Morita equivalence classes of $2$-blocks with abelian defect groups of rank $4$

TL;DR

This paper classifies all 2-blocks with abelian defect groups of rank 4 up to Morita equivalence, confirming Broué's conjecture for these cases and providing a comprehensive understanding of their structure.

Contribution

It provides a complete classification of 2-blocks with abelian defect groups of rank 4 up to Morita equivalence, extending to both algebraically closed fields and discrete valuation rings.

Findings

All such blocks are classified up to Morita equivalence.

Broué's abelian defect group conjecture is verified for these blocks.

The classification applies over various algebraic settings.

Abstract

We classify all $2$-blocks with abelian defect groups of rank $4$ up to Morita equivalence. The classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field. An application is that Brou\'{e}'s abelian defect group conjecture holds for all blocks under consideration here.