\topmatter  
\title  Fusion algebras and Alperin's weight conjecture 
\endtitle
\rightheadtext{Fusion algebras}
\author Markus Linckelmann \endauthor
\address             \vbox{\hbox{Markus Linckelmann}
                           \hbox{\hskip 3mm CNRS, Universit\'e Paris 7}
                           \hbox{\hskip 3mm UFR Math\'ematiques}
                           \hbox{\hskip 3mm 2, place Jussieu}
                           \hbox{\hskip 3mm 75251 Paris Cedex 05}
                           \hbox{FRANCE}}
\endaddress
\date{June 2002}\enddate

\abstract 
The fusion system $\CF$ of a block $b$ of a finite group $G$ over a suitable $p$
-adic ring $\CO$ 
does not in general determine the
number $l(b)$ of isomorphism classes of simple modules of the block. We show tha
t conjecturally
the missing information should be encoded in a single second cohomology class $\
alpha$ of
the constant functor with value $\CO^\times$ on the orbit category $\bar\CF^c$ o
f centric 
subgroups in $\CF$. By work of Broto, Levi, Oliver [\BLOb], the existence of a c
lassifying
space of the block $b$ is equivalent to the existence of a certain
second cohomology class $\zeta$
of the center functor $\Cal Z$ on $\CF^c$.
If both invariants $\alpha$, $\zeta$ exist 
we show that there is an $\CO$-algebra $\Cal L(b)$
associated with $b$ such that Alperin's weight conjecture becomes now equivalent
to the equality $l(b) = l(\Cal L(b))$. If $b$ has an abelian defect group, 
$\Cal L(b)$ is isomorphic to a source algebra of the Brauer correspondent
of $b$.
\endabstract