$K$-rigidity in modular representation theory.

Abstract

The theory of $K$-rigidity aims to study representation rings (a.k.a `$K$-rings') up to a `deformation of the tensor product', in a certain sense. Calculating representation rings for a group of tame or finite representation type over some field is a difficult problem, and if a group is of wild representation type over some field, it is a quantifiably untenable problem. For every nonabelian 2-group of tame representation type over a field of characteristic $2$, calculating the ring of representations remains an open problem. By studying which calculations hold `up to a deformation of the tensor product', one gains deeper insight into the role of certain homological tools in said calculations. In this thesis, a relationship between representation type and representation ring calculations is established for certain families of finite group schemes. Those of finite representation type are shown to have `total $K$-rigidity', meaning that upon any deformation of their tensor product, the ring of representations is unchanged. Those of tame representation type are shown to have the more delicate property of `noble $K$-rigidity', meaning that they obey a geometric criterion for which tensor products of representations are unchanged in their isomorphism class, upon deformation. Those of wild representation type are shown to be `not noble $K$-rigid', so that even under nice geometric conditions, one can still produce representations whose tensor product changes isomorphism class upon deformation.