Editing Character table (section)

==Orthogonality relations==
{{Main|Schur orthogonality relations}}
The space of complex-valued class functions of a finite group ''G'' has a natural [[inner product]]:

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

where <math>\overline{\beta(g)}</math> denotes the [[complex conjugate]] of the value of <math>\beta</math> on <math>g</math>.  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 G</math> the orthogonality relation for columns is as follows:

:<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>\chi_i</math> of ''G'' and the symbol <math>\left| C_G(g) \right|</math> denotes the [[order of a group|order]] of the [[centralizer]] of <math>g</math>.

For an arbitrary character <math>\chi_i</math>, it is irreducible [[if and only if]] <math>\left\langle \chi_i, \chi_i \right\rangle = 1</math>.

The orthogonality relations can aid many computations including:
* Decomposing an unknown character as a [[linear combination]] of irreducible characters, i.e. # of copies of irreducible representation ''V''<sub>''i''</sub> in <math>V = \left\langle \chi, \chi_i \right\rangle</math>.
* 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, <math>\left| G \right| = \left| Cl(g) \right| * \sum_{\chi_i} \chi_i(g) \overline{\chi_i(g)}</math>, for any ''g'' in ''G''.

If the irreducible representation ''V'' is non-trivial, then <math>\sum_g \chi(g) = 0.</math>

More specifically, consider the [[regular representation]] which is the permutation obtained from a finite group ''G'' acting on (the [[free module|free vector space]] spanned by) itself. The characters of this representation are <math>\chi(e) = \left| G \right|</math> and <math>\chi(g) = 0</math> for <math>g</math> not the identity.  Then given an irreducible representation <math>V_i</math>,

:<math>\left\langle \chi_{\text{reg}}, \chi_i \right\rangle = \frac{1}{\left| G \right|}\sum_{g \in G} \chi_i(g) \overline{\chi_{\text{reg}}(g)} = \frac{1}{\left| G \right|} \chi_i(1) \overline{\chi_{\text{reg}}(1)} = \operatorname{dim} V_i</math>.

Then decomposing the regular representations as a sum of irreducible representations of ''G'', we get <math>V_{\text{reg}} = \bigoplus V_i^{\operatorname{dim} V_i}</math>, from which we conclude

:<math>|G| = \operatorname{dim} V_{\text{reg}} = \sum(\operatorname{dim} V_i)^2</math>

over all irreducible representations <math>V_i</math>. This sum can help narrow down the dimensions of the irreducible representations in a character table. For example, if the group has order 10 and 4 conjugacy classes (for instance, the [[dihedral group]] of order 10) then the only way to express the order of the group as a sum of four squares is <math>10 = 1^2 + 1^2 + 2^2 + 2^2</math>, so we know the dimensions of all the irreducible representations.