Graded tame blocks of group algebras

TL;DR

This paper classifies gradings on tame dihedral blocks of group algebras, computes automorphism groups fixing simple modules, and explores grading transfer via derived equivalences.

Contribution

It provides a complete classification of gradings on dihedral blocks up to graded Morita equivalence and details methods for transferring gradings.

Findings

Classified gradings on dihedral blocks up to graded Morita equivalence

Computed the group of outer automorphisms fixing simple modules

Demonstrated grading transfer via derived equivalences

Abstract

In this paper we investigate gradings on tame blocks of group algebras whose defect group is dihedral. We classify gradings on an arbitrary dihedral block up to graded Morita equivalence. We do this by computing the group of outer automorphisms that fix the isomorphism classes of simple modules. We also show how to grade these blocks via transfer of gradings via dervied equivalences.