Explicit tilting complexes for the Broué conjecture on 3-blocks

Summary

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 =⊕ 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.

Broué [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.