ char Keq p$. Suppose that a classical group $G$ acts on the $F$-vector space $V$. Then it can induce the actions on the vector space $Voplus V$ and on the group algebra $K[Voplus V]$, respectively. In this paper we determine the structure of $G$-invariant ideals of the group algebra $K[Voplus V]$, and establish the relationship between the invariant ideals of $K[V]$ and the vector invariant ideals of $K[Voplus V]$, 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.