Let G be a finite group, let A be a block algebra of the group algebra RG that has an abelian defect group D, and let B be the Brauer correspondent of A, a block algebra of RN_{G}(D), the group algebra of the normalizer of D in G.
Then Broué conjectured in the 1990 paper [Bro2] that
Broué's Abelian Defect Group Conjecture |
The derived categories D^{b}(mod-A) and D^{b}(mod-B) of bounded complexes of finitely generated modules for A and B are equivalent as triangulated categories. |
If this is true, it has many consequences. For example:
The character-theoretic consequences are subsumed by the statement that there is a `perfect isometry' between the blocks. This means an isometry between the groups of virtual characters for the two blocks that preserves the subgroups spanned by characters of projective modules. An isometry amounts to a bijection `with signs' between the sets of irreducible ordinary characters; the other condition can be restated as an arithmetical condition about character values.