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

Zeng, Lingli, and Nan, Jizhu. "Vector invariant ideals of abelian group algebras under the actions of the unitary groups and orthogonal groups." Czechoslovak Mathematical Journal 66.4 (2016): 1059-1078. <http://eudml.org/doc/287525>.

@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 -