| Let G be a finite group acting on a vector space V of dimension n over a finite field F. The induced action on the dual space V* extends to the symmetric algebra S(V*) of the polynomial functions on V which we denote by F[V]. An ideal I (?)F[V] is called a G-invariant (stable) ideal if it is closed under the action of G, i.e., gf ∈ I, Vg ∈ G,f ∈ I. In this thesis we study the structures of the vector G-invariant ideals of group algebra K[V (?) V] in the non-modular case, where K is a field with charK ≠charF. In addition, we determine the structures of the transfer ideals under the actions of orthogonal group, unitary group and dihedral group in the modular case. The main frames are as follows.The first chapter gives a general introduction into invariant theory. In particular, we focus on the background and significance for the G-invariant ideals.In Chapter 2, we restrict our attention to the non-modular case. In this case, the structures of the invariant ideals of group algebras K[V] and K[V (?) V] are considered under the actions of symplectic group, unitary group and orthogonal group, respectively. Furthermore, we establish the relations between the invariant ideals of K[V] and the vector invariant ideals of K[V (?) V]. More precisely, every invariant ideal of K[V] can be obtained by the natural projection of certain vector invariant ideals of K[V (?) V].The Chapter 3 is devoted to study the transfer ideal, which is a special G-invariant ideal in the modular case. By using the matrix method, we prove that there are three types of 2-codimensional invariant subspaces determined by the elements of order p in the orthogonal group O2v(Fq, S). Then the structure of transfer variety is obtained under the action of orthogonal group by Hilbert zero-point theorem. Finally, we determine the primary decomposition of the radical ideal of transfer, the height of transfer ideal and its chains of prime ideals. Similarly, we also present the structure of the transfer ideal for the unitary group Un(Fq2, H).In the last chapter, we study the transfer ideal under the action of dihedral group D2p, and demonstrate that this transfer ideal is a principal ideal generated by xp-1 in the invariant ring Fp[x,y]D2p... |
