A Glimpse of Representation and Character Theory

A finite group can act on a vector space by invertible linear maps, and representation theory studies those linear actions. They turn questions about group structure into questions about matrices, modules, and eventually a small table of complex numbers. The theory is an application of the modules-over-a-ring viewpoint, with one particular ring — the group ring — playing the role that the polynomial ring played in commutative algebra. Throughout, $G$ is a finite group, $F$ is a field, and $V$ is a vector space over $F$ ; the deepest results assume $F = C$ .

Representations

The homomorphism condition $φ (g h) = φ (g) φ (h)$ is the entire content: group multiplication is realized as matrix multiplication. Every group has several standard representations.¹

Trivial representation. Degree $1$ , with $φ (g) = 1$ for all $g$ . Never faithful when $∣ G ∣ > 1$ .
Regular representation. Let $G$ act on itself by left multiplication; the permutation of basis vectors $g \cdot g_{i} = g g_{i}$ gives a faithful degree- $∣ G ∣$ representation whose matrices are permutation matrices.
Dihedral and quaternion. The symmetries of the $n$ -gon give a faithful degree- $2$ real representation of $D_{2 n}$ by an explicit rotation matrix and a reflection matrix; the quaternion group $Q_{8}$ has a faithful degree- $2$ complex representation.

A representation sends each group element to an invertible matrix and respects products, so the group's multiplication becomes matrix multiplication.

The group ring

A single ring built from $G$ and $F$ holds all representations of $G$ at once.

As an $F$ -vector space $F G$ has the elements of $G$ as a basis, so $dim_{F} F G = ∣ G ∣$ , and $F$ sits in its center, making $F G$ an $F$ -algebra.¹ It is commutative exactly when $G$ is abelian, and it has zero divisors whenever $∣ G ∣ > 1$ : for a nonidentity $g$ of order $n$ , the element $r = 1 + g + \dots + g^{n - 1}$ satisfies $r (1 - g) = 0$ . The group ring plays the same structural role here that $F [x]$ played for a single linear operator in the rational canonical form.

A representation and an $F G$ -module are the same data. Given $φ : G \to G L (V)$ , define a module action by $(\sum a_{i} g_{i}) \cdot v = \sum a_{i} φ (g_{i}) (v)$ ; the module axioms hold precisely because $φ$ is a homomorphism. Conversely an $F G$ -module $V$ is an $F$ -vector space on which each $g$ acts invertibly and $F$ -linearly, which is a representation.¹

Giving a representation is the same as making the vector space a module over the group ring; the submodules coincide with the invariant subspaces.

Under this dictionary, a subspace $U \subseteq V$ is a submodule exactly when it is $G$ -invariant ( $g \cdot u \in U$ for all $g, u$ ), and two representations are equivalent when their modules are isomorphic — equivalently, related by a single change of basis $P$ with $φ (g) = P ψ (g) P^{- 1}$ for all $g$ .¹

Complete reducibility

We break a representation into indivisible pieces.

In matrix terms, a reducible representation has a basis putting every $φ (g)$ in block upper-triangular form, and a decomposable one puts them all in block diagonal form simultaneously. Degree- $1$ representations are automatically irreducible. Whether a representation can be fully broken apart depends on the field.

The proof averages a vector-space projection $π_{0} : V \to U$ over the group to make it $G$ -equivariant:

π = \frac{1}{∣ G ∣} g \in G \sum g π_{0} g^{- 1},

which requires exactly that $∣ G ∣$ be invertible in $F$ .¹ The result is an $F G$ -module projection onto $U$ whose kernel is the complement $W$ . When $F$ has characteristic $0$ — in particular $F = C$ — the hypothesis is automatic.

Under Maschke's hypothesis every representation splits; one change of basis puts every matrix in the same block-diagonal form built from irreducibles.

The hypothesis is essential. Over $F_{p}$ with $g$ of order $p$ acting on a $2$ -dimensional space by a single Jordan block, the invariant line has no invariant complement: the module is reducible but indecomposable, so not completely reducible.¹ Modular representation theory, over fields whose characteristic divides $∣ G ∣$ , is a separate and harder subject.

The Wedderburn count

Complete reducibility says $C G$ is semisimple, and Wedderburn's theorem describes such a ring as a direct product of matrix algebras over $C$ . Character theory uses only two numerical consequences, both constraining the irreducibles.²

The sum-of-squares identity comes from decomposing the regular representation, which contains each irreducible exactly $d_{i}$ times, so $\sum d_{i} \cdot d_{i} = dim C G = ∣ G ∣$ . Separately, the number of degree- $1$ representations equals the index $∣ G : G^{'} ∣$ of the commutator subgroup, since a degree- $1$ representation factors through the abelianization $G / G^{'}$ .²

The squared degrees of the irreducibles tile a region of area equal to the group order; for the symmetric group on three letters this is

1 + 1 + 4 = 6

These constraints often force a unique list of degrees.

Characters

The trace of $φ (g)$ compresses a whole matrix into one scalar, and over $C$ the resulting function on $G$ already determines the representation up to equivalence. Writing down matrices of degree $100$ is unnecessary.

Because trace is conjugation-invariant, $χ (h^{- 1} g h) = χ (g)$ , so every character is a class function. Two basic values are read off immediately: $χ (1)$ is the degree of the representation, and equivalent representations have equal characters, since similar matrices have equal traces.³ The character of the trivial representation is the constant $1$ , called the principal character.

Class functions form a vector space of dimension $r$ (one basis function per conjugacy class), and the number of irreducible characters is also $r$ . Putting a Hermitian inner product on this space makes the irreducibles an orthonormal basis.

The irreducible characters are orthonormal; the inner product of two of them is one when they are equal and zero otherwise.

Orthonormality gives an immediate irreducibility test: a character $χ$ has $⟨ χ, χ ⟩ = 1$ if and only if it is irreducible, and the multiplicity of $χ_{i}$ in any representation with character $θ$ is $⟨ θ, χ_{i} ⟩$ . The companion statement about columns is the second orthogonality relation: distinct columns of the table below are orthogonal, and the squared column norm at $g$ is $∣ C_{G} (g) ∣$ .³

The character table

The character table of $G$ arranges the values $χ_{i} (g_{j})$ in a grid: one row per irreducible character, one column per conjugacy class. It is a square $r \times r$ array, and it is a near-complete fingerprint of the group.

For the symmetric group $S_{3}$ , there are three conjugacy classes — the identity, the three transpositions, and the two $3$ -cycles — hence three irreducible characters. The degrees are $1, 1, 2$ from the count above. The two degree- $1$ characters are the trivial and sign homomorphisms, and the degree- $2$ character is found by orthogonality.⁴

$S_{3}$	$1$	$(12)$	$(123)$
class size	$1$	$3$	$2$
$χ_{triv}$	$1$	$1$	$1$
$χ_{sign}$	$1$	$- 1$	$1$
$χ_{2}$	$2$	$0$	$- 1$

Worked example (character table of $S_{3}$ ). Build the three rows from the degree- $1$ homomorphisms and the permutation representation, using the class sizes $1, 3, 2$ .

Rows $1$ and $2$ (degree $1$ ). The trivial character is $χ_{triv} = (1, 1, 1)$ . The sign character sends even permutations to $+ 1$ and odd ones to $- 1$ ; transpositions are odd and $3$ -cycles even, so $χ_{sign} = (1, - 1, 1)$ .

Row $3$ (degree $2$ ). Let $S_{3}$ act on the three symbols. The character $π$ of this degree- $3$ permutation representation counts fixed points — the identity fixes three, a transposition fixes one, a $3$ -cycle fixes none:

π (1) = 3, π ((12)) = 1, π ((123)) = 0.

The all-ones vector spans a trivial subrepresentation, so $π = χ_{triv} + χ_{2}$ and

χ_{2} = π - χ_{triv} = (3 - 1, 1 - 1, 0 - 1) = (2, 0, - 1) .

Orthonormality. With the class-weighted inner product $⟨ θ, ψ ⟩ = \frac{1}{6} \sum_{c} ∣ c ∣ θ (c) \overline{ψ (c)}$ ,

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

so $χ_{2}$ is irreducible, while

⟨ χ_{triv}, χ_{2} ⟩ = \frac{1}{6} (1 \cdot 1 \cdot 2 + 3 \cdot 1 \cdot 0 + 2 \cdot 1 \cdot (- 1)) = \frac{1}{6} (2 - 2) = 0,

⟨ χ_{triv}, χ_{sign} ⟩ = \frac{1}{6} (1 - 3 + 2) = 0.

The three rows are orthonormal, and $1^{2} + 1^{2} + 2^{2} = 6 = ∣ G ∣$ confirms there are no further irreducibles.⁴

The same method builds the tables of $D_{8}$ and $Q_{8}$ , each with four degree- $1$ characters and one degree- $2$ character. The two groups share an identical character table even though they are not isomorphic, so the table does not always determine the group.⁴

Multiplicities read off the same inner product. The regular representation decomposes with each irreducible appearing as often as its degree.

Further directions

Character theory converts group-theoretic questions into linear algebra over a finite table. Counting solutions, detecting normal subgroups (a normal subgroup is a union of conjugacy classes, visible as a set of columns), and Burnside's theorem that groups of order $p^{a} q^{b}$ are solvable all reduce to manipulating characters. Two generalizations extend the theory: induced characters, which build representations of $G$ from those of a subgroup, and modular representation theory over fields where Maschke's theorem fails and the group ring is no longer semisimple. Both study a group through the modules over its group ring — the same module-theoretic lens the commutative-algebra capstone applied to polynomial rings.

Dummit & Foote, §18.1 — linear and matrix representations, the group ring $F G$ as an $F$ -algebra, the equivalence between representations and $F G$ -modules, invariant subspaces as submodules, and Maschke's theorem with the averaging projection. ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶
Dummit & Foote, §18.2 — Wedderburn's theorem, semisimplicity of $C G$ , the finiteness of the irreducibles, the identity $\sum d_{i}^{2} = ∣ G ∣$ , and the count $∣ G : G^{'} ∣$ of degree- $1$ representations. ↩ ↩² ↩³
Dummit & Foote, §18.3 — characters as traces, class functions, the equality of the number of irreducible characters with the number of conjugacy classes, the Hermitian inner product, and the first and second orthogonality relations. ↩ ↩² ↩³
Dummit & Foote, §19.1 — the character table, the tables of $Z_{n}$ , $S_{3}$ , $D_{8}$ , and $Q_{8}$ , and the fact that $D_{8}$ and $Q_{8}$ share a character table without being isomorphic. ↩ ↩² ↩³

Representations

The group ring

Complete reducibility

The Wedderburn count

Characters

The character table

Further directions

Footnotes