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SOURCE ALGEBRAS AND SOURCE MODULES J. L. Alperin, Markus Linckelmann, Rapha el Rouquier
 

Summary: SOURCE ALGEBRAS AND SOURCE MODULES
J. L. Alperin, Markus Linckelmann, Rapha el Rouquier
May 1998
The aim of this note is to give a self-contained approach in module theoretic
terms to two fundamental results in block theory, due to L. Puig [6, 14.6]: rst,
there is an embedding of the source algebra of the Brauer correspondent of a block
of some nite group into a source algebra of that block, and second, the source
algebras of the Brauer correspondent can be described explicitly. Our proof of the
rst result - Theorem 5 and its Corollary 6 below - is essentially the translation
to our terminology of the proof in [1, 4.10].
The second result, describing the source algebras of the Brauer correspondent,
follows also from [3], but the proofs in [3] use both the main result on the structure
of nilpotent blocks in [5] as well as techniques from [6, 4, 6], while our approach -
from Proposition 9 onwards - requires only some classical results on the structure
of blocks with a central defect group.
An account of the results in [6] on source algebras of blocks with a normal defect
group can also be found in [7, 45], except for the explicit description of the central
extension ^
E (see Section 10 below) as being de ned in terms of the multiplicity
algebra of the block, which is quoted in [7, (45.10)(b)] without proof, referring

  

Source: Alperin, Jon L. - Department of Mathematics, University of Chicago

 

Collections: Mathematics