Editing Character theory (section)

===Orthogonality relations===
{{main|Schur orthogonality relations}}
The space of complex-valued [[class function]]s of a finite group {{mvar|G}} has a natural [[inner product]]:

:<math>\left \langle \alpha, \beta\right \rangle := \frac{1}{|G|}\sum_{g \in G} \alpha(g) \overline{\beta(g)}</math>

where {{math|{{overline|''β''(''g'')}}}} is the [[complex conjugate]] of {{math|''β''(''g'')}}. With respect to this inner product, the irreducible characters form an [[orthonormal basis]] for the space of class-functions, and this yields the orthogonality relation for the rows of the character table:

:<math>\left \langle \chi_i, \chi_j \right \rangle  = \begin{cases} 0 & \mbox{ if } i \ne j, \\ 1 & \mbox{ if } i = j. \end{cases}</math>

For {{math|''g'', ''h''}} in {{mvar|G}}, applying the same inner product to the columns of the character table yields:

:<math>\sum_{\chi_i} \chi_i(g) \overline{\chi_i(h)} = \begin{cases} \left | C_G(g) \right |, & \mbox{ if } g, h \mbox{ are conjugate } \\ 0 & \mbox{ otherwise.}\end{cases}</math>

where the sum is over all of the irreducible characters {{math|''χ<sub>i</sub>''}} of {{mvar|G}} and the symbol {{math|{{pipe}}''C<sub>G</sub>''(''g''){{pipe}}}} denotes the order of the [[centralizer]] of {{mvar|g}}. Note that since {{mvar|g}} and {{mvar|h}} are conjugate iff they are in the same column of the character table, this implies that the columns of the character table are orthogonal.

The orthogonality relations can aid many computations including:
* Decomposing an unknown character as a linear combination of irreducible characters.
* Constructing the complete character table when only some of the irreducible characters are known.
* Finding the orders of the centralizers of representatives of the conjugacy classes of a group.
* Finding the order of the group.