Charaktertabellen

Charaktertabellen, in English known as character tables, are tabular summaries of the irreducible representations of a finite group. Each table lists the group's conjugacy classes as columns and the irreducible characters as rows. The entry at row χ and column C is the character χ(C) — the trace of any group element in C under the corresponding irreducible representation. The rows encode the representation degrees, while the columns reflect the group's class structure.

Character tables satisfy two key orthogonality relations: row orthogonality and column orthogonality. Concretely, the sum over all conjugacy classes of the class size times χ_i(C) times the complex conjugate of χ_j(C) equals the order of the group if i = j and is zero otherwise. The sum of the squares of the degrees of the irreducible representations also equals the order of the group. With these relations, the table determines all irreducible representations and helps decompose arbitrary representations into irreducibles.

Construction typically begins by determining the conjugacy classes, then using the sum-of-squares formula to find possible degrees, and finally applying orthogonality relations to fill in the remaining characters, often aided by knowledge of subgroups and known representations.

Example: for the symmetric group S3 (order 6), there are three irreducible representations: two one-dimensional and one two-dimensional. The conjugacy classes are e, transpositions, and 3-cycles. The character table is:

trivial: 1 1 1

sign: 1 -1 1

standard: 2 0 -1

Applications occur in mathematics and physics: character tables classify symmetries, aid in decomposing representations, and provide selection rules in spectroscopy.