Group Representations — Mathematics Knowledge System

Overview

A representation of a group G is a homomorphism ρ: G → GL(V) from G into the group of invertible linear maps on a vector space V. Representation theory translates abstract group-theoretic problems into linear algebra, where powerful computational and structural tools are available. It has applications in quantum mechanics (symmetry groups of physical systems), number theory (Galois representations), combinatorics (symmetric group representations), and harmonic analysis.

Intuition

A representation realises an abstract group as a group of matrices. Every symmetry of a geometric or physical system defines a representation: rotating a vector in ℝ³ gives a representation of SO(3); permuting basis vectors gives a representation of the symmetric group. Studying the representation tells you about both the group and the object it acts on.

Formal Definition

Definition

Let G be a group and V a vector space over a field k. A representation of G on V is a group homomorphism ρ: G → GL(V). We say V is a G-module (or a kG-module when the field is specified).

\rho : G \to \mathrm{GL}(V), \quad \rho(gh) = \rho(g)\rho(h) \quad \forall g,h \in G

Representation: group homomorphism into GL(V)

\rho(e) = \mathrm{Id}_V, \quad \rho(g^{-1}) = \rho(g)^{-1}

Identity and inverse preserved

\mathrm{Hom}_G(V, W) = \{\phi : V \to W \mid \phi(\rho_V(g)v) = \rho_W(g)\phi(v),\; \forall g \in G\}

G-equivariant maps (intertwining operators)

V \otimes_k W, \quad V^* = \mathrm{Hom}_k(V,k) \quad \text{— tensor and dual representations}

Constructions on representations

Notation

Notation	Meaning
$\rho$	Representation homomorphism G → GL(V)
$\mathrm{GL}(V)$	Group of invertible linear maps on V
$\mathrm{Hom}_G(V,W)$	Space of G-equivariant maps from V to W
$V^G$	Subspace of G-fixed vectors

Properties

Trivial representation

\rho(g) = 1 \in k^\times \text{ for all } g \in G \text{ — the trivial 1-dimensional representation}

Regular representation

The group ring kG acts on itself by left multiplication, giving the regular representation of dimension |G|.

Dual representation

\rho^*(g) = (\rho(g^{-1}))^T \text{ is a representation on } V^*

Theorems

Theorem 1: Maschke's Theorem

If G is a finite group and \mathrm{char}(k) \nmid |G|, then every representation of G over k is completely reducible (a direct sum of irreducible representations).

Theorem 2: Decomposition over ℂ

Every finite-dimensional complex representation of a finite group G decomposes as V \cong \bigoplus_{\rho \text{ irred.}} V_\rho^{\oplus m_\rho}

Worked Examples

1
Let G = ℤ/nℤ with generator g. Define ρ(g^k) = rotation by 2πk/n.
$\rho(g^k) = \begin{pmatrix}\cos(2\pi k/n) & -\sin(2\pi k/n) \\ \sin(2\pi k/n) & \cos(2\pi k/n)\end{pmatrix}$
2
Check: ρ(g^j)ρ(g^k) = rotation by 2πj/n + 2πk/n = rotation by 2π(j+k)/n = ρ(g^{j+k}).
$\rho(g^j)\rho(g^k) = \rho(g^{j+k})$

✓ Answer

ρ is a 2-dimensional real representation of ℤ/nℤ by rotations through multiples of 2π/n.

Practice Problems

Mediumfree response

List all 1-dimensional representations of ℤ/4ℤ over ℂ.

Mediumproof writing

Show that if char(k) divides |G|, Maschke's theorem can fail by giving an example.

Common Mistakes

Common Mistake

Confusing a representation with a group action on a set

A group action on a set S is a homomorphism G → Bij(S); a representation is a homomorphism G → GL(V) into linear automorphisms of a vector space. The latter has additional linear structure.

Common Mistake

Assuming all representations are over ℂ

Representations can be over any field k. The theory over ℂ (or any algebraically closed field of characteristic 0) is the cleanest, but real and modular (positive characteristic) representations are also important.

Quiz

A representation ρ: G → GL(V) is a:

Group action on a setGroup homomorphism into invertible linear mapsBijection from G to VLinear map V → V

Maschke's theorem guarantees complete reducibility when:

G is abelianV is finite-dimensionalchar(k) does not divide |G|G is a simple group

Historical Background

The systematic theory of group representations began with Frobenius's work on finite groups in the 1890s. Frobenius, motivated by a question of Dedekind about factoring group determinants, introduced characters and proved the fundamental orthogonality relations. Schur and Burnside extended the theory in the early 1900s. Weyl's work on Lie group representations in the 1920s extended the theory to continuous groups, while Brauer developed modular representation theory (over fields of positive characteristic) in the 1930s–1950s.

1896
Frobenius introduces group characters and proves orthogonality
Georg Frobenius
1905
Schur develops the theory of projective representations and Schur's lemma
Issai Schur
1925
Weyl's work on representations of semisimple Lie groups
Hermann Weyl
1935
Brauer begins modular representation theory
Richard Brauer

Summary

A representation of G is a homomorphism ρ: G → GL(V); V is then called a G-module.
G-equivariant maps (intertwining operators) form the morphisms in the category of G-representations.
Maschke's theorem: over a field of characteristic not dividing |G|, every representation is completely reducible.
Basic constructions: direct sum, tensor product, dual, symmetric/exterior powers.
The regular representation kG contains every irreducible representation with multiplicity equal to its dimension.

References

BookSerre, J.-P. — Linear Representations of Finite Groups (1977), Chapters 1–2
BookFulton, W. & Harris, J. — Representation Theory: A First Course (1991), Lectures 1–3
WebsiteWikipedia — Group representation
WebsiteMathWorld — Group Representation