Modular representation theory

Summary

Modular representation theory studies representations of a finite group $G$ over a field whose characteristic $p$ divides the order of $G$. In this case Maschke's theorem fails, so representations need not split into irreducible pieces, and the group algebra is not semisimple. Brauer developed the theory, with tools such as Brauer characters, blocks, defect groups and the Cartan matrix. It links ordinary characters in characteristic $0$ with representations over finite fields through reduction modulo $p$.