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

Zeng, Lingli ; Nan, Jizhu

Czechoslovak Mathematical Journal, Tome 66 (2016) no. 4, pp. 1059-1078.

Résumé

Let $F$ be a finite field of characteristic $p$ and $K$ a field which contains a primitive $p$th root of unity and $\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}K\ne p$. 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.

Détail

DOI : 10.1007/s10587-016-0310-x

Classification : 16S34, 20G40
Mots-clés : vector invariant ideal; group algebra; unitary group; orthogonal group