Splendid Morita equivalences for principal 2-blocks with dihedral defect groups

TL;DR

This paper classifies splendid Morita equivalence classes of principal 2-blocks with dihedral defect groups, constructing explicit equivalences and computing decomposition numbers to deepen understanding of modular representation theory.

Contribution

It provides a complete classification of splendid Morita equivalences for principal 2-blocks with dihedral defect groups, including explicit constructions and criteria for Morita equivalences.

Findings

Classification of equivalence classes achieved

Explicit stable Morita equivalences constructed

Generalized decomposition numbers computed

Abstract

Given a dihedral $2$-group $P$ of order at least~8, we classify the splendid Morita equivalence classes of principal $2$-blocks with defect groups isomorphic to $P$. To this end we construct explicit stable equivalences of Morita type induced by specific Scott modules using Brauer indecomposability and gluing methods; we then determine when these stable equivalences are actually Morita equivalences, and hence automatically splendid Morita equivalences. Finally, we compute the generalised decomposition numbers in each case.