Modular Representation Theory and Physics

Modular Representation Theory and Physics: A Homological-Categorical Handbook. The notes build modular representation theory for finite groups over a p-modular system (K,R,k), introduce Brauer characters, blocks, decomposition and Cartan matrices, and analyze defect groups. They connect number-theoretic invariants (Schur indices, Frobenius–Schur indicators) to division algebras and Galois cohomology, then develop homological tools: projective resolutions, Ext/Tor, stable categories, Auslander–Reiten theory, and support varieties via group cohomology. Derived and Rickard equivalences are presented as categorical/physical dualities, with worked examples (Cp, S3, S4, D8, Q8) and a glossary plus exercises linking algebraic structures to topological and quantum-theoretic interpretations.