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

@article{Zeng2016,
abstract = {Let $F$ be a finite field of characteristic $p$ and $K$ a field which contains a primitive $p$th root of unity and $\{\rm char\} K\ne p$. 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.},
author = {Zeng, Lingli, Nan, Jizhu},
journal = {Czechoslovak Mathematical Journal},
keywords = {vector invariant ideal; group algebra; unitary group; orthogonal group; vector invariant ideal; group algebra; unitary group; orthogonal group},
language = {eng},
number = {4},
pages = {1059-1078},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Vector invariant ideals of abelian group algebras under the actions of the unitary groups and orthogonal groups},
url = {http://eudml.org/doc/287525},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Zeng, Lingli
AU - Nan, Jizhu
TI - Vector invariant ideals of abelian group algebras under the actions of the unitary groups and orthogonal groups
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 4
SP - 1059
EP - 1078
AB - Let $F$ be a finite field of characteristic $p$ and $K$ a field which contains a primitive $p$th root of unity and ${\rm char} K\ne p$. 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.
LA - eng
KW - vector invariant ideal; group algebra; unitary group; orthogonal group; vector invariant ideal; group algebra; unitary group; orthogonal group
UR - http://eudml.org/doc/287525
ER -