Abstract

The Broué conjecture, that a block with abelian defect group is derived equivalent to its Brauer correspondent, has been proven for blocks of cyclic defect group and verified for many other blocks, mostly with defect group C3 × C3 or C5 × C5. In this paper, we exhibit explicit tilting complexes from the Brauer correspondent to the global block B for a number of Morita equivalence classes of blocks of defect group C3 × C3. We also describe a database with data sheets for over a thousand blocks of abelian defect group in the ATLAS group and their subgroups.

Introduction

Let G be a finite group and let k be a field of characteristic p, where p divides |G|. Let kG =⊕ Bi be a decomposition of the group algebra into blocks, and let Di be the defect group of the block Bi, of order. By Brauer's Main Theorems (see [1] for an accessible exposition) there is a one-to-one correspondence between blocks of kG with defect group Di and blocks of kNG(Di) with defect group Di. Let bi be the block corresponding to Bi, called its Brauer correspondent.

Broué [5] has conjectured that if Di is abelian and Bi is a principal block, then Bi and bi are derived equivalent, i.e., the bounded derived categories Db(Bi) and Db(bi) are equivalent. In fact, it is generally believed by researchers in the field that the hypothesis that Bi be principal is unnecessary.