Character Tables — Mathematics Knowledge System

Overview

The character table of a finite group G is a square matrix whose rows are indexed by the irreducible complex representations (up to isomorphism) and whose columns are indexed by the conjugacy classes. The entry in row ρ and column [g] is χ_ρ(g) = tr(ρ(g)). Character tables encode the representation theory of G completely: they determine all irreducible representations up to isomorphism, allow decomposition of any representation into irreducibles, and reveal structural properties of G such as normal subgroups and the abelianisation.

Intuition

The character table of G is a square matrix (rows = irreducible representations, columns = conjugacy classes). Its rows are orthogonal (first orthogonality relation) and its columns are also orthogonal (second orthogonality relation). You can read off: the number of irreducibles, their dimensions (from the first column), which representations are real, and which normal subgroups G has (from the structure of the table).

Formal Definition

Definition

Let G have conjugacy classes C₁, …, Cₖ (with C₁ = {e}) and irreducible complex representations V₁, …, Vₖ. The character table is the k×k matrix X with entries X_{ij} = χᵢ(gⱼ) where gⱼ ∈ Cⱼ.

X_{ij} = \chi_i(g_j), \quad i,j = 1,\ldots,k

Character table entries

\frac{1}{|G|}\sum_{j=1}^k |C_j|\,\chi_i(g_j)\,\overline{\chi_{i'}(g_j)} = \delta_{ii'}

Row orthogonality (first orthogonality relation)

\sum_{i=1}^k \chi_i(g_j)\,\overline{\chi_i(g_{j'})} = \frac{|G|}{|C_j|}\,\delta_{jj'}

Column orthogonality (second orthogonality relation)

\sum_{i=1}^k (\dim V_i)^2 = |G|, \quad \dim V_i = \chi_i(e) = X_{i1}

Sum of squares identity; dimensions from first column

Notation

Notation	Meaning
$X$	Character table matrix
$\chi_i(C_j)$	Value of character χᵢ on conjugacy class Cⱼ
$k$	Number of conjugacy classes = number of irreducible representations

Properties

First column gives dimensions

\chi_i(e) = \dim V_i \geq 1 \text{ for all } i

Complex conjugate rows

If \chi_i is an irreducible character, so is \overline{\chi_i} = \chi_{i^*} \text{ (the dual representation)}

Trivial representation row

One row consists entirely of 1s (the trivial representation)

Theorems

Theorem 1: Orthogonality of rows and columns

The character table X satisfies \overline{X}^T D X = |G| I and X D \overline{X}^T = |G| I where D = \mathrm{diag}(|C_1|, \ldots, |C_k|).

Theorem 2: Normal subgroup detection

N \trianglelefteq G \iff N = \bigcap_{\chi_i : \chi_i|_N = \dim(V_i)} \ker \chi_i

Theorem 3: Burnside's theorem

If |G| = p^a q^b for primes p, q, then G is solvable.

Worked Examples

1
S₃ has 3 conjugacy classes: {e}, {(12),(13),(23)}, {(123),(132)}. Three irreducibles: trivial (1), sign (1), standard (2).
$\text{Classes: } \{e\},\; \{3\text{-transpositions}\},\; \{2\text{ 3-cycles}\}$
2
Trivial rep: all values 1. Sign rep: +1 on e and 3-cycles, -1 on transpositions. Standard rep: 2, 0, -1.
3
Arranging into a table (rows = irreps, columns = conjugacy classes):
$\begin{pmatrix} 1 & 1 & 1 \\ 1 & -1 & 1 \\ 2 & 0 & -1 \end{pmatrix}$

✓ Answer

The 3×3 character table of S₃; orthogonality of rows and columns can be verified directly.

Practice Problems

Mediumfree response

Construct the character table of ℤ/4ℤ.

Hardproof writing

Prove that if G has exactly two conjugacy classes then G ≅ ℤ/2ℤ.

Common Mistakes

Common Mistake

Thinking the character table determines G up to isomorphism

Non-isomorphic groups can have the same character table. The smallest example is D₄ (dihedral group of order 8) and Q₈ (quaternion group) — they have identical character tables but are non-isomorphic.

Common Mistake

Confusing rows (irreps) and columns (conjugacy classes)

The character table is rows = irreps, columns = conjugacy classes. Both row orthogonality and column orthogonality hold, but with different weight factors.

Quiz

The number of rows (and columns) of the character table of G equals:

The order |G|The number of conjugacy classes of GThe number of subgroups of GThe rank of the centre Z(G)

The entry in the first column (identity class) of row i of the character table gives:

The order of the representationThe dimension of the ith irreducible representationZero alwaysThe character of the trivial representation

Historical Background

Character tables were first computed by Frobenius for symmetric groups in the 1890s–1900s. The character table of S₄ was known to Frobenius by 1900. The character table of the Monster group — the largest sporadic simple group, of order approximately 8×10⁵³ — was a major achievement of the 1970s–1980s, completed by Conway, Norton, and others, and played a central role in Monstrous Moonshine. The Atlas of Finite Groups (1985) tabulates character tables of all finite simple groups.

1896–1900
Frobenius computes character tables of symmetric groups
Georg Frobenius
1985
The Atlas of Finite Groups published, cataloguing character tables of finite simple groups
John Conway, Robert Curtis, Simon Norton, Richard Parker, Robert Wilson
1992
Borcherds proves the Moonshine conjecture, relating Monster character table to modular forms
Richard Borcherds

Summary

The character table is a square matrix X with X_{ij} = χᵢ(gⱼ), rows indexed by irreps, columns by conjugacy classes.
Row orthogonality: distinct irreducible characters are orthogonal; each has norm 1.
Column orthogonality: distinct conjugacy classes are orthogonal in a dual sense.
Normal subgroups are exactly the intersections of kernels of irreducible characters.
Non-isomorphic groups can share the same character table (e.g., D₄ and Q₈).

References

BookSerre, J.-P. — Linear Representations of Finite Groups (1977), Chapter 2
BookJames, G. & Liebeck, M. — Representations and Characters of Groups, 2nd ed. (2001)
WebsiteWikipedia — Character table