Jordan decomposition for weights and the blockwise Alperin weight conjecture

Abstract

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.

The Inductive Blockwise Alperin Weight Condition for Type and the Prime

“…Thus by the arguments above (similar as [23,Thm. 3.16]), there exists a blockwise Aequivariant bijection Ω : IBr( B) → Alp( B).…”

“…For the local situation, Ruhstorfer [48] gave an equivariant Morita equivalence between certain blocks of N G (Q) and N N ′ (Q). Using the similar methods of [23], we show that this also induces a Morita equivalence between the corresponding blocks of the quotient groups…”

“…In this section, we recollect some notation and concepts related to the representation theory of finite groups, which mainly follow those used in [8], [46] and [23]. Throughout, we fix a prime number ℓ and consider modular representations with respect to ℓ.…”

“…When focusing on groups of type A, we let G = SL n (F p ) so that G F = G = SL n (ǫq). In [23], the authors reduced the inductive BAW condition to the so-called strictly quasi-isolated blocks, and we may assume that ℓ ∤ q − ǫ. Now let s be strictly quasi-isolated, i.e., C G * F (s) C • G * (s) is not contained in a proper Levi subgroup of G * .…”

“…Thus, to give an equivariant bijection between IBr(G, e G s ) and Alp(G, e G s ), it suffices to establish such bijection between IBr(N ′ , e L ′ F s ) and Alp(N ′ , e L ′F s ). We improve the methods in the proof of [23,Thm. 1], and reduce the verification of the inductive BAW condition in type A to isolated blocks.…”