Representations of Groups

A representation of a group $G$ is a homomorphism $ρ : G \to G L (V)$ that realizes abstract group elements as invertible linear transformations of a vector space $V$ . This turns group theory into linear algebra, letting matrices carry the symmetry. Representations appear everywhere symmetry acts: quantum mechanics, crystallography, number theory, and Fourier analysis. The theory was forged by Frobenius, Schur, and Burnside around 1896–1905.

A representation of a group $G$ on a vector space $V$ over a field $k$ is a group homomorphism $ρ : G \to G L (V)$ , the group of invertible linear maps on $V$ . Equivalently, it endows $V$ with an action of $G$ by linear transformations, so that the algebraic structure of $G$ is mirrored by concrete matrices. This matters because it converts hard questions about abstract groups into tractable problems in linear algebra, while simultaneously decomposing the symmetry of physical and mathematical systems into irreducible "atoms." The surprising fact: over the complex numbers, a finite group has only finitely many irreducible representations, their number equals the number of conjugacy classes, and the character — a single function recording traces — captures a representation completely up to isomorphism.

The Core Insight

Instead of studying a group $G$ in the abstract, let it act linearly. Every group element becomes a matrix, multiplication becomes matrix multiplication, and the entire representation can be broken into indecomposable pieces (irreducibles). The miracle — Maschke's theorem plus Schur's lemma — is that for finite groups over $C$ , this decomposition is complete and rigid, and the bookkeeping reduces to characters, columns of complex numbers obeying beautiful orthogonality relations.

Formal Statement

Linear Representation

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 \to G L (V), ρ (g h) = ρ (g) ρ (h), ρ (e) = id_{V} .

⟳The homomorphism condition is the defining axiom; that

ρ (e) = id

and

ρ (g^{- 1}) = ρ (g)^{- 1}

follow automatically from it.

The dimension (or degree) of $ρ$ is $dim_{k} V$ . Choosing a basis of an $n$ -dimensional $V$ identifies $G L (V)$ with $G L_{n} (k)$ , so $ρ$ becomes a homomorphism $G \to G L_{n} (k)$ , a matrix representation. Here:

$G$ is the group (often finite);
$V$ is the representation space, $k$ the ground field (usually $C$ );
$G L (V)$ is the group of invertible linear maps $V \to V$ ;
$ρ (g)$ is the linear operator assigned to $g$ .

An equivalent formulation: a $G$ -module is a $k$ -vector space $V$ with a map $G \times V \to V$ , $(g, v) \mapsto g \cdot v$ , satisfying $e \cdot v = v$ , $(g h) \cdot v = g \cdot (h \cdot v)$ , and linearity in $v$ .

Subrepresentation, Irreducibility, Equivalence

A subspace $W \subseteq V$ is a subrepresentation (or $G$ -invariant subspace) if $ρ (g) W \subseteq W$ for all $g \in G$ . A nonzero representation $V$ is irreducible if its only subrepresentations are $0$ and $V$ ; otherwise it is reducible. Two representations $ρ : G \to G L (V)$ and $σ : G \to G L (W)$ are equivalent (isomorphic) if there is an invertible linear $T : V \to W$ with

T ρ (g) = σ (g) T for all g \in G .

⟳Equivalence is conjugation by a fixed basis-change

T

; in matrix terms

σ (g) = T ρ (g) T^{- 1}

, which is exactly "same representation, different coordinates."

Character

The character of a representation $ρ : G \to G L (V)$ is the function

χ_{ρ} : G \to k, χ_{ρ} (g) = tr (ρ (g)) .

⟳The trace is basis-independent because

tr (T A T^{- 1}) = tr (A)

; hence equivalent representations share a character, and

χ

is a class function — constant on conjugacy classes since

χ (h g h^{- 1}) = tr (ρ (h) ρ (g) ρ (h)^{- 1}) = χ (g)

Note $χ_{ρ} (e) = dim V$ , the degree.

Maschke's Theorem

Let $G$ be a finite group and $k$ a field whose characteristic does not divide $∣ G ∣$ (e.g. $k = C$ ). Then every representation of $G$ is completely reducible: every subrepresentation $W \subseteq V$ has a $G$ -invariant complement $W^{'}$ with $V = W \oplus W^{'}$ . Consequently every finite-dimensional representation is a direct sum of irreducibles.

Schur's Lemma

Let $V, W$ be irreducible representations of $G$ over an algebraically closed field $k$ , and let $T : V \to W$ be a $G$ -equivariant linear map (i.e. $T ρ_{V} (g) = ρ_{W} (g) T$ ). Then:

$T$ is either $0$ or an isomorphism.
If $V = W$ , then $T = λ id_{V}$ for some scalar $λ \in k$ .

Orthogonality and Completeness for Finite Groups over $\mathbb{C}$

Let $G$ be finite. Define the Hermitian inner product on class functions

⟨ χ, ψ ⟩ = \frac{1}{∣ G ∣} g \in G \sum χ (g) \overline{ψ (g)} .

⟳This is the averaged

L^{2}

inner product over the group; the normalizing

1/∣ G ∣

makes the trivial character have norm 1.

Then the irreducible characters ${χ_{1}, \dots, χ_{r}}$ form an orthonormal basis of the space of class functions:

⟨ χ_{i}, χ_{j} ⟩ = δ_{ij}, r = # {conjugacy classes of G} .

⟳Orthonormality comes from Schur's lemma applied to the averaging projector; the count

r

follows because both sides have dimension equal to the number of classes.

Moreover, if $d_{1}, \dots, d_{r}$ are the degrees of the irreducibles,

i = 1 \sum r d_{i}^{2} = ∣ G ∣.

⟳Decompose the regular representation

C [G]

into irreducibles; each irreducible of degree

d_{i}

appears exactly

d_{i}

times, and the total dimension is

∣ G ∣

Why It Holds / Derivation

Maschke's Theorem via Averaging

We construct a $G$ -invariant complement to a subrepresentation $W \subseteq V$ .

Step 1. Choose any linear projection

π_{0} : V \to W

with

π_{0} ∣_{W} = id_{W}

(exists by picking a vector-space complement, ignoring

G

). Step 2. Average

π_{0}

over the group to make it

G

-equivariant:

π = \frac{1}{∣ G ∣} g \in G \sum ρ (g) π_{0} ρ (g)^{- 1} .

⟳The factor

1/∣ G ∣

requires

char k ∤ ∣ G ∣

∣ G ∣

is invertible in

k

; this averaging trick is the workhorse of finite-group representation theory.

Step 3. Check

π

maps into

W

: since each

ρ (g)

preserves

W

and

π_{0}

lands in

W

, every summand lands in

W

, so

im π \subseteq W

. Step 4. Check

π ∣_{W} = id_{W}

: for

w \in W

ρ (g)^{- 1} w \in W

, so

π_{0} ρ (g)^{- 1} w = ρ (g)^{- 1} w

, hence

ρ (g) π_{0} ρ (g)^{- 1} w = w

, and averaging gives

π w = w

. Step 5. Check

π

G

-equivariant: for any

h \in G

ρ (h) π ρ (h)^{- 1} = \frac{1}{∣ G ∣} g \sum ρ (h g) π_{0} ρ (h g)^{- 1} = π,

by re-indexing

g \mapsto h g

. Step 6. Then

W^{'} = ker π

G

-invariant (kernel of an equivariant map) and

V = W \oplus W^{'}

. Iterating gives the full decomposition into irreducibles.

■

Schur's Lemma

Part 1. Let

T : V \to W

be equivariant with

V, W

irreducible. Then

ker T

is a subrepresentation of

V

: if

v \in ker T

, then

T (ρ_{V} (g) v) = ρ_{W} (g) T v = 0

. By irreducibility

ker T = 0

V

. Likewise

im T

is a subrepresentation of

W

, so it is

0

W

. If

T \neq = 0

, then

ker T = 0

and

im T = W

, so

T

is an isomorphism. Part 2. Now

V = W

and

k

algebraically closed. Since

V

is finite-dimensional,

T

has an eigenvalue

λ \in k

. The map

T - λ id

is equivariant and has nonzero kernel (the

λ

-eigenspace). By Part 1,

T - λ id = 0

, so

T = λ id

■

Why algebraic closure is needed: the eigenvalue must exist in $k$ ; over $R$ , rotation matrices are irreducible $2 \times 2$ representations whose commutant is $C$ , not just scalars.

Ferdinand Georg Frobenius | 1849–1917

Frobenius, working in Berlin, single-handedly created the character theory of finite groups in a burst of papers from 1896 to 1903, prompted by a letter from Dedekind about factoring the "group determinant." He discovered the orthogonality relations, the link between characters and conjugacy classes, and induced representations — all before the matrix interpretation was standard, working instead with the abstract characters directly. His student Issai Schur and contemporary William Burnside rapidly extended the theory; Burnside's $p^{a} q^{b}$ theorem (every group of such order is solvable), proved using character theory, was an early triumph showing representations could settle questions pure group theory could not.

Building Intuition

Think of a group as a collection of abstract symmetries — rotations of a molecule, shufflings of a deck, the structure of integers mod $n$ . A representation is a way of seeing those symmetries as honest matrices acting on space. The point of irreducibility is to find the smallest invariant "screens" on which the group acts indecomposably: every other action is just a stack of these.

Characters are the radically efficient shortcut. Rather than carry around whole matrices, you keep only their traces — one number per conjugacy class. Astonishingly, this lossy-looking compression loses nothing: two representations with the same character are isomorphic. The orthogonality relations [5] then say the irreducible characters are orthonormal unit vectors, so decomposing any representation into irreducibles is just computing inner products — projecting onto an orthonormal basis, exactly like a Fourier expansion. Indeed, representation theory of finite groups is the generalization of Fourier analysis from the circle to arbitrary symmetry groups.

Decomposition of a representation into irreducible components, projected via character inner products

Worked Examples

Irreducible Representations of $S_3$

Given: the symmetric group

S_{3}

(order 6) with 3 conjugacy classes:

{e}

, the transpositions

{(12), (13), (23)}

, and the 3-cycles

{(123), (132)}

. Step 1 — count and constrain. Number of irreducibles

r = 3

(one per conjugacy class). Degrees

d_{i}

satisfy

\sum d_{i}^{2} = ∣ G ∣ = 6

. The only solution in positive integers with one

d_{i} = 1

(trivial rep always exists) is

1^{2} + 1^{2} + 2^{2} = 6

. Step 2 — name them.

Trivial $χ_{triv}$ : every $g \mapsto 1$ .
Sign $χ_{sgn}$ : $g \mapsto sgn (g) = \pm 1$ .
Standard $χ_{std}$ : the 2-dimensional rep on ${(x_{1}, x_{2}, x_{3}) : \sum x_{i} = 0}$ inside the permutation action on $C^{3}$ .

Step 3 — character table (columns indexed by classes, with class sizes):

| | $e$ (1) | transp. (3) | 3-cycle (2) | |---|---|---|---| | $χ_{triv}$ | 1 | 1 | 1 | | $χ_{sgn}$ | 1 | $- 1$ | 1 | | $χ_{std}$ | 2 | 0 | $- 1$ |

Step 4 — verify orthonormality of

χ_{std}

⟨ χ_{std}, χ_{std} ⟩ = \frac{1}{6} (1 \cdot 2^{2} + 3 \cdot 0^{2} + 2 \cdot (- 1)^{2}) = \frac{1}{6} (4 + 0 + 2) = 1. ✓

S_{3} has irreducibles of degrees 1, 1, 2; character table above.

Decomposing the Permutation Representation of $S_3$

Given:

S_{3}

acts on

C^{3}

by permuting coordinates; this is the permutation representation

ρ_{perm}

. Step 1 — compute its character.

χ_{perm} (g) = # {fixed points of g}

$χ_{perm} (e) = 3$ (all fixed)
$χ_{perm} (transposition) = 1$ (one fixed point)
$χ_{perm} (3-cycle) = 0$ (no fixed points)

Step 2 — find multiplicities

a_{i} = ⟨ χ_{perm}, χ_{i} ⟩

using class sizes

(1, 3, 2)

a_{triv} = \frac{1}{6} (1 \cdot 3 \cdot 1 + 3 \cdot 1 \cdot 1 + 2 \cdot 0 \cdot 1) = \frac{1}{6} (3 + 3 + 0) = 1.

a_{sgn} = \frac{1}{6} (1 \cdot 3 \cdot 1 + 3 \cdot 1 \cdot (- 1) + 2 \cdot 0 \cdot 1) = \frac{1}{6} (3 - 3 + 0) = 0.

a_{std} = \frac{1}{6} (1 \cdot 3 \cdot 2 + 3 \cdot 1 \cdot 0 + 2 \cdot 0 \cdot (- 1)) = \frac{1}{6} (6 + 0 + 0) = 1.

Step 3 — check dimensions:

1 \cdot 1 + 0 \cdot 1 + 1 \cdot 2 = 3 = dim C^{3} . ✓

ρ_{perm} ≅ χ_{triv} \oplus χ_{std}

The trivial part is the diagonal line $C (1, 1, 1)$ ; the standard 2-dim part is the sum-zero plane.

Common Misconceptions

"Reducible" does not mean "decomposes as a direct sum." Over fields where Maschke fails (e.g.

char k ∣ ∣ G ∣

, called the modular case), a representation can have a proper invariant subspace with no invariant complement — it is reducible but indecomposable. Complete reducibility is a special feature of the non-modular setting, not a general law. Also beware: an irreducible complex representation need not stay irreducible over

R

, and vice versa.

Maschke Fails in Characteristic $p$

Let $G = Z / p$ act on $V = k^{2}$ over $k = F_{p}$ via

ρ (1) = (1011) .

The line

W = span {(1, 0)}

is invariant. But the only invariant line is

W

itself (the matrix is a single Jordan block), so there is no invariant complement. Thus

V

is reducible yet indecomposable, and Maschke's theorem genuinely requires

char k ∤ ∣ G ∣ = p

. The averaging proof breaks because

1/∣ G ∣ = 1/ p

does not exist in

F_{p}

The character does not determine the representation over arbitrary fields, only over $C$ (or fields where Maschke holds and that are large enough). Moreover, the character is a class function, so it sees only conjugacy-class data — it cannot distinguish

ρ (g)

as an operator beyond its trace. Two genuinely different elements can have equal character values without that meaning anything is wrong.

Connections

Representation theory sits at a busy crossroads. It is the linear shadow of Group Theory: abstract groups studied through matrices. The whole apparatus of characters is literally Fourier Analysis generalized — for $G = Z / n$ the irreducible characters are the discrete exponentials $e^{2 π ik m / n}$ , and Fourier transform = decomposition into irreducibles. The decomposition theory is governed by Linear Algebra (eigenvalues, traces, inner products) and is cleanly packaged by the Group Algebra $C [G]$ , where Maschke's theorem becomes the statement that $C [G]$ is semisimple, a special case of Wedderburn's Theorem. Continuous symmetries push the same ideas into Lie Groups and Lie Algebras, and arithmetic ones into Galois Representations.

Real-World Applications

Quantum mechanics & selection rules. The symmetry group of a Hamiltonian forces energy eigenspaces to carry irreducible representations; degeneracies and which transitions are forbidden (selection rules) are read directly from character tables.
Molecular vibrational spectroscopy. The vibrational modes of a molecule decompose into irreducibles of its point group, predicting infrared/Raman activity — a standard chemistry computation.
Crystallography. The 230 space groups and their irreducible representations classify electronic band structure and phonons in solids (Bloch's theorem is rep theory of the translation group).
Particle physics. Quarks and the eightfold way arise from irreducibles of $S U (3)$ ; the entire Standard Model is organized by representations of gauge groups.
Error-correcting codes & combinatorial designs. Character sums and representation-theoretic constraints (e.g. via the group algebra) build and analyze symmetric codes.
Fast algorithms. The Fast Fourier Transform and its non-abelian generalizations are efficient changes of basis into the irreducible decomposition of $C [G]$ .
Number theory. $L$ -functions of Artin and Galois representations encode deep arithmetic; the Langlands program is, in part, representation theory at industrial scale.

Historical Development

1837 — Cauchy and others study substitution (permutation) groups concretely, an implicit precursor to matrix actions.
1872–1880s — Klein's Erlangen program elevates symmetry/transformation groups to the central organizing principle of geometry.
1896 — Frobenius, prompted by Dedekind's group-determinant problem, defines group characters and proves the orthogonality relations, founding the subject.
1897 — Frobenius introduces induced representations and the reciprocity (Frobenius reciprocity) governing induction and restriction.
1897–1900 — Burnside develops representation theory in parallel; his textbook (1897, 2nd ed. 1911) becomes the standard reference.
1904 — Burnside proves the $p^{a} q^{b}$ theorem (such groups are solvable) using character theory — a landmark application.
1905 — Schur proves Schur's lemma and clarifies the matrix-theoretic foundations; introduces the Schur orthogonality relations and projective representations.
1929 — Emmy Noether recasts the whole theory in terms of modules over the group algebra, making Maschke and Wedderburn structural statements about semisimple algebras.
1939–1962 — Weyl, Wigner, and Mackey extend the theory to Lie groups and infinite groups, cementing its role in physics and harmonic analysis.

Subtleties & Limits

Modular representation theory — the study of representations over fields whose characteristic divides

∣ G ∣

— abandons complete reducibility and replaces clean orthogonality with the subtle machinery of blocks, defect groups, and Brauer characters. Deep conjectures remain open here: Brauer's $k (B)$ conjecture, the Alperin weight conjecture, and Broué's abelian defect group conjecture all probe how local subgroup structure controls global representation theory and are unproven in general. Even in characteristic zero, classifying all irreducible representations of an arbitrary infinite or profinite group is wide open, and the representation theory underlying the Langlands program connects these questions to some of the hardest problems in number theory.

Representations of Groups

The Core Insight

Formal Statement

Linear Representation

Subrepresentation, Irreducibility, Equivalence

Character

Maschke's Theorem

Schur's Lemma

Orthogonality and Completeness for Finite Groups over $\mathbb{C}$

Why It Holds / Derivation

Maschke's Theorem via Averaging

Schur's Lemma

Ferdinand Georg Frobenius | 1849–1917

Building Intuition

Decomposition of a representation into irreducible components, projected via character inner products

Worked Examples

Irreducible Representations of $S_3$

Decomposing the Permutation Representation of $S_3$

Common Misconceptions

Maschke Fails in Characteristic $p$

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also

Before this

Where this leads