Throughout, \(G\) is a finite group and representations are finite-dimensional over \(\CC\).
Character of a representation (Tag D-CTAKB)
Let \(G\) be a group, let \(k\) be a field, and let \(M\) be a finite-dimensional representation of \(G\) over \(k\), that is, a finite-dimensional \(k\)-vector space with a linear action \(G\actson M\). For \(g\in G\), let \[ T_g\colon M \to M, \qquad m \mapsto g\actson m. \] The character of \(M\) is the function \(\chi_M\colon G\to k\), \(\chi_M(g) \coloneqq \tr T_g\).
(Characters determine representations)
The character \(\chi_V(g) \da \tr\rho_V(g)\) determines \(V\) up to isomorphism: by Maschke's theorem \(V\) is a direct sum of irreducible representations, and by Schur's lemma and the orthogonality relations the multiplicity of each irreducible summand is an inner product of characters.
\(\chi_V\) is a class function, since the trace is invariant under conjugation.
\(\chi_V(1) = \dim V\).
\(\chi_{V\oplus W} = \chi_V + \chi_W\) and \(\chi_{V\tensor W} = \chi_V\chi_W\).
Orthogonality
For representations \(V\) and \(W\), \[ \inner{\chi_V}{\chi_W} \da {1\over\size G}\sum_{g\in G} \chi_V(g)\overline{\chi_W(g)} = \dim \Hom_G(V,W), \] and the irreducible characters form an orthonormal basis of the space of class functions on \(G\). Consequently:
\(V\) is irreducible if and only if \(\inner{\chi_V}{\chi_V} = 1\).
The multiplicity of an irreducible \(V_i\) as a summand of \(W\) is \(\inner{\chi_W}{\chi_{V_i}}\).
Building a character table
The number of irreducible characters equals the number of conjugacy classes, so the character table is square.
The degree-one characters are the characters of the abelianization \(G/[G,G]\), pulled back to \(G\); they include the trivial character, and for \(S_n\) the sign character.
The degrees \(d_i\) of the irreducible characters divide \(\size G\) and satisfy \(\size G = \sum_i d_i^2\).
For an action of \(G\) on a finite set \(X\), the permutation character is \(\chi(g) = \size{\Fix(g)}\); if the action is \(2\)-transitive, then \(\chi-\chi_{\text{triv}}\) is irreducible.
The remaining entries are determined by the column orthogonality relations \(\sum_i \chi_i(g)\overline{\chi_i(h)} = \size{C_G(g)}\) if \(g\) and \(h\) are conjugate and \(0\) otherwise; for \(g=h=1\) this is \(\size G = \sum_i d_i^2\).
(Permutation characters and Burnside's lemma)
For the permutation character \(\chi(g) = \size{\Fix(g)}\) of an action on \(X\), Burnside’s lemma states that the number of orbits is \(\inner{\chi}{\chi_{\text{triv}}}\); see The class equation.