On the invariant theory of finite unipotent groups generated by bireflections

Let k be a field of characteristic p and let V be a k-vector space. In Chapter 2 of this thesis we classify all unipotent groups G ? GL(V ) consisting of bireflections for p not equal to 2: we show that unipotent groups consisting of bireflections are either two-row groups, two-column groups, hook groups or one of two types of exceptional group.

The well known theorem of Chevalley-Shephard-Todd shows the importance of (pseudo-)reflection groups to invariant theory. Our interest in bireflection groups is motivated by the theorem of Kemper which tells us if G ? GL(V ) is a p-group and the invariant ring k[V ] G is Cohen-Macaulay then G is generated by bireflections. We use our classification to investigate which groups consisting of bireflections have Cohen-Macaulay or complete intersection invariant rings.

In Chapter 3 we introduce techniques and notation which we use later to find invariant rings of groups by viewing them as subgroups of Nakajima groups. In Chapter 4 we show that for k = Fp there is a family of hook groups, including all non-abelian hook groups, which have complete intersection invariant rings.

It is well known that Cohen-Macaulay invariant rings of p-groups in characteristic p are Gorenstein. There has been speculation by experts in the area, that they might in fact be complete intersections. In Chapter 5 we settle this negatively by giving an example of a p-group which has Cohen-Macaulay but non complete intersection invariant ring. To the best of our knowledge this is the first example of that kind.

Finally in Chapter 6 we show that when k = F_p both types of exceptional group have complete intersection invariant rings.