Blocks of profinite groups with cyclic defect group

Abstract

We demonstrate that the blocks of a profinite group whose defect groups are cyclic have a Brauer tree algebra structure analogous to the case of finite groups. We show further that the Brauer tree of a block with defect group ℤ 𝑝 is of star type.M S C ( 2 0 2 0 ) 20C20 (primary), 20E18 (secondary) INTRODUCTIONThe modular representation theory of a finite group 𝐺 may loosely be described as the study of the category of 𝑘𝐺-modules and their relationship with the group 𝐺, where 𝑘 for us will be an algebraically closed field whose characteristic 𝑝 divides the order of 𝐺. A simple but effective approach to modular representation theory is to write 𝑘𝐺 as a direct product of indecomposable algebras, known in the theory as the blocks of 𝐺, and study the representation theory of each block separately. The difficulty of a given block 𝐵 may be measured by a 𝑝-subgroup of 𝐺, called the defect group of 𝐵. The only general family of defect groups whose corresponding blocks are completely understood, is the class of cyclic 𝑝-groups. Blocks with cyclic defect group have an explicit description as so-called 'Brauer tree algebras'. For a clear and encyclopedic discussion of the block theory of finite groups we recommend [10,11].The study of the modular representation theory of profinite groups was begun in [12,13], while the study of blocks and defect groups has been initiated recently in [5]. In this article we classify, in Theorem 4.3, the blocks of an arbitrary profinite group whose defect groups are cyclic (meaning either a finite cyclic 𝑝-group or the 𝑝-adic integers ℤ 𝑝 ): they are Brauer tree algebras in strict analogy with the finite case. Our approach is to use limit arguments and invoke the corresponding theory for finite groups, thus avoiding explicit mention of the most technical arguments required for finite groups. The results are quite striking: a block of the profinite group 𝐺 with finite cyclic defect group has finite dimension (Theorem 4.4) and is thus a block of a finite quotient of 𝐺

Irreducible Characters with Cyclic Anchor Group

“…The converse of the previous result, which states that "if B is a p-block of FG having a simple module with a cyclic vertex then the defect group of B is cyclic", was shown by K. Erdmann in [32]. The p-block with a cyclic defect group is a renewed topic which appeared recently in many articles; some of them can be seen in [38,39]. In this work, our first main objective is to show that the anchor group A ψ of irreducible character ψ of G is cyclic if and only if the defect group of the p-block which contains ψ is cyclic.…”