Classifying blocks with abelian defect groups of rank $3$ for the prime $2$

TL;DR

This paper classifies all blocks with a specific abelian defect group of rank 3 for prime 2 up to Morita equivalence, completing the classification for such blocks of order dividing 16 and confirming Broue's conjecture for them.

Contribution

It provides a complete Morita classification for blocks with defect group $C_{2^n} imes C_2 imes C_2$, including novel reduction techniques for cases involving a subgroup of index 2.

Findings

Complete classification of blocks with defect group $C_{2^n} imes C_2 imes C_2$

Verification of Broue's abelian defect group conjecture for these blocks

Development of new reduction methods for Morita equivalence analysis

Abstract

In this paper we classify all blocks with defect group $C_{2^n}\times C_2\times C_2$ up to Morita equivalence. Together with a recent paper of Wu, Zhang and Zhou, this completes the classification of Morita equivalence classes of $2$-blocks with abelian defect groups of rank at most $3$. The classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field. The case considered in this paper is significant because it involves comparison of Morita equivalence classes between a group and a normal subgroup of index $2$, so requires novel reduction techniques which we hope will be of wider interest. We note that this also completes the classification of blocks with abelian defect groups of order dividing $16$ up to Morita equivalence. A consequence is that Broue's abelian defect group conjecture holds for all blocks mentioned above.