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

Czechoslovak Mathematical Journal

Lingli Zeng¹  Nan³  Jizhu ⁵ 

[1] (corresponding author), Department of Mathematics, Northwest University, 229 North Taibai Road, Xi'an 710127, Shaanxi, P. , School of Mathematical Sciences, Dalian University of Technology, No. 2 Linggong Road, Dalian 116024, Ganjingzi, Liaoning P. 2 Linggong Road, Dalian 116024, Ganjingzi, Liaoning, P. R. China (current address), and School of Mathematical Sciences, Dalian University of Technology, No. R. China (primary address),;R. China,

关键词: vector invariant ideal;    group algebra;    unitary group;    orthogonal group;   

DOI  :  

学科分类：数学（综合）

来源: Akademie Ved Ceske Republiky

【 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 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.

【 授权许可】

Unknown