Blocks with only one irreducible Brauer character orbit

2026/04/30 by Fuming Jiang, Kun Zhang, Yuanyang Zhou
#math.GR #math.RT

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Abstract

In this paper, we confirm the Kessar-Linckelmann conjecture for blocks with abelian defect groups. More generally, we prove that if a covered block with abelian defect groups has its irreducible Brauer characters forming a single orbit under the block stabilizer, then the covering block is inertial; we term these blocks with a single irreducible Brauer character orbit. Consequently, we establish Broué's abelian defect conjecture for such blocks. As a byproduct, we show that blocks of finite quasisimple groups with a single irreducible Brauer character orbit necessarily have abelian defect groups. As applications, we verify the blockwise Alperin weight conjecture for blocks with a unique irreducible Brauer character and prove Puig's long-standing conjecture in full generality. All results rely essentially on the Classification of finite simple groups.

Citations

• On stable equivalences with endopermutation source and Külshammer--Puig classes
• Jordan correspondence and block distribution of characters
• The Alperin Weight Conjecture and the Glauberman correspondence via character triples
• Brauer's Height Zero Conjecture
• Morita equivalences and the inductive blockwise Alperin weight condition for type \mathsf A
• Block extensions, local categories, and basic Morita equivalences
• Finite simple groups with some abelian Sylow subgroups
• On Blocks with One Modular Character
• The inductive Alperin-McKay and blockwise Alperin weight conditions for blocks with cyclic defect groups
• 2-blocks with abelian defect groups
• Quasi-isolated blocks and Brauer's height zero conjecture
• Conjectures of Alperin and Broue for 2-blocks with elementary abelian defect groups of order 8
• Nilpotent extensions of blocks
• On the Decomposition Numbers of the Ree Groups 2F4(q²) in Non-Defining Characteristic
• Blocks with Equal Height Zero Degrees
• Quasi-isolated elements in reductive groups

Related

• Blocks with abelian defect groups of rank 2 and one simple module
• Cartan matrices and Brauer's k(B)-Conjecture III
• Cartan matrices and Brauer's k(B)-Conjecture IV
• Brauer's problem 21 for principal blocks
• The blocks with four irreducible characters