Representations of Groups

⚡ Key Takeaways

A representation of a group $G$ is a homomorphism $\rho: G \to GL(V)$ 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.

💡 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 $\mathbb{C}$ , this decomposition is complete and rigid, and the bookkeeping reduces to characters, columns of complex numbers obeying beautiful orthogonality relations.

DefinitionLinear 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

1

\quad \rho: G \to GL(V), \qquad \rho(gh) = \rho(g)\,\rho(h),\quad \rho(e) = \mathrm{id}_V.

Derivation hint: The homomorphism condition is the defining axiom; that $\rho(e)=\mathrm{id}$ and $\rho(g^{-1})=\rho(g)^{-1}$ follow automatically from it. The dimension (or degree) of $\rho$ is $\dim_k V$ . Choosing a basis of an $n$ -dimensional $V$ identifies $GL(V)$ with $GL_n(k)$ , so $\rho$ becomes a homomorphism $G \to GL_n(k)$ , a matrix representation. Here:

$G$ is the group (often finite);
$V$ is the representation space, $k$ the ground field (usually $\mathbb{C}$ );
$GL(V)$ is the group of invertible linear maps $V \to V$ ;
$\rho(g)$ 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 $e\cdot v = v$ , $(gh)\cdot v = g\cdot(h\cdot v)$ , and linearity in $v$ .

DefinitionSubrepresentation, Irreducibility, Equivalence

A subspace $W \subseteq V$ is a subrepresentation (or $G$ -invariant subspace) if $\rho(g)W \subseteq W$ 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 $\rho: G\to GL(V)$ and $\sigma: G\to GL(W)$ are equivalent (isomorphic) if there is an invertible linear $T: V\to W$ with

2

\quad T\,\rho(g) = \sigma(g)\,T \quad\text{for all } g\in G.

Derivation hint: Equivalence is conjugation by a fixed basis-change $T$ ; in matrix terms $\sigma(g) = T\rho(g)T^{-1}$ , which is exactly "same representation, different coordinates."

DefinitionCharacter

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

3

\quad \chi_\rho: G \to k, \qquad \chi_\rho(g) = \operatorname{tr}\bigl(\rho(g)\bigr).

Derivation hint: The trace is basis-independent because $\operatorname{tr}(TAT^{-1})=\operatorname{tr}(A)$ ; hence equivalent representations share a character, and $\chi$ is a class function — constant on conjugacy classes since $\chi(hgh^{-1}) = \operatorname{tr}(\rho(h)\rho(g)\rho(h)^{-1}) = \chi(g)$ . Note $\chi_\rho(e) = \dim V$ , the degree.

TheoremMaschke's Theorem

Let $G$ be a finite group and $k$ a field whose characteristic does not divide $|G|$ (e.g. $k=\mathbb{C}$ ). 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.

TheoremSchur'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. $T\rho_V(g) = \rho_W(g)T$ ). Then: 1. $T$ is either $0$ or an isomorphism. 2. If $V = W$ , then $T = \lambda\,\mathrm{id}_V$ for some scalar $\lambda \in k$ .

TheoremOrthogonality and Completeness for Finite Groups over

\mathbb{C}

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

4

\quad \langle \chi, \psi\rangle = \frac{1}{|G|}\sum_{g\in G} \chi(g)\,\overline{\psi(g)}.

Derivation hint: 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 $\{\chi_1,\dots,\chi_r\}$ form an orthonormal basis of the space of class functions:

5

\quad \langle \chi_i, \chi_j\rangle = \delta_{ij}, \qquad r = \#\{\text{conjugacy classes of }G\}.

Derivation hint: 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,

6

\quad \sum_{i=1}^{r} d_i^{\,2} = |G|.

Derivation hint: Decompose the regular representation $\mathbb{C}[G]$ into irreducibles; each irreducible of degree $d_i$ appears exactly $d_i$ times, and the total dimension is $|G|$ .

ProofMaschke's Theorem via Averaging

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

Step 1. Choose any linear projection $\pi_0: V \to W$ with $\pi_0|_W = \mathrm{id}_W$ (exists by picking a vector-space complement, ignoring $G$ ).

Step 2. Average $\pi_0$ over the group to make it $G$ -equivariant:

7

\quad \pi = \frac{1}{|G|}\sum_{g\in G} \rho(g)\,\pi_0\,\rho(g)^{-1}.

Derivation hint: The factor $1/|G|$ requires $\operatorname{char} k \nmid |G|$ so $|G|$ is invertible in $k$ ; this averaging trick is the workhorse of finite-group representation theory.

Step 3. Check $\pi$ maps into $W$ : since each $\rho(g)$ preserves $W$ and $\pi_0$ lands in $W$ , every summand lands in $W$ , so $\operatorname{im}\pi \subseteq W$ .

Step 4. Check $\pi|_W = \mathrm{id}_W$ : for $w\in W$ , $\rho(g)^{-1}w \in W$ , so $\pi_0\rho(g)^{-1}w = \rho(g)^{-1}w$ , hence $\rho(g)\pi_0\rho(g)^{-1}w = w$ , and averaging gives $\pi w = w$ .

Step 5. Check $\pi$ is $G$ -equivariant: for any $h\in G$ ,

\rho(h)\pi\rho(h)^{-1} = \frac{1}{|G|}\sum_g \rho(hg)\pi_0\rho(hg)^{-1} = \pi,

by re-indexing $g \mapsto hg$ .

Step 6. Then $W' = \ker\pi$ is $G$ -invariant (kernel of an equivariant map) and $V = W \oplus W'$ . Iterating gives the full decomposition into irreducibles. $\blacksquare$

∎

ProofSchur'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 $T(\rho_V(g)v) = \rho_W(g)Tv = 0$ . By irreducibility $\ker T = 0$ or $V$ . Likewise $\operatorname{im} T$ is a subrepresentation of $W$ , so it is $0$ or $W$ . If $T\neq 0$ , then $\ker T = 0$ and $\operatorname{im}T = W$ , so $T$ is an isomorphism.

Part 2. Now $V = W$ and $k$ algebraically closed. Since $V$ is finite-dimensional, $T$ has an eigenvalue $\lambda\in k$ . The map $T - \lambda\,\mathrm{id}$ is equivariant and has nonzero kernel (the $\lambda$ -eigenspace). By Part 1, $T - \lambda\,\mathrm{id} = 0$ , so $T = \lambda\,\mathrm{id}$ . $\blacksquare$ Why algebraic closure is needed: the eigenvalue must exist in $k$ ; over $\mathbb{R}$ , rotation matrices are irreducible $2\times 2$ representations whose commutant is $\mathbb{C}$ , not just scalars.

∎

mathematician · 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.

🧠 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.

Figure

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

ExampleIrreducible 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 $\chi_{\mathrm{triv}}$ : every $g \mapsto 1$ .
Sign $\chi_{\mathrm{sgn}}$ : $g \mapsto \operatorname{sgn}(g) = \pm 1$ .
Standard $\chi_{\mathrm{std}}$ : the 2-dimensional rep on $\{(x_1,x_2,x_3): \sum x_i = 0\}$ inside the permutation action on $\mathbb{C}^3$ .

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

	$e$ (1)	transp. (3)	3-cycle (2)
$\chi_{\mathrm{triv}}$	1	1	1
$\chi_{\mathrm{sgn}}$	1	$-1$	1
$\chi_{\mathrm{std}}$	2	0	$-1$

Step 4 — verify orthonormality of $\chi_{\mathrm{std}}$ :

\langle\chi_{\mathrm{std}},\chi_{\mathrm{std}}\rangle = \tfrac{1}{6}\bigl(1\cdot 2^2 + 3\cdot 0^2 + 2\cdot(-1)^2\bigr) = \tfrac{1}{6}(4+0+2) = 1.\ \checkmark

\boxed{S_3 \text{ has irreducibles of degrees } 1,\,1,\,2;\ \text{character table above.}}

ExampleDecomposing the Permutation Representation of

S_3

Given: $S_3$ acts on $\mathbb{C}^3$ by permuting coordinates; this is the permutation representation $\rho_{\mathrm{perm}}$ .

Step 1 — compute its character. $\chi_{\mathrm{perm}}(g) = \#\{\text{fixed points of }g\}$ .

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

Step 2 — find multiplicities $a_i = \langle\chi_{\mathrm{perm}}, \chi_i\rangle$ using class sizes $(1,3,2)$ :

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

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

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

Step 3 — check dimensions: $1\cdot 1 + 0\cdot 1 + 1\cdot 2 = 3 = \dim\mathbb{C}^3.\ \checkmark$

\boxed{\rho_{\mathrm{perm}} \cong \chi_{\mathrm{triv}} \oplus \chi_{\mathrm{std}}}

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

⚠️ Common mistake

"Reducible" does not mean "decomposes as a direct sum." Over fields where Maschke fails (e.g. $\operatorname{char} k \mid |G|$ , 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 $\mathbb{R}$ , and vice versa.

CounterexampleMaschke Fails in Characteristic

p

Let $G = \mathbb{Z}/p$ act on $V = k^2$ over $k = \mathbb{F}_p$ via

\rho(1) = \begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}.

The line $W = \operatorname{span}\{(1,0)\}$ 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 $\operatorname{char} k \nmid |G| = p$ . The averaging proof breaks because $1/|G| = 1/p$ does not exist in $\mathbb{F}_p$ .

⚠️ Common mistake

The character does not determine the representation over arbitrary fields, only over $\mathbb{C}$ (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 $\rho(g)$ as an operator beyond its trace. Two genuinely different elements can have equal character values without that meaning anything is wrong.

connection

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 $G = \mathbb{Z}/n$ the irreducible characters are the discrete exponentials $e^{2\pi i k m/n}$ , 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 $\mathbb{C}[G]$ , where Maschke's theorem becomes the statement that $\mathbb{C}[G]$ 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.

🔬 Where this appears

⏳ History

- 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.

open-problem

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.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also