Modular Representations — Mathematics Knowledge System

Overview

Modular representation theory studies representations of finite groups (or algebraic groups) over fields of prime characteristic p. When p divides the group order, Maschke's theorem fails — the group algebra is no longer semisimple — and rich new phenomena emerge: indecomposable but non-irreducible modules, Brauer characters, defect groups, and blocks. The theory, initiated by Richard Brauer in the 1930s–40s, has deep connections to number theory, algebraic K-theory, and the classification of finite simple groups.

Intuition

In ordinary representation theory (characteristic 0), group algebras are semisimple: everything breaks into irreducibles. Over a field of prime characteristic p dividing |G|, some representations cannot be completely decomposed — they have 'trapped' composition factors that cannot be split apart. Brauer's theory uses modular arithmetic to extract information about ordinary representations from these obstructed ones.

Formal Definition

Definition

Let G be a finite group and k an algebraically closed field of characteristic p > 0. The group algebra kG is the k-algebra with basis G and multiplication extended linearly from the group multiplication.

kG = \left\{\sum_{g \in G} a_g g : a_g \in k\right\}

Group algebra over k

kG \text{ is semisimple} \iff p \nmid |G| \quad \text{(Maschke's theorem)}

Maschke's theorem

kG \cong \prod_i B_i \quad \text{(block decomposition as two-sided ideals)}

Block decomposition

\varphi_M(g) = \sum_{\lambda \in k^\times} m_\lambda \hat{\lambda}, \quad g \text{ p-regular}

Brauer character of a module M

Notation

Notation	Meaning
$kG$	Group algebra of G over field k
$p$	Prime characteristic of the field k
$\varphi$	Brauer character (on p-regular conjugacy classes)
$d_{\chi\varphi}$	Decomposition number: multiplicity of φ in reduction of ordinary χ
$D(B)$	Defect group of a block B

Properties

Maschke's theorem

kG \text{ is semisimple} \iff \gcd(|G|, \operatorname{char} k) = 1

Number of simple modules

\text{The number of simple } kG\text{-modules equals the number of } p\text{-regular conjugacy classes of } G

Brauer reciprocity

d_{\chi\varphi} = [P_\varphi : V_\chi] \quad \text{(decomposition matrix = Cartan matrix connection)}

Block orthogonality

\text{Modules in different blocks have no non-trivial extensions between them}

Worked Examples

1
G = ℤ/2ℤ = {1, g} with g² = 1. k = 𝔽₂ has characteristic 2 = |G|.
$k = \mathbb{F}_2,\quad |G| = 2,\quad \operatorname{char} k = 2$
2
By Maschke's theorem, kG is semisimple iff char k does not divide |G|. Since 2 | 2, kG is NOT semisimple.
$2 \mid |G| \implies kG \text{ not semisimple}$
3
Concretely: 𝔽₂[ℤ/2ℤ] ≅ 𝔽₂[x]/(x²-1) = 𝔽₂[x]/(x-1)² (since x²-1=(x-1)² over 𝔽₂). The ideal (x-1) is a non-trivial nilpotent ideal.
$\mathbb{F}_2[x]/(x+1)^2$

✓ Answer

𝔽₂[ℤ/2ℤ] is a local ring with a unique simple module and a non-split extension — not semisimple.

Practice Problems

Hardfree response

Let G = ℤ/pℤ and k = 𝔽p. Show that kG has a unique simple module and classify all indecomposable kG-modules.

Common Mistakes

Common Mistake

Assuming Weyl's complete reducibility theorem applies in prime characteristic

Complete reducibility fails when char k divides |G|. Even for semisimple algebraic groups in characteristic p, representations can be indecomposable but reducible.

Common Mistake

Confusing decomposition numbers with composition multiplicities

Decomposition numbers d_{χφ} count how many times the reduction mod p of an ordinary irreducible χ has the modular irreducible φ as a composition factor — not as a direct summand.

Quiz

Maschke's theorem states that kG is semisimple when:

k has characteristic 0G is abelianchar(k) does not divide |G|G is a p-group

In modular representation theory, Brauer characters are defined on:

All conjugacy classesOnly p-singular elementsOnly p-regular elementsOnly the identity element

Historical Background

Richard Brauer developed modular representation theory systematically in a series of papers from 1935 to 1956, introducing Brauer characters, blocks, and the theory of defect groups. His three 'main theorems' (1944, 1956, 1959) relating ordinary and modular characters became cornerstones of the subject. The local-global philosophy — relating global representation theory to local data at a prime p — continues to drive research today.

1935
Brauer introduces modular characters to study ordinary characters
Richard Brauer
1940
Brauer defines blocks and defect groups
Richard Brauer
1956
Brauer's second main theorem on blocks and local subgroups
Richard Brauer
1970s
Alperin formulates his weight conjecture connecting blocks and local structure
Jonathan Alperin

Summary

Over a field of characteristic p dividing |G|, the group algebra kG is not semisimple — Maschke's theorem fails.
Brauer characters are the modular analogue of ordinary characters, defined on p-regular conjugacy classes.
Simple kG-modules correspond to p-regular conjugacy classes; their number equals the number of p-regular classes.
Decomposition numbers relate ordinary and modular characters; the defect group of a block controls its complexity.

References

BookAlperin, J.L. — Local Representation Theory (1986), Cambridge University Press
BookSerre, J.-P. — Linear Representations of Finite Groups (1977), §15–18
WebsiteWikipedia — Modular representation theory