A topic in the Open Knowledge Graph — a free, open map of 15,290 topics and the order to learn them in.
A character table is a square matrix whose rows correspond to irreducible representations and whose columns to conjugacy classes of a finite group G, with entries χᵢ(Cⱼ). The orthogonality relations constrain these entries so tightly that the table encodes the full representation theory of G. Computing the character table is often the first concrete goal when studying a group's representations.
The character table of a finite group G organizes all irreducible character values into a single matrix. The rows are indexed by the non-isomorphic irreducible representations ρ₁, …, ρₖ, the columns by the conjugacy classes C₁, …, Cₖ (with C₁ = {e} by convention), and the entry in row i, column j is χᵢ(Cⱼ). Since the number of irreducible representations equals the number of conjugacy classes, this table is always square.
The orthogonality relations provide powerful constraints for computing the table. Row orthogonality says Σⱼ |Cⱼ|/|G| · χᵢ(Cⱼ) conjugate(χₘ(Cⱼ)) = δᵢₘ, and column orthogonality gives Σᵢ χᵢ(Cⱼ) conjugate(χᵢ(Cₗ)) = |G|/|Cⱼ| · δⱼₗ. Combined with the sum-of-squares formula Σ dᵢ² = |G| (where dᵢ = χᵢ(e) is the dimension) and the fact that entries are sums of roots of unity, these constraints often determine the table completely or reduce it to a small number of cases.
For S₃, the table has three rows and three columns. The conjugacy classes are {e}, {(12),(13),(23)}, {(123),(132)} with sizes 1, 3, 2. The trivial representation gives row (1, 1, 1). The sign representation gives row (1, −1, 1). The remaining irreducible has dimension 2 (since 1² + 1² + d² = 6 forces d = 2), and the orthogonality relations determine its character values: (2, 0, −1). The complete table is a 3×3 matrix that encodes everything about how S₃ acts on vector spaces.
A subtle point: the character table does not uniquely determine the group. The dihedral group D₄ and the quaternion group Q₈ are non-isomorphic groups of order 8 with identical character tables. The table captures the representation-theoretic structure faithfully but loses information about the multiplication table of the group. Nevertheless, the character table determines many group-theoretic properties: the order of the group, the sizes of conjugacy classes, whether the group is abelian (all irreducibles are one-dimensional), whether it is simple (no row is a sum of the trivial character and another), and more.