This paper has two main parts. First, we give a classification of the ℓ-blocks of finite special linear and unitary groups SL n (ǫq) in the non-defining characteristic ℓ ≥ 3. Second, we describe how the ℓ-weights of SL n (ǫq) can be obtained from the ℓ-weights of GL n (ǫq) when ℓ ∤ gcd(n, q − ǫ), and verify the Alperin weight conjecture for SL n (ǫq) under the condition ℓ ∤ gcd(n, q − ǫ). As a step to establish the Alperin weight conjecture for all finite groups, we prove the inductive blockwise Alperin weight condition for any unipotent ℓ-block of SL n (ǫq) if ℓ ∤ gcd(n, q − ǫ).
We establish the inductive blockwise Alperin weight condition for simple groups of Lie type
$\mathsf C$
and the bad prime
$2$
. As a main step, we derive a labelling set for the irreducible
$2$
-Brauer characters of the finite symplectic groups
$\operatorname {Sp}_{2n}(q)$
(with odd q), together with the action of automorphisms. As a further important ingredient, we prove a Jordan decomposition for weights.
Recently, there has been substantial progress on the Alperin weight conjecture. As a step to establish the Alperin weight conjecture for all finite groups, we prove the inductive blockwise Alperin weight condition for simple groups of classical type under some additional assumption.
The Alperin weight conjecture was reduced to simple groups by the work of Navarro, Tiep and Späth. To prove Alperin weight conjecture, it suffices to show that all finite non-abelian simple groups are BAW-good. We reduce the verification of the inductive conditons for groups of Lie type in non-defining characteristic to quasi-isolated blocks.2010 Mathematics Subject Classification. 20C20, 20C33.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.