Explicit tilting complexes for the Broué conjecture on 3-blocks

Abstract

Abstract The Broue 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 C 3 × C 3 or C 5 × C 5 . 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 C 3 × C 3. 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 =⊕ B i be a decomposition of the group algebra into blocks, and let D i be the defect group of the block B i , 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 D i and blocks of kNG(D i ) with defect group D i . Let b i be the block corresponding to B i , called its Brauer correspondent. Broue [5] has conjectured that if D i is abelian and B i is a principal block, then B i and b i are derived equivalent, i.e., the bounded derived categories D b (B i ) and D b (b i ) are equivalent. In fact, it is generally believed by researchers in the field that the hypothesis that B i be principal is unnecessary.