Extensions of tame algebras and finite group schemes of domestic representation type

TL;DR

This paper investigates how polynomial growth in representation theory behaves under algebra extensions and characterizes certain blocks of finite group schemes of odd characteristic as having domestic representation type, linking them to tame hereditary algebras.

Contribution

It provides criteria for the preservation of polynomial growth under algebra extensions and characterizes principal blocks of finite group schemes of odd characteristic as domestic, relating them to tame hereditary algebras.

Findings

Polynomial growth is preserved under certain algebra extensions.

Principal blocks of finite group schemes of odd characteristic are of polynomial growth iff they are Morita equivalent to trivial extensions of tame hereditary algebras.

Such blocks are of domestic representation type and linked to binary polyhedral group schemes.

Abstract

Let k be an algebraically closed field. Given an extension A : B of finite-dimensional k- algebras, we establish criteria ensuring that the representation-theoretic notion of polynomial growth is preserved under ascent and descent. These results are then used to show that principal blocks of finite group schemes of odd characteristic are of polynomial growth if and only if they are Morita equivalent to trivial extensions of radical square zero tame hereditary algebras. In that case, the blocks are of domestic representation type and the underlying group schemes are closely related to binary polyhedral group schemes.