---
tag: G28W7J8G2D
kind: concept
name: Modular representation theory
msc: [20C20, 20C05]
specializes: [WYGRXCG2RN]
sources:
  - url: https://en.wikipedia.org/wiki/Modular_representation_theory
same_as:
  wikidata: Q6889739
  wikipedia: Modular representation theory
---

## Summary

Modular representation theory studies [[representations|WYGRXCG2RN]] of a finite group $G$ over a field whose characteristic $p$ divides the order of $G$.
In this case [[Maschke's theorem|4SBZ4EMPBZ]] fails, so representations need not split into irreducible pieces, and the [[group algebra|MNP2HBWB5Q]] is not semisimple.
Brauer developed the theory, with tools such as Brauer characters, blocks, defect groups and the [[Cartan matrix|ZBQ4KRC98K]].
It links ordinary characters in characteristic $0$ with representations over finite fields through reduction modulo $p$.

## Explanations

### L1 human

A representation lets a group act on a vector space by linear maps.
Over the complex numbers, every representation of a finite group breaks into irreducible pieces, like a number into primes.
Over a field such as $\F_p$, where adding $p$ ones gives zero, this breaking can fail when $p$ divides the size of the group.
For the cyclic group of order $p$ acting on $\F_p^2$ by $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$, the line spanned by the first basis vector is preserved, but no complementary line is.
Modular representation theory is the study of this richer and harder situation.