Irreducible Representations — Mathematics Knowledge System

Overview

An irreducible representation (irrep) of a group G is a non-zero representation V that has no proper non-zero G-invariant subspaces. Irreducible representations are the 'atoms' of representation theory: by Maschke's theorem, every finite-dimensional representation over ℂ (or any field of characteristic not dividing |G|) decomposes uniquely as a direct sum of irreducibles. The classification of irreducible representations of a group is one of the central problems of representation theory.

Intuition

An irreducible representation is one you cannot 'block-diagonalise': no matter what basis you choose, you cannot write all ρ(g) simultaneously in block form. These are the indivisible building blocks. Knowing all irreps of G is like knowing the prime factorisation of every G-module.

Formal Definition

Definition

Let ρ: G → GL(V) be a representation. V is irreducible if it is non-zero and its only G-invariant subspaces are 0 and V itself.

V \text{ irreducible} \iff V \ne 0 \text{ and } \{W \subseteq V : W \text{ is G-stable}\} = \{0, V\}

Definition of irreducibility

V \cong \bigoplus_{i} V_i^{\oplus m_i}, \quad m_i = \langle \chi_V, \chi_i \rangle

Decomposition into irreducibles; multiplicities via characters

\sum_{V_i \text{ irred.}} (\dim V_i)^2 = |G|

Sum of squares of dimensions equals |G|

Notation

Notation	Meaning
$\mathrm{Irr}(G)$	Set of isomorphism classes of irreducible complex representations of G
$m_i$	Multiplicity of irreducible Vᵢ in a representation V
$V_i$	ith irreducible representation (often indexed by conjugacy class)

Properties

Dimension bound

\dim V_i \leq \sqrt{|G|} \text{ for any irreducible } V_i

Dimension divides group order

\dim V_i \mid |G| \text{ for all irreducible complex representations of a finite group}

1-dimensional irreps

The 1-dimensional representations of G are in bijection with the characters G/[G,G] \to k^\times of the abelianisation.

Theorems

Theorem 1: Schur's Lemma

If V and W are irreducible G-representations, then \mathrm{Hom}_G(V,W) = 0 if V \not\cong W, and = k\cdot\mathrm{id}_V if V \cong W (over algebraically closed k).

Theorem 2: Complete reducibility (Maschke)

Every representation of a finite group over k (with \mathrm{char}(k) \nmid |G|) is a direct sum of irreducibles.

Theorem 3: Unique decomposition

The decomposition V \cong \bigoplus_i V_i^{\oplus m_i} is unique up to isomorphism and reordering.

Worked Examples

1
ℤ/3ℤ is abelian, so all irreducible complex representations are 1-dimensional.
$\dim V_i = 1 \text{ for all } i$
2
A 1-dim rep sends the generator g to some cube root of unity ω^k.
$\rho_k(g) = \omega^k, \quad \omega = e^{2\pi i/3},\; k = 0,1,2$
3
There are 3 conjugacy classes ({e}, {g}, {g²}), hence exactly 3 irreducibles, confirmed.
$|\mathrm{Irr}(\mathbb{Z}/3\mathbb{Z})| = 3$

✓ Answer

Three 1-dimensional irreps: ρₖ(gⁿ) = e^{2πink/3} for k = 0,1,2.

Practice Problems

Mediumfree response

How many irreducible complex representations does A₄ have, and what are their dimensions?

Mediumproof writing

Show that every irreducible representation of an abelian group over ℂ is 1-dimensional.

Common Mistakes

Common Mistake

Thinking irreducible means indecomposable over any field

Over algebraically closed fields of characteristic 0, irreducible and indecomposable coincide (Maschke). But over other fields (e.g., ℝ or 𝔽_p), an indecomposable representation need not be irreducible.

Common Mistake

Assuming all irreducible representations are 1-dimensional

1-dim irreps exist and are simple, but groups like S₃ have a 2-dimensional irreducible (the standard representation), and S_n for n ≥ 5 has irreducibles of much larger dimension.

Quiz

An irreducible representation of G is one that:

Has no G-equivariant automorphismsHas no proper non-zero G-invariant subspacesIs 1-dimensionalHas trivial character

For a finite group G over ℂ, the number of irreducible representations equals:

The order |G|The number of subgroupsThe number of conjugacy classesThe rank of G

Historical Background

The notion of irreducibility is implicit in Frobenius's character theory and was formalised by Burnside and Schur in the early 1900s. The classification of irreducible representations of symmetric groups (via Young tableaux) was completed by Frobenius (1900) and Young (1900–1902). For compact Lie groups, Weyl's character formula (1925) gives the dimensions and characters of all irreducibles. The full classification of irreducible representations of real reductive Lie groups is the program of Harish-Chandra, which occupied much of the 20th century.

1896
Frobenius classifies irreducible representations of symmetric groups
Georg Frobenius
1905
Schur's lemma characterises irreducibility via Hom_G spaces
Issai Schur
1925
Weyl's character formula for irreducible representations of compact Lie groups
Hermann Weyl
1950s
Harish-Chandra develops theory of irreps of real reductive groups
Harish-Chandra

Summary

An irreducible representation has no proper non-zero G-invariant subspaces.
Over ℂ, every finite-dimensional representation decomposes uniquely into irreducibles (Maschke + Schur).
The number of irreducible complex representations equals the number of conjugacy classes.
Dimensions of irreducibles satisfy: Σ (dim Vᵢ)² = |G| and dim Vᵢ | |G|.
Every irreducible representation of an abelian group over ℂ is 1-dimensional.

References

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