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

Abstract

Let F be a finite field of characteristic p and K a field which contains a primitive pth root of unity and char K ≠ 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 V and on the group algebra K[V V], respectively. In this paper we determine the structure of G-invariant ideals of the group algebra K[V V], and establish the relationship between the invariant ideals of K[V] and the vector invariant ideals of K[V 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.