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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 AutF¯(Q) with values in k× in K¨ulshammer-Puig [13, 1.8]. 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 Broto, Levi, Oliver , the existence of a classifying space of the block b is equivalent to the existence of a certain extension category L of Fc 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.
|Subjects:||Q Science > QA Mathematics|
|Divisions:||School of Engineering & Mathematical Sciences > Department of Mathematical Science|
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