[Paper Review] Jordan decomposition for weights and the blockwise Alperin weight conjecture

Zhicheng Feng, Zhenye Li+1 more|arXiv (Cornell University)|Feb 28, 2021

Finite Group Theory Research57 references4 citations

TL;DR

This paper establishes a reduction of the inductive blockwise Alperin weight (BAW) condition for finite simple groups of Lie type in non-defining characteristic to quasi-isolated blocks, using Jordan decomposition of weights and Morita equivalences via Bonnafé–Rouquier theory. The key result shows that if quasi-isolated ℓ-blocks of quasi-simple groups of Lie type satisfy the inductive BAW condition and Assumption 5.3 holds, then the corresponding simple group is BAW-good at ℓ.

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.

Motivation & Objective

• To reduce the verification of the inductive BAW condition for finite simple groups of Lie type in non-defining characteristic to quasi-isolated blocks.
• To establish a Jordan decomposition for weights in the context of blockwise Alperin weight conjecture.
• To extend the Bonnafé–Rouquier equivalence to local subgroups and quotient groups, enabling a natural correspondence of conjugacy classes of weights.
• To verify the inductive BAW condition under Assumption 5.3 and the assumption that quasi-isolated ℓ-blocks satisfy the inductive BAW condition.
• To provide a framework for proving the blockwise Alperin weight conjecture by reducing it to the more tractable case of quasi-isolated blocks.

Proposed method

• Utilizes the Bonnafé–Rouquier equivalence and its extension by Ruhstorfer to establish Morita equivalences between block algebras of normalizer quotients in finite groups of Lie type.
• Applies a local version of the Bonnafé–Rouquier equivalence to ℓ-subgroups Q, constructing Morita equivalences between block algebras of N_G^F(Q)/Q and N_L^F(Q)/Q.
• Establishes a natural, automorphism-equivariant correspondence between conjugacy classes of weights in e_s^L^F and e_s^G^F via the Morita equivalence.
• Employs the criterion of Brough–Späth for the inductive BAW condition, relying on Assumption 5.3 on stabilizers and extendibility of irreducible Brauer characters.
• Uses the unitriangularity of decomposition matrices—known for unipotent characters at good primes via Brunat, Dudas, and Taylor—as a sufficient condition for Assumption 5.3 to hold.
• Applies Theorem 3.15 and Lemma 3.12 to construct blockwise equivariant bijections between Brauer characters and weights, verifying the inductive BAW condition step-by-step.

Experimental results

Research questions

• RQ1 Can the inductive BAW condition for finite simple groups of Lie type in non-defining characteristic be reduced to quasi-isolated blocks?
• RQ2 Does a Jordan decomposition for weights exist in the context of the blockwise Alperin weight conjecture for groups of Lie type?
• RQ3 Can the Bonnafé–Rouquier equivalence be extended to local subgroups and quotient groups to preserve weight conjugacy classes?
• RQ4 Under what conditions does Assumption 5.3 on stabilizers and extendibility of Brauer characters imply the inductive BAW condition?
• RQ5 Is the inductive BAW condition for unipotent blocks of finite groups of Lie type sufficient to imply the full BAW-goodness of the corresponding simple group?

Key findings

• The inductive BAW condition for a finite simple group S of Lie type over a field of characteristic p ≠ ℓ is reduced to the verification of the condition on quasi-isolated ℓ-blocks of its quasi-simple cover.
• A Morita equivalence is established between block algebras of N_G^F(Q)/Q and N_L^F(Q)/Q for any ℓ-subgroup Q, preserving the conjugacy classes of weights.
• The natural correspondence of weights induced by the Morita equivalence is equivariant under the action of automorphism groups, ensuring compatibility with the inductive condition.
• Assumption 5.3—concerning stabilizers and extendibility of Brauer characters—suffices to ensure the inductive BAW condition holds if the quasi-isolated blocks satisfy it.
• The proof confirms that the inductive BAW condition holds for S if all quasi-isolated ℓ-blocks of its quasi-simple cover satisfy the inductive BAW condition and Assumption 5.3 is valid.
• The result provides a uniform reduction strategy for proving the blockwise Alperin weight conjecture, particularly for the 16 infinite families of finite simple groups of Lie type.