Block theory for profinite groups

Abstract

The objective of this work is to study the techniques developed in the theory of blocks for finite groups and then, using the machinery of profinite groups and results from the modular representation theory of profinite groups, to extend the fundamental results of the theory of blocks of finite groups to profinite groups. We are thus interested in studying the block structure of the complete group algebra k[[G]] of a profinite group G, where k is a field of characteristic p. Our approach is as follows. We extend the concepts and fundamental properties of relative projectivity and vertices from profinite k[[G]]-modules to pseudocompact k[[G]]-modules. We introduce the concept of blocks of profinite groups, characterizing a block of a profinite group G as the inverse limit of blocks of finite groups G/N, where N is a open normal subgroup of G. Then we introduce the concept of defect group for a block of a profinite group, developing the basic properties and characterizations of these groups analogous to those existing for the finite case. We demonstrate a version of Brauer’s Correspondence Theorem for virtually pro-p groups. Finally, we study the structure of the blocks of a profinite group with cyclic defect group. We demonstrate that these blocks have a Brauer tree algebra structure analogous to the finite case and we demonstrate that the Brauer trees for these blocks are all star type trees when the cyclic defect group is Zp.