Character tables and class functions

A class function has one freely chosen complex value per conjugacy class, so the dimension of the class-function space equals the number of classes. To count irreducibles, inspect the center of the complex group algebra. A group-algebra element is central precisely when its coefficients are constant on conjugacy classes, giving a basis of class sums. On the other hand, the matrix-block decomposition gives one central scalar per irreducible block. These two computations of the center dimension prove the required equality.

Irreducible characters are already orthonormal, hence independent. Their number now equals the dimension of the class-function space, so they are a basis. A table built from some known irreducibles is complete when its independent rows exhaust this dimension, with the sum-of-squares formula supplying a useful extra check. Dimension data alone do not determine every character value. In particular, conjugate one-dimensional characters of a cyclic group have the same dimension but differ at nonidentity elements.

Multiply each table column by the square root of its class size divided by the group order. Row orthogonality says the resulting square matrix has orthonormal rows, so it is unitary and its columns are orthonormal too. Undoing those scale factors gives zero inner product between distinct class columns. The squared norm of a column is the group order divided by that class size, which is the size of the element's centralizer by orbit–stabilizer for conjugation.

For the symmetric group on three letters, the identity, transposition, and three-cycle columns have squared norms six, two, and three. The table rows are trivial, sign, and standard, with values respectively one everywhere, the permutation signs, and two/zero/minus one. The transposition column has entries one, minus one, and zero, producing norm two. The three-cycle column has entries one, one, and minus one, producing norm three. These checks catch errors that might escape a dimension calculation at the identity.

The cyclic group of order three has three singleton classes and three one-dimensional characters. If its generator has primitive cube root value omega, the rows are the constant row, the successive powers of omega, and their complex conjugates. The finite geometric identity makes distinct rows orthogonal. Complex conjugation cannot be dropped: the sum of squares of the values of a nontrivial row is zero, whereas its squared Hermitian norm is one. A positive inner product requires the conjugate.

A class function is a genuine complex representation character exactly when its expansion in irreducible rows has nonnegative integer coefficients. The necessity follows from decomposition, and the sufficiency follows by taking the indicated direct sum. A norm-one class function alone need not be genuine: the negative of an irreducible character also has norm one. Similarly, a class indicator is a useful function but usually has fractional or negative expansion coefficients. The irreducibility norm criterion assumes at the outset that the function is already an actual character.