Editing Character theory (section)

==Properties==
* Characters are [[class function]]s, that is, they each take a constant value on a given [[conjugacy class]]. More precisely, the set of irreducible characters of a given group {{mvar|G}} into a field {{mvar|F}} form a [[basis (linear algebra)|basis]] of the {{mvar|F}}-vector space of all class functions {{math|''G'' → ''F''}}.
* [[Representation_theory#Equivariant_maps_and_isomorphisms|Isomorphic]] representations have the same characters. Over a field of [[characteristic (algebra)|characteristic]] {{math|0}}, two representations are isomorphic [[if and only if]] they have the same character.<ref>Nicolas Bourbaki, ''Algèbre'', Springer-Verlag, 2012, Chap. 8, p392</ref>
* If a representation is the [[direct sum of representations|direct sum]] of [[subrepresentation]]s, then the corresponding character is the sum of the characters of those subrepresentations.
* If a character of the finite group {{mvar|G}} is restricted to a [[subgroup]] {{mvar|H}}, then the result is also a character of {{mvar|H}}.
* Every character value {{math|''χ''(''g'')}} is a sum of {{mvar|n}} {{mvar|m}}-th [[roots of unity]], where {{mvar|n}} is the degree (that is, the dimension of the associated vector space) of the representation with character {{mvar|χ}} and {{mvar|m}} is the [[order (group theory)|order]] of {{mvar|g}}. In particular, when {{math|1=''F'' = '''C'''}}, every such character value is an [[algebraic integer]].
* If {{math|1=''F'' = '''C'''}} and {{mvar|χ}} is irreducible, then  <math display="block">[G:C_G(x)]\frac{\chi(x)}{\chi(1)}</math> is an [[algebraic integer]] for all {{mvar|x}} in {{mvar|G}}.
* If {{mvar|F}} is [[algebraically closed]] and {{math|[[characteristic of a ring|char]](''F'')}} does not divide the [[order of a group|order]] of {{mvar|G}}, then the number of irreducible characters of {{mvar|G}} is equal to the number of [[conjugacy class]]es of {{mvar|G}}. Furthermore, in this case, the degrees of the irreducible characters are divisors of the order of {{mvar|G}} (and they even divide {{math|[''G'' : ''Z''(''G'')]}} if {{math|''F'' {{=}} '''C'''}}).

===Arithmetic properties===
Let ρ and σ be representations of {{mvar|G}}. Then the following identities hold:
*<math>\chi_{\rho \oplus \sigma} = \chi_\rho + \chi_\sigma</math>
*<math>\chi_{\rho \otimes \sigma} = \chi_\rho \cdot \chi_\sigma</math>
*<math>\chi_{\rho^*} = \overline {\chi_\rho}</math>
*<math>\chi_{{\scriptscriptstyle \rm{Alt}^2} \rho}(g) = \tfrac{1}{2}\! \left[ \left(\chi_\rho (g) \right)^2 - \chi_\rho (g^2) \right]</math>
*<math>\chi_{{\scriptscriptstyle \rm{Sym}^2} \rho}(g) = \tfrac{1}{2}\! \left[ \left(\chi_\rho (g) \right)^2 + \chi_\rho (g^2) \right]</math>
where {{math|''ρ''⊕''σ''}} is the [[direct sum of representations|direct sum]], {{math|''ρ''⊗''σ''}} is the [[tensor product]], {{math|''ρ''<sup>∗</sup>}} denotes the [[conjugate transpose]] of {{mvar|ρ}}, and {{math|Alt<sup>2</sup>}} is the [[exterior algebra|alternating product]] {{math|Alt<sup>2</sup>''ρ'' {{=}} ''ρ'' ∧ ''ρ''}} and {{math|Sym<sup>2</sup>}} is the [[symmetric square]], which is determined by
<math display="block">\rho \otimes \rho = \left(\rho \wedge \rho \right) \oplus \textrm{Sym}^2 \rho.</math>