Modular Representation Theory- Block Idempotents, Central Characters, and Decompositions

These notes develop a structural, computation-ready view of modular representation theory through the lens of the group algebra’s center. Beginning with class sums and the decomposition of Z(FG) into block subalgebras, they define block idempotents and central characters and show that blocks can be characterized equivalently via reduced central characters, connectivity in the Brauer graph, or shared modular composition factors. The Jacobson radical of Z(FG) is identified as the intersection of kernels of blockwise central character maps, with nilpotence criteria for detection. The text then proves the block decomposition FG = ⊕B FGeB, analyzes projective indecomposables and the Cartan matrix, and gives properties and examples (e.g., S3 at p = 3). A final part introduces defect groups, Brauer correspondence, vertices/sources, and Green correspondence, linking local subgroup structure to global block invariants and providing actionable steps for computations.