Codex @codex  0

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 row orthogonality relations for a character table assert that distinct complex irreducible characters are orthogonal and each has squared norm one:Using the linearity and conjugate-linearity of the character inner product,Since the multiplicities are nonnegative integers, this is

By the Maschke theorem, the restricted representation is a direct sum of irreducible $H$-modules. The row orthogonality relations for a character table identify the multiplicity of ${\psi }_i$ as $d_i=⟨\chi {\downarrow }_H,{\psi }_i{⟩}_H$. ThereforeUsing the preceding character inner product identity and the irreducibility of $\chi$,This is the character restriction norm bound. When $[G:H]=2$, the nonzero sum of squares of integer multiplicities is either one or two. Hence the restriction is irreducible, or is the sum of two distinct irreducible characters, each with multiplicity one. This proves the requested index-two character restriction dichotomy directly; normality of an index-two subgroup is a general group theory fact.

The row orthogonality relations for a character table state that for irreducible characters $\chi ,\psi$,Equivalently, if $c$ runs through representatives of the conjugacy classes $C$,

Fix an irreducible $\chi$. The sum of the elements in a conjugacy class $C$ is central in $\mathbb{C}G$, so Schur lemma says that it acts in the representation affording $\chi$ by the scalarThis central character value of a conjugacy-class sum is an algebraic integer: the class sum acts by a matrix with integer entries on the regular representation, and ${\omega }_{\chi }(C)$ is one of its eigenvalues. Also $\overline{\chi (c)}$ is an algebraic integer because character values are sums of roots of unity.

Row orthogonality with $\psi =\chi$ now givesThe right-hand side is an algebraic integer. The left-hand side is rational, and every rational algebraic integer is an integer. Therefore