The representation theory of finite groups is also treated on Representations; this page adds induced representations, representations of the symmetric group, and the Frobenius–Schur indicator.
Maschke and Schur
By Maschke's theorem, every finite-dimensional complex representation of a finite group \(G\) is a direct sum of irreducible representations, and by Schur's lemma the multiplicities are inner products of characters. If \(d_1,\ldots,d_r\) are the dimensions of the irreducible representations, then \[ \size G = \sum_i d_i^2, \] and \(r\) is the number of conjugacy classes of \(G\).
Building a character table
The number of irreducible characters equals the number of conjugacy classes, so the table is square.
The degree-one characters are the characters of \(G/[G,G]\); they include the trivial character.
The degrees satisfy \(\size G = \sum_i d_i^2\) and divide \(\size G\).
For an action of \(G\) on a finite set, the permutation character is \(\chi(g) = \size{\Fix(g)}\); if the action is \(2\)-transitive, \(\chi-\chi_{\text{triv}}\) is irreducible.
The remaining entries are determined by the column orthogonality relations.
Induced representations, the symmetric group, and indicators
Induced and restricted representations. For \(H\le G\), a character \(\psi\) of \(H\), and a character \(\chi\) of \(G\), Frobenius reciprocity gives \(\inner{\Ind_H^G \psi}{\chi}_G = \inner{\psi}{\Res^G_H\chi}_H\); characters of \(G\) are obtained by inducing characters of subgroups and decomposing.
The symmetric group. The irreducible representations of \(S_n\) are indexed by the partitions of \(n\), their characters are computed by the Murnaghan–Nakayama rule, and their dimensions by the hook length formula; see Symmetric functions.
Real and quaternionic representations. The Frobenius–Schur indicator \({1\over\size G}\sum_{g\in G}\chi(g^2)\) of an irreducible character \(\chi\) is \(1\), \(0\), or \(-1\) according as the representation is realizable over \(\RR\), has a non-real character, or is quaternionic.