Fusion category algebras

Abstract

The fusion system F on a defect group P of a block b of a finite group G over a suitable p -adic ring O 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 F c of F -centric subgroups Q of P of b which “glues together” the second cohomology classes α ( Q ) of Aut F (Q) 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 F (b) such that Alperin's weight conjecture becomes equivalent to the equality l(b)=l( F (b)) . 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 L of F c by the center functor Z . If both invariants α , L exist, we show that there is an O -algebra L (b) associated with b having F (b) as quotient such that Alperin's weight conjecture becomes again equivalent to the equality l(b)=l( L (b)) ; furthermore, if b has an abelian defect group, L (b) is isomorphic to a source algebra of the Brauer correspondent of b .