Editing Character table (section)

==Definition and example==
The irreducible [[complex number|complex]] characters of a [[finite group]] form a '''character table''' which encodes much useful information about the [[group (mathematics)|group]] ''G'' in a concise form.  Each row is labelled by an [[irreducible character]] and the entries in the row are the values of that character on any representative of the respective [[conjugacy class]] of ''G'' (because characters are [[class function]]s). The columns are labelled by (representatives of) the conjugacy classes of ''G''. It is customary to label the first row by the character of the '''[[trivial representation]]''', which is the trivial action of {{mvar|G}} on a [[dimension (vector space)|1-dimensional]] [[vector space]] by <math>\rho(g)=1</math> for all <math>g\in G</math>. Each entry in the first row is therefore 1. Similarly, it is customary to label the first column by the [[identity element|identity]]. The entries of the first column are the values of the irreducible characters at the identity, the [[Degree of a character|degree]]s of the irreducible characters. Characters of degree 1 are known as '''linear characters'''.

Here is the character table of ''C''<sub>3</sub> = ''<nowiki><u></nowiki>'', the [[cyclic group]] with three elements and [[cyclic group|generator]] ''u'':

{| class="wikitable"
|-
|&nbsp;
|(1)
|(''u'')
|(''u''<sup>2</sup>)
|-
|'''1'''
|1
|1
|1
|-
|χ<sub>1</sub>
|1
|ω
|ω<sup>2</sup>
|-
|χ<sub>2</sub>
|1
|ω<sup>2</sup>
|ω
|-
|}

where ω is a primitive cube [[root of unity]]. The character table for general cyclic groups is (a scalar multiple of) the [[DFT matrix]].

Another example is the character table of <math>S_3</math>:

{| class="wikitable"
|-
|&nbsp;
|(1)
|(12)
|(123)
|-
|χ<sub>triv</sub>
|1
|1
|1
|-
|χ<sub>sgn</sub>
|1
|−1
|1
|-
|χ<sub>stand</sub>
|2
|0
|−1
|-
|}

where (12) represents the conjugacy class consisting of (12), (13), (23), while (123) represents the conjugacy class consisting of (123), (132). To learn more about character table of symmetric groups, see [http://groupprops.subwiki.org/wiki/Linear_representation_theory_of_symmetric_groups].

The first row of the character table always consists of 1s, and corresponds to the '''[[trivial representation]]''' (the 1-dimensional representation consisting of 1&times;1 matrices containing the entry 1). Further, the character table is always square because (1) irreducible characters are pairwise orthogonal, and (2) no other non-trivial class function is orthogonal to every character. (A class function is one that is constant on conjugacy classes.) This is tied to the important fact that the irreducible representations of a finite group ''G'' are in [[bijection]] with its conjugacy classes.  This bijection also follows by showing that the class sums form a [[basis (linear algebra)|basis]] for the [[center (ring theory)|center]] of the [[group ring|group algebra]] of ''G'', which has dimension equal to the number of irreducible representations of ''G''.