Representations of Finite Groups

⚡ Key Takeaways

A representation of a finite group $G$ is a homomorphism $\rho: G \to GL(V)$ that realizes abstract group elements as invertible linear transformations of a vector space $V$ . The central result is that every representation of a finite group over $\mathbb{C}$ decomposes uniquely into irreducible pieces, and these irreducibles are completely captured by their characters. Applies wherever symmetry acts linearly: quantum mechanics, crystallography, number theory, error-correcting codes. Developed by Frobenius, Schur, and Burnside around 1896–1905.

💡 The Core Insight

The group algebra $\mathbb{C}[G]$ of a finite group is semisimple: it splits as a product of matrix algebras, one for each irreducible representation. This single algebraic fact (Maschke + Wedderburn) forces complete reducibility, orthogonality of characters, and the dimension formula $|G| = \sum d_i^2$ . The whole subject is the systematic exploitation of this decomposition.

DefinitionRepresentation

Let $G$ be a finite group and $k$ a field. A representation of $G$ over $k$ is a finite-dimensional $k$ -vector space $V$ together with a group homomorphism

1

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

Derivation hint: The homomorphism property is the entire content; $\rho(e)=\mathrm{id}$ and $\rho(g^{-1})=\rho(g)^{-1}$ follow automatically from it. The integer $\dim_k V = n$ is the degree (or dimension) of the representation. Choosing a basis of $V$ identifies $GL(V) \cong GL_n(k)$ , so $\rho$ becomes a homomorphism into invertible $n\times n$ matrices. We often write $g\cdot v$ for $\rho(g)v$ and call $V$ a $G$ -module.

DefinitionSubrepresentation, Irreducibility, Equivalence

A subrepresentation is a subspace $W \subseteq V$ with $\rho(g)W \subseteq W$ for all $g$ . A nonzero representation $V$ is irreducible (or simple) if its only subrepresentations are $0$ and $V$ . Two representations $\rho:G\to GL(V)$ and $\sigma:G\to GL(W)$ are equivalent (isomorphic) if there is a linear isomorphism $T:V\to W$ with

2

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

Derivation hint: $T$ is a change of basis intertwining the two actions; equivalence means "same representation written in different coordinates."

DefinitionCharacter

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

3

\quad \chi_\rho : G \to \mathbb{C}, \qquad \chi_\rho(g) = \operatorname{tr}\big(\rho(g)\big).

Derivation hint: Trace is basis-independent and invariant under conjugation, $\operatorname{tr}(ABA^{-1})=\operatorname{tr}(B)$ , so $\chi$ is a class function. Because $\operatorname{tr}(\rho(hgh^{-1})) = \operatorname{tr}(\rho(g))$ , characters are constant on conjugacy classes; they are class functions. Note $\chi_\rho(e) = \dim V$ .

TheoremMaschke's Theorem

Let $G$ be a finite group and $k$ a field whose characteristic does not divide $|G|$ (in particular $k=\mathbb{C}$ ). Then every representation of $G$ over $k$ is completely reducible: it decomposes as a direct sum of irreducible subrepresentations,

4

\quad V \cong V_1 \oplus V_2 \oplus \cdots \oplus V_m, \qquad V_i \text{ irreducible}.

Derivation hint: Average any projection onto a subrepresentation over the group to produce a $G$ -equivariant projection; its kernel is a complement.

TheoremSchur's Lemma

Let $V,W$ be irreducible complex representations and $T:V\to W$ a $G$ -equivariant linear map ( $T\rho(g)=\sigma(g)T$ ). Then: 1. If $V\not\cong W$ , then $T=0$ . 2. If $V = W$ , then $T = \lambda\,\mathrm{id}$ for some scalar $\lambda\in\mathbb{C}$ .

TheoremOrthogonality and the Character Table

Define the inner product on class functions

5

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

Derivation hint: This is the $L^2$ inner product on functions $G\to\mathbb C$ with normalized counting measure; the bar is complex conjugation. Then the irreducible characters $\chi_1,\dots,\chi_r$ form an orthonormal basis of the space of class functions:

6

\quad \langle \chi_i, \chi_j\rangle = \delta_{ij}.

Derivation hint: Apply Schur's Lemma to the averaged intertwiner between irreducibles $i$ and $j$ ; the trace bookkeeping yields the Kronecker delta. Consequently the number $r$ of irreducible representations equals the number of conjugacy classes, and

7

\quad |G| = \sum_{i=1}^{r} d_i^2, \qquad d_i = \chi_i(e) = \dim V_i.

Derivation hint: Decompose the regular representation $\mathbb{C}[G]$ ; the irreducible $V_i$ appears exactly $d_i$ times, and $\dim\mathbb{C}[G]=|G|$ .

ProofMaschke's Theorem via Averaging

We show every subrepresentation has a $G$ -invariant complement.

Step 1. Let $W \subseteq V$ be a subrepresentation. Choose any linear projection $\pi_0: V \to W$ (so $\pi_0|_W = \mathrm{id}$ , $\operatorname{im}\pi_0 = W$ ), ignoring the group for now.

Step 2. Average it over $G$ to make it equivariant:

8

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

Derivation hint: The factor $1/|G|$ requires $|G|$ invertible in $k$ , i.e. $\operatorname{char} k \nmid |G|$ .

Step 3. Since each $\rho(g)$ preserves $W$ and $\pi_0$ lands in $W$ , the map $\pi$ still has image $W$ and restricts to the identity on $W$ ; hence $\pi$ is a projection onto $W$ .

Step 4. Check equivariance: for $h\in G$ ,

9

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

Derivation hint: Re-index the sum by $g' = hg$ ; summing over $G$ is invariant under left translation. so $\pi$ commutes with all $\rho(h)$ , i.e. $\rho(h)\pi = \pi\rho(h)$ .

Step 5. Then $U = \ker\pi$ is a subrepresentation (because $\pi$ is equivariant) and $V = W \oplus U$ . Iterating on the pieces gives the full decomposition into irreducibles. $\qquad\blacksquare$

∎

ProofSchur's Lemma

Let $T:V\to W$ be equivariant, $V,W$ irreducible over $\mathbb{C}$ .

Step 1. $\ker T$ is a subrepresentation of $V$ (if $v\in\ker T$ , then $T\rho(g)v = \sigma(g)Tv = 0$ ). By irreducibility $\ker T = 0$ or $V$ .

Step 2. Likewise $\operatorname{im}T$ is a subrepresentation of $W$ , so it is $0$ or $W$ . Hence $T$ is either $0$ or an isomorphism, proving part (1).

Step 3. For part (2) with $V=W$ : $\mathbb{C}$ is algebraically closed, so $T$ has an eigenvalue $\lambda$ . Then $T - \lambda\,\mathrm{id}$ is equivariant with nontrivial kernel, hence is $0$ by Step 1. Thus $T = \lambda\,\mathrm{id}$ . $\qquad\blacksquare$

∎

mathematician · Ferdinand Georg Frobenius | 1849–1917

Frobenius single-handedly founded the character theory of finite groups in a burst of papers from 1896 to 1900, motivated by a question Dedekind posed about factoring the "group determinant." Without yet having matrices in mind, he discovered orthogonality relations, the dimension formula $|G|=\sum d_i^2$ , and the induced character. Issai Schur, his student, recast everything in the cleaner language of matrices and modules, proving Schur's Lemma and the orthogonality relations in the form used today. Burnside, working in parallel in England, wrote the first influential textbook and proved his celebrated $p^aq^b$ theorem — that no nonabelian simple group has order divisible by only two primes — using characters in a way no purely group-theoretic argument matched for decades.

🧠 Intuition

Think of a group $G$ as a set of abstract symmetries and a representation as a way of seeing those symmetries acting on space. The trivial representation sees nothing (everything maps to $1$ ); a faithful representation sees everything. Irreducible representations are the "atoms" — the smallest pieces of space that $G$ can shuffle without breaking into smaller invariant chunks, and Maschke says every action is a clean stack of these atoms with no entanglement (no nontrivial extensions). The character is a brilliant compression: instead of recording entire matrices, you record only their traces. Astonishingly, this loses no information for distinguishing irreducibles — two complex representations are equivalent iff they have equal characters. The character table is a finite square grid ( $r$ irreducibles $\times$ $r$ conjugacy classes), and orthogonality makes its rows and columns orthonormal vectors. Computing a representation's "irreducible content" becomes computing inner products — projecting a vector onto an orthonormal basis, exactly like Fourier analysis. Indeed representation theory is the nonabelian generalization of Fourier series.

Figure

Decomposition of a representation into irreducible components, analogous to projecting onto an orthonormal basis

ExampleCharacter Table of

S_3

Given: $G = S_3$ , the symmetric group on $3$ letters, $|G| = 6$ . Conjugacy classes are $\{e\}$ (size 1), transpositions $\{(12),(13),(23)\}$ (size 3), and $3$ -cycles $\{(123),(132)\}$ (size 2). So there are $r = 3$ conjugacy classes, hence $3$ irreducibles.

Step 1 — Dimensions. Solve $\sum d_i^2 = 6$ with $r=3$ positive integers, one of them $1$ (trivial rep). The unique solution is $1^2 + 1^2 + 2^2 = 6$ . So $d_1=1, d_2=1, d_3=2$ .

Step 2 — The two $1$ -dimensional reps. The trivial $\chi_1$ sends every $g\mapsto 1$ . The sign $\chi_2 = \operatorname{sgn}$ sends transpositions to $-1$ , $3$ -cycles to $+1$ .

Step 3 — The $2$ -dimensional rep. Use orthogonality. Write $\chi_3 = (a,b,c)$ on classes $(e,\text{transp},\text{3-cyc})$ with $a = \dim = 2$ . From $\langle \chi_3,\chi_1\rangle = 0$ :

10

\quad \frac{1}{6}\big(1\cdot 2 + 3\cdot b + 2\cdot c\big) = 0.

Derivation hint: weight each class value by class size; orthogonality with trivial gives one linear equation. From $\langle\chi_3,\chi_2\rangle=0$ : $\tfrac16(2 - 3b + 2c)=0$ . Subtracting gives $6b = 0 \Rightarrow b=0$ , then $2 + 2c = 0 \Rightarrow c = -1$ .

Step 4 — Verify. $\langle\chi_3,\chi_3\rangle = \tfrac16(1\cdot4 + 3\cdot0 + 2\cdot1) = \tfrac16(6) = 1$ . ✓ irreducible.

Boxed answer:

\boxed{\begin{array}{c|ccc} & e & (12) & (123)\\\hline \chi_1 & 1 & 1 & 1\\ \chi_2 & 1 & -1 & 1\\ \chi_3 & 2 & 0 & -1 \end{array}}

ExampleDecomposing the Permutation Representation of

S_3

Given: $S_3$ acts on $\mathbb{C}^3$ by permuting coordinates: $\rho(g)e_i = e_{g(i)}$ . Decompose this representation into irreducibles using the $S_3$ table above.

Step 1 — Compute its character. The trace of a permutation matrix counts fixed points: $\chi_{\text{perm}}(g) = \#\{i : g(i)=i\}$ .

$e$ : fixes $3$ points $\Rightarrow \chi = 3$ .
transposition: fixes $1$ point $\Rightarrow \chi = 1$ .
$3$ -cycle: fixes $0$ points $\Rightarrow \chi = 0$ .

So $\chi_{\text{perm}} = (3,1,0)$ .

Step 2 — Multiplicities via inner products (class sizes $1,3,2$ ):

11

\quad m_i = \langle \chi_{\text{perm}},\chi_i\rangle = \frac{1}{6}\sum_{\text{classes}} (\text{size})\,\chi_{\text{perm}}\,\overline{\chi_i}.

Derivation hint: project the permutation character onto each orthonormal irreducible character.

$m_1 = \tfrac16(1\cdot3\cdot1 + 3\cdot1\cdot1 + 2\cdot0\cdot1) = \tfrac16(6) = 1$ .
$m_2 = \tfrac16(1\cdot3\cdot1 + 3\cdot1\cdot(-1) + 2\cdot0\cdot1) = \tfrac16(0) = 0$ .
$m_3 = \tfrac16(1\cdot3\cdot2 + 3\cdot1\cdot0 + 2\cdot0\cdot(-1)) = \tfrac16(6) = 1$ .

Step 3 — Conclude.

\boxed{\mathbb{C}^3 \cong \chi_1 \oplus \chi_3 = (\text{trivial}) \oplus (\text{2-dim standard})}

This is the familiar fact: $\mathbb{C}^3$ splits into the line $\langle e_1+e_2+e_3\rangle$ (fixed by all permutations = trivial) and its $2$ -dimensional complement $\{x : \sum x_i = 0\}$ (the standard representation). Verify: $1\cdot 1 + 1\cdot 2 = 3 = \dim\mathbb{C}^3$ . ✓

⚠️ Common mistake

Characters are class functions, not homomorphisms. $\chi_\rho(gh) \neq \chi_\rho(g)\chi_\rho(h)$ in general — that fails already for $\chi_3$ of $S_3$ . Only $1$ -dimensional characters are themselves homomorphisms. Also, $\chi_\rho(g^{-1}) = \overline{\chi_\rho(g)}$ (over $\mathbb{C}$ ), not $\chi_\rho(g)^{-1}$ .

"Equal dimension" does not mean "equivalent." Two representations can have the same degree yet be inequivalent; you must compare full characters. Conversely, equal characters do force equivalence over $\mathbb{C}$ — that is the strength of the theory.

Maschke needs the characteristic condition. Over a field of characteristic $p$ dividing $|G|$ (modular representation theory), the averaging trick divides by zero, complete reducibility fails, and the whole semisimple picture collapses.

Counterexample

Maschke fails in characteristic $p$ . Let $G = \mathbb{Z}/p$ over $k = \mathbb{F}_p$ , generated by the matrix

12

\quad \rho(1) = \begin{pmatrix} 1 & 1\\ 0 & 1\end{pmatrix} \in GL_2(\mathbb{F}_p),

Derivation hint: this has order $p$ since its $p$ -th power is the identity by the binomial computation $(I+N)^p = I + pN = I$ in characteristic $p$ . The line $W = \langle e_1\rangle$ is invariant, but it has no invariant complement: any $G$ -stable line must be an eigenspace, and $\rho(1)$ has only the single eigenvalue $1$ with one-dimensional eigenspace $W$ . So $V = \mathbb{F}_p^2$ is reducible but not completely reducible — Maschke genuinely requires $p \nmid |G|$ .

connection

Representation theory sits at a crossroads of algebra and analysis. The decomposition $\mathbb{C}[G] \cong \prod_i M_{d_i}(\mathbb{C})$ is an instance of the Wedderburn–Artin Theorem for semisimple algebras, and Maschke's theorem is precisely the statement that $\mathbb{C}[G]$ is semisimple. Characters generalize the Discrete Fourier Transform: for abelian $G$ , the irreducible characters are exactly the group homomorphisms $G \to \mathbb{C}^\times$ and form the Pontryagin Dual, recovering ordinary Fourier analysis on $\mathbb{Z}/n$ . The orthogonality relations are an instance of the Peter–Weyl Theorem for the finite (compact) group $G$ . The induced representation construction connects to Frobenius Reciprocity and underlies Galois Representations in number theory.

🔬 Where this appears

⏳ History

- 1837 — Gauss and Dirichlet implicitly use characters of abelian groups in studying quadratic reciprocity and primes in arithmetic progressions (Dirichlet characters).

- 1896 — Frobenius, prompted by Dedekind's letters about the "group determinant," defines characters of nonabelian finite groups and proves the orthogonality relations and dimension formula.

- 1897 — Frobenius introduces induced representations and proves Frobenius reciprocity.

- 1898 — Heinrich Maschke proves complete reducibility for representations over fields of suitable characteristic.

- 1905 — Issai Schur recasts the theory in matrix language, proves Schur's Lemma, and gives the clean modern derivation of orthogonality.

- 1904 — William Burnside proves the

p^aq^b

theorem via character theory, a result with no elementary proof for over 60 years.

- 1911 — Burnside's second edition of Theory of Groups of Finite Order makes representation theory a standard tool.

- 1935 — Richard Brauer founds modular representation theory (characteristic

p \mid |G|

), where Maschke fails.

- 1955–63 — Brauer's and others' character-theoretic methods become indispensable in the classification of finite simple groups.

open-problem

Over $\mathbb{C}$ the theory is essentially complete, but several frontiers remain. Modular representation theory (characteristic $p \mid |G|$ ) is far subtler: decomposition matrices and the structure of indecomposable (not irreducible) modules are governed by deep conjectures, including Broué's abelian defect conjecture and Donovan's conjecture, still open in general. For infinite families of groups, computing character tables explicitly is hard; the character tables of the finite groups of Lie type were determined by Deligne–Lusztig theory (1976), a profound achievement, yet "Lusztig's conjecture" on modular characters resisted full proof for decades. Even rationality questions — whether a representation can be realized over $\mathbb{Q}$ or $\mathbb{R}$ rather than $\mathbb{C}$ , controlled by the Frobenius–Schur indicator — connect to subtle arithmetic (the Brauer–Witt and Schur index theory). Finally, the precise relationship between a group's character table and the group itself is incomplete: nonisomorphic groups can share a character table (e.g. the order- $8$ groups $D_4$ and $Q_8$ ), so the table does not determine $G$ .

Formal Statement

Why It Holds

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also