Characters compress representations into trace functions. For finite groups over characteristic zero, they retain enough information to decompose representations and build character tables.
Characters and class functions
Definition. If \(\rho:G\to\operatorname{GL}(V)\) is a finite-dimensional complex representation, its character is \[ \chi_\rho(g)=\operatorname{tr}(\rho(g)). \] A class function is a function constant on conjugacy classes.
Proposition. Characters are class functions.
Proof. Since \(\rho(hgh^{-1})=\rho(h)\rho(g)\rho(h)^{-1}\), the matrices \(\rho(g)\) and \(\rho(hgh^{-1})\) are similar and have the same trace.
Inner product of class functions
Definition. For class functions \(\chi,\psi:G\to\mathbb C\), define \[ \langle\chi,\psi\rangle=\frac{1}{|G|}\sum_{g\in G}\chi(g)\overline{\psi(g)}. \]
Theorem. Irreducible characters form an orthonormal basis for the vector space of complex class functions on \(G\).
Decomposing representations
Proposition. If \(V\cong\bigoplus_i m_iV_i\) is a decomposition into irreducibles with characters \(\chi_i\), then \[ \chi_V=\sum_i m_i\chi_i, \qquad m_i=\langle\chi_V,\chi_i\rangle. \]
Example. The permutation representation of \(S_3\) on \(\mathbb C^3\) has character values \(3,1,0\) on the conjugacy classes \(1\), transpositions, and \(3\)-cycles. It decomposes as the trivial representation plus the two-dimensional standard representation.
Row and column orthogonality
Theorem. Rows of the character table are orthonormal under the character inner product. Columns satisfy a dual orthogonality relation weighted by centralizer sizes.
Remark. Row orthogonality distinguishes irreducible representations. Column orthogonality reveals constraints on conjugacy classes and centralizers.
Regular character
Proposition. The regular character \(\chi_{\mathrm{reg}}\) satisfies \(\chi_{\mathrm{reg}}(e)=|G|\) and \(\chi_{\mathrm{reg}}(g)=0\) for \(g\ne e\). If \(V_i\) are irreducible, then \[ \chi_{\mathrm{reg}}=\sum_i (\dim V_i)\chi_i, \qquad |G|=\sum_i(\dim V_i)^2. \]
A small character table
Example. For \(S_3\), the irreducible dimensions are \(1,1,2\). The conjugacy classes are \(1\), transpositions, and \(3\)-cycles. The characters are \[ \begin{array}{c|ccc} &1&(12)&(123)\\\hline \mathbf 1&1&1&1\\ \operatorname{sgn}&1&-1&1\\ \operatorname{std}&2&0&-1 \end{array} \] and the squares of dimensions sum to \(1+1+4=6\).
Schur orthogonality
Theorem. For irreducible unitary representations \(\rho,\sigma\) of a finite group, \[ \frac1{|G|}\sum_{g\in G} \rho(g)_{ij}\overline{\sigma(g)_{k\ell}} =\frac{\delta_{\rho\sigma}\delta_{ik}\delta_{j\ell}}{\dim\rho}. \] In particular, irreducible characters are orthonormal class functions.
Proof. For a linear map \(A:V_\sigma\to V_\rho\), average \[ \mathcal A(A)=\frac1{|G|}\sum_g\rho(g)A\sigma(g)^{-1}. \] This is an intertwiner. Schur's lemma makes it zero when \(\rho\not\cong\sigma\), and a scalar multiple of the identity when they are equivalent. Taking traces determines the scalar. Apply this to matrix units to obtain coefficient orthogonality; summing diagonal coefficients gives character orthogonality.
Character table consequences
The number of irreducible complex representations equals the number of conjugacy classes. The column orthogonality relation is \[ \sum_{\chi\in\widehat G}\chi(g)\overline{\chi(h)} = \begin{cases} |C_G(g)|,&g,h\text{ conjugate},\\ 0,&\text{otherwise}. \end{cases} \] Characters detect equivalence because semisimple representations are determined by their irreducible multiplicities \[ m_\chi=\langle\chi_V,\chi\rangle. \]
Example (\(S_3\)). The conjugacy classes have sizes \(1,3,2\). The irreducible characters are \[ \begin{array}{c|ccc} &e&(12)&(123)\\\hline \mathbf1&1&1&1\\ \operatorname{sgn}&1&-1&1\\ \mathrm{std}&2&0&-1 \end{array} \] and \(1^2+1^2+2^2=6=|S_3|\).