character table

A character table is the compact data sheet of a finite group's complex representation theory — a single square grid that encodes every irreducible character at once. Reading it well is like reading a periodic table: a small array of numbers from which a surprising amount of group structure can be recovered.

For a finite group G over the complex numbers, the character table is the square array whose rows are indexed by the irreducible characters chi_1, ..., chi_r and whose columns are indexed by the conjugacy classes C_1, ..., C_r; the entry in row i, column j is chi_i(g_j) for a representative g_j of class C_j. The table is square precisely because the number of irreducible representations equals the number of conjugacy classes. The first row is the trivial character (all 1s) and the first column lists the character degrees chi_i(e).

The table is governed by the orthogonality relations: rows are orthonormal under the inner product weighted by class sizes (row orthogonality), and columns are orthogonal too (column orthogonality). These constraints, together with the sum-of-squares rule that the degrees squared sum to |G|, often determine the entire table by pure arithmetic. Nonetheless the table does not determine the group: the dihedral group D_8 and the quaternion group Q_8, both of order 8, share an identical character table.