The blocks and weights of finite special linear and unitary groups

Abstract

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).

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

“…For unipotent blocks Feng has established in [Fen18] the inductive BAW condition for unipotent blocks and Li-Zhang have treated in [LZ18] other blocks under additional assumptions on the outer automorphism group. 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.…”

Unitriangular basic sets, Brauer characters and coprime actions

“…The parametrization of IBr( G) is given in [KT09] for ǫ = 1, called the admissible symbols, while the case ǫ = −1 was considered in [D17]. Note that an exposition can be found in [F19,Rmk. 3.4].…”

On the inductive blockwise Alperin weight condition for classical groups

“…To the present, the (iBAW) condition has been verified for some cases, such as simple alternating groups, many of the sporadic groups, simple groups of Lie type in the defining characteristic, Suzuki groups and Ree groups, simple groups of type G 2 and 3 D 4 , some cases of blocks of groups of type A; see for instance [8], [13], [17], [21], [38], [42], [52] and [54]. Unfortunately, it still seems a far way to deal with the case of simple groups of Lie type in non-defining characteristic in general.…”

“…In this sense, the unipotent blocks would play a fundamental and important role when considering the (iBAW) condition for finite quasi-simple groups of Lie type. In [21], the author verified the (iBAW) condition for unipotent blocks of groups of type A, untwisted or twisted, under some additional assumption on the prime involved. Considering classical type, the following is our main result for unipotent blocks.…”

“…This follows by [32, Lem. 3.7 and 3.8] (see[21, Prop. 4.5] for details).Note that we haveL = Z(G)L in Proposition 5.6.…”