Rings of Invariants which are Hypersurfaces

Abstract

As we have seen, we seek to characterize those representations V of groups G whose rings of invariants are well-behaved. The best behaved rings of invariants, $\mathbb{K}[V]^{G}$, are those which are polynomial rings, that is, $\mathbb{K}[V]^{G}$ is generated by dim (V) many invariants. A slightly less well behaved class of examples is provided by those rings of invariants which are hypersurfaces, that is, $\mathbb{K}[V]^{G}$ is generated by dim (V)+1 many invariants. Those representations with this property have been extensively studied in characteristic 0 by Nakajima (1983). Less is known for modular groups. When G is a Nakajima group with maximal proper subgroup H, the following proposition shows that the ring of H-invariants is a hypersurface (or a polynomial) ring.