Hussain, Fawad
(2011)
Homological properties of invariant rings of finite groups.
PhD thesis, University of Glasgow.
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Abstract
Let $V$ be a nonzero finite dimensional vector space over the finite field $F_q$. We take the left action of $G \le GL(V)$ on $V$ and this induces a right action of $G$ on the dual of $V$ which can be extended to the symmetric algebra $F_q[V]$ by ring automorphisms. In this thesis we find the explicit generators and relations among these generators for the ring of invariants $F_q[V]^G$. The main body of the research is in chapters 4, 5 and 6. In chapter 4, we consider three subgroups of the general linear group which preserve singular alternating, singular hermitian and singular quadratic forms respectively, and find rings of invariants for these groups. We then go on to consider,
in chapter 5, a subgroup of the symplectic group. We take two special cases for this subgroup. In the first case we find the ring of invariants for this group. In the second
case we progress to the ring of invariants for this group but the problem is still open. Finally, in chapter 6, we consider the orthogonal groups in even characteristic. We
generalize some of the results of [24]. This generalization is important because it will help to calculate the rings of invariants of the orthogonal groups over any finite field of even characteristic.
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