Editing Character theory (section)

==Character tables==
{{Further|Character table}}
The irreducible [[complex number|complex]] characters of a finite group form a '''character table''' which encodes much useful information about the group {{mvar|G}} in a compact form.  Each row is labelled by an irreducible representation and the entries in the row are the characters of the representation on the respective conjugacy class of {{mvar|G}}. The columns are labelled by (representatives of) the conjugacy classes of {{mvar|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 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. Therefore, the first column contains the degree of each irreducible character.

Here is the character table of

:<math>C_3 = \langle u \mid u^{3} = 1 \rangle,</math>

the [[cyclic group]] with three elements and generator ''u'':

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

where {{mvar|ω}} is a [[primitive root of unity|primitive]] third root of unity.

The character table is always square, because the number of irreducible representations is equal to the number of conjugacy classes.<ref>Serre, §2.5</ref>

===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.

===Character table properties===
Certain properties of the group {{mvar|G}} can be deduced from its character table:

* The order of {{mvar|G}} is given by the sum of the squares of the entries of the first column (the degrees of the irreducible characters). More generally, the sum of the squares of the [[absolute value]]s of the entries in any column gives the order of the centralizer of an element of the corresponding conjugacy class.
*All normal subgroups of {{mvar|G}} (and thus whether or not {{mvar|G}} is simple) can be recognised from its character table. The [[Kernel (group theory)|kernel]] of a character {{mvar|χ}} is the set of elements {{mvar|g}} in {{mvar|G}} for which {{math|''χ''(''g'') {{=}} ''χ''(1)}}; this is a normal subgroup of {{mvar|G}}. Each normal subgroup of {{mvar|G}} is the intersection of the kernels of some of the irreducible characters of {{mvar|G}}.
*The [[commutator subgroup]] of {{mvar|G}} is the intersection of the kernels of the linear characters of {{mvar|G}}.
*If {{mvar|G}} is finite, then since the character table is square and has as many rows as conjugacy classes, it follows that {{mvar|G}} is [[abelian group|abelian]] iff each conjugacy class is a singleton iff the character table of {{mvar|G}} is <math>|G| \!\times\! |G|</math> iff each irreducible character is linear.
*It follows, using some results of [[Richard Brauer]] from [[modular representation theory]], that the prime divisors of the orders of the elements of each conjugacy class of a finite group can be deduced from its character table (an observation of [[Graham Higman]]).

The character table does not in general determine the group [[up to]] [[group isomorphism|isomorphism]]: for example, the [[quaternion group]] {{mvar|Q}} and the [[dihedral group]] of {{math|8}} elements, {{math|''D''<sub>4</sub>}}, have the same character table. Brauer asked whether the character table, together with the knowledge of how the powers of elements of its conjugacy classes are distributed, determines a finite group up to isomorphism. In 1964, this was answered in the negative by [[E. C. Dade]].

The linear representations of {{mvar|G}} are themselves a group under the [[tensor product]], since the tensor product of 1-dimensional vector spaces is again 1-dimensional. That is, if <math>\rho_1:G\to V_1</math> and <math> \rho_2:G\to V_2</math> are linear representations, then <math> \rho_1\otimes\rho_2 (g)=(\rho_1(g)\otimes\rho_2(g))</math> defines a new linear representation. This gives rise to a group of linear characters, called the [[character group]] under the operation <math> [\chi_1*\chi_2](g)=\chi_1(g)\chi_2(g)</math>. This group is connected to [[Dirichlet character]]s and [[Fourier analysis]].