Lecture 2- Modular Representations and Brauer Characters

Introduces modular systems (K, O, k), defines p-regular vs p-singular elements, and develops Brauer characters (defined on p-regular classes) with the Brauer–Nesbitt theorem. Explains reduction mod p, the decomposition matrix D linking ordinary and Brauer characters, and that the number of Brauer irreducibles equals the number of p-regular conjugacy classes. Worked examples include S₃ at p=3 and the kernel phenomenon in passing from ordinary to modular characters; a preview points to Frobenius actions and field-of-definition issues next.