Fusion category algebras

The fusion system on a defect group P of a block b of a finite group G over a suitable p-adic ring does not in general determine the number l(b) of isomorphism classes of simple modules of the block. We show that conjecturally the missing information should be encoded in a single second cohomology class α of the constant functor with value k× on the orbit category of -centric subgroups Q of P of b which “glues together” the second cohomology classes α(Q) of with values in k× in Külshammer–Puig [Invent. Math. 102 (1990) 17–71]. We show that if α exists, there is a canonical quasi-hereditary k-algebra such that Alperinʹs weight conjecture becomes equivalent to the equality . By work of C. Broto, R. Levi, B. Oliver [J. Amer. Math. Soc. 16 (2003) 779–856], the existence of a classifying space of the block b is equivalent to the existence of a certain extension category of by the center functor . If both invariants α, exist, we show that there is an -algebra associated with b having as quotient such that Alperinʹs weight conjecture becomes again equivalent to the equality ; furthermore, if b has an abelian defect group, is isomorphic to a source algebra of the Brauer correspondent of b.