 Modular representation theory

Modular representation theory studies representations of a finite group $G$ over a field $k$ whose characteristic divides $|G|$. In that case the averaging argument of Maschke's theorem cannot divide by $|G|$, so invariant subspaces need not have invariant complements.

For example, a generator of $C_p$ can act over ${\mathbb{F}}_p$ by $J=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$, since $J^p=I$. The first coordinate line is invariant but has no invariant complementary line. This forces the study of indecomposable modules, projective covers, radicals, and blocks, beyond the irreducible decomposition sufficient in characteristic zero.

Modular representation theory is a branch of representation theory that deals with the study of algebraic structures, particularly groups, over fields with finite characteristic. This area of mathematics arises in various contexts, particularly in the representation theory of finite groups and modular forms in algebra. Here's a breakdown of key concepts in modular representation theory: 1. **Representation Theory**: This is the study of how algebraic structures (like groups, rings, or algebras) can be represented through matrices and linear transformations.