The inductive blockwise Alperin weight condition for G2(q) and D43(q)

Abstract

The inductive blockwise Alperin weight condition is a system of conditions whose verification for all non-abelian finite simple groups would imply the blockwise Alperin weight conjecture. We establish this condition for the groups G 2 (q), q 5, and 3 D 4 (q) for all primes dividing their order.

On the Alperin-McKay conjecture for simple groups of type A

“…Also Li constructed in [Li18] an equivariant bijection for the inductive BAW condition in symplectic groups under some assumption on the ℓ-modular decomposition matrix. More particular cases of simple groups of small rank were checked in [FLL17,Sch16,SF14].…”

On the inductive blockwise Alperin weight condition for the unipotent blocks of finite groups of Lie type E_6

“…Du), pcli17@pku.edu.cn (P. Li), zsy0509@pku.edu.cn (S. Zhao). groups, simple groups of type 2 F 4 , see [37]; simple groups of Lie type in the defining characteristic, simple groups of type 2 B 2 and 2 G 2 , see [41]; simple groups of type G 2 and 3 D 4 , see [40]; some cases of type A, see [16], [17] and [19]; the case of type C under the assumption that the decomposition matrix with respect to E(G, l ′ ) is unitriangular, see [20] and [29]; unipotent blocks of classical groups, see [18].…”

“…We say that the inductive blockwise Alperin weight (iBAW) condition holds for S and the l if the (iBAW) condition holds for every l-block of X. Lemma 2.1. ( [40,Lemma 2.11]) Let l be a prime, S a finite non-abelian simple group and X the universal l ′ -covering group of S . Let B be an l-block of X.…”

The Inductive Blockwise Alperin Weight Condition for Type and the Prime

“…The reduction gives hope that the Alperin weight conjecture can be proved by use of the classification of the finite simple groups. In several cases the inductive BAW condition has been verified: simple alternating groups (Malle [36]), groups of Lie type in their defining characteristic (Späth [51]), Suzuki and Ree groups (Malle [36]), groups of types G 2 and 3 D 4 (Schulte [49]), groups of type C at the prime 2 (Feng-Malle [24]), groups of type F 4 at odd primes (An-Hiß-Lübeck [6]). Under the unitriangularity assumption of the decomposition matrices, the inductive BAW condition was verified for groups of type B (Feng-C. Li-Zhang [22]) and type C (C. Li [33]) at odd primes.…”