Lanne's \textbf{T}-functor and Hypersurfaces

Abstract:

      Through discussing the transformation of the invariant ideals, we firstly prove that the \textbf{T}-functor can only decrease the embedding dimension in the category of unstable algebras over the Steenrod algebra. As a corollary we obtain that the \textbf{T}-functor preserves the hypersurfaces in the category of unstable algebras. Then with the applications of these results to invariant theory, we provide an alternative proof that if the invariant of a finite group is a hypersurface, then so are its stabilizer subgroups. Moreover, by several counter-examples we demonstrate that if the invariants of the stabilizer subgroups or Sylow $p$-subgroups are hypersurfaces, the invariant of the group itself is not necessarily a hypersurface.