Character orthogonality

Irreducible characters are orthonormal for $⟨\chi ,\psi ⟩=|G{|}^{-1}{\sum }_g\chi (g)\overline{\psi (g)}$, and ${\sum }_{\chi }\chi (1{)}^2=|G|$.

For a finite group, the sum of products of irreducible character values is zero when $g,h$ are not conjugate, and equals $|C_G(g)|$ when $g,h$ are conjugate. To prove this from row orthogonality relations for a character table, choose representatives $g_j$ of its conjugacy classes and form $U_{ij}=\sqrt{|g_j^G|/|G|}{\chi }_i(g_j)$. Row orthogonality says $U{U}^{\ast }=I$. There are equally many irreducible characters and conjugacy classes, so this is a square unitary matrix and $U^{\ast }U=I$ as well. Its column equations give the result, using $|G|/|g_j^G|=|C_G(g_j)|$.

The irreducible characters of a finite group form a basis of its complex class functions. Equality of every irreducible character value therefore forces equality of every class-function value, including indicators of conjugacy classes, and hence conjugacy. The converse follows from the class-function property.

For a finite group, the regular representation contains $d_{\rho }$ copies of each irreducible representation $\rho$. Its character of a representation is $|G|$ at the identity and zero elsewhere. Thus ${\sum }_{\rho }{d}_{\rho }{\chi }_{\rho }(x)=|G|1_{\{x=e\}}$, including the trivial representation. This formula converts independent uniform expectations into constrained configuration counts.

For irreducible complex characters $\chi ,\psi$ of a finite group,Equivalently, summing over conjugacy classes $C$ with representatives $c$ gives

For an irreducible character $\chi$ and a conjugacy class $C$ represented by $c$, the central group-algebra element ${\sum }_{x\in C}x$ acts as the scalarThis scalar is an algebraic integer, since the class sum acts by an integer matrix on the regular representation.

For every irreducible complex character $\chi$ of a finite group $G$,Indeed, row orthogonality expresses $|G|/\chi (1)$ as a sum of products of algebraic integers:It is a rational algebraic integer and therefore an integer.

A degree-two irreducible representation of a nonabelian finite simple group would be faithful. Its determinant is a linear character and hence trivial, so its image lies in ${\mathrm{SL}}_2(\mathbb{C})$. Degree divisibility makes the group order even; an involution must map to the unique nonidentity involution $-I$ in ${\mathrm{SL}}_2(\mathbb{C})$ and would therefore be central, a contradiction.

For a prime $p$, either a group of order $2p$ is abelian and all irreducible character degrees are one, or $p$ is odd and its degrees are

If $p>q$ are prime and $G$ is nonabelian of order $pq$, then it has $q$ linear characters and $(p-1)/q$ irreducible characters of degree $q$. It consequently hasconjugacy classes.

For complex class functions on a finite group $G$, the character inner product isIf $\psi$ is irreducible, this inner product is its multiplicity in the representation with character $\chi$.