Vector invariant ideals of abelian group algebras under the actions of the unitary groups and orthogonal groups. (English) Zbl 1397.13013
Summary: Let \(F\) be a finite field of characteristic \(p\) and \(K\) a field which contains a primitive \(p\)th root of unity and \(\text{char}\, K\neq p\). 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 J. Nan and L. Zeng [J. Algebra Appl. 12, No. 8, Article ID 1350046, 12 p. (2013; Zbl 1282.16030)], we solve the problem of vector invariant ideals for all classical groups over finite fields.
MSC:
| 13A50 | Actions of groups on commutative rings; invariant theory |
| 16S34 | Group rings |
| 20C05 | Group rings of finite groups and their modules (group-theoretic aspects) |
Citations:
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