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Character theory is an essential tool in the Classification Of Finite Simple Groups . Close to half of the proof of the Feit-Thompson Theorem involves intricate calculations with character values. Easier, but still essential, results relying on character theory include the Burnside Theorem , and a theorem of Richard Brauer and Michio Suzuki stating that a finite Simple Group cannot have a Sylow 2 Subgroup that is a generalized Quaternion Group .


DEFINITIONS


The character of a group representation
: ho\ :\ G\ o\ GL(n,\mathbb{C})
of a group ''G'' is the function
:\chi_{ ho}\ :\ G\ o\ \mathbb{C}
which sends ''g'' in ''G'' to the Trace (the sum of the diagonal elements) of the matrix ρ(''g''). A character χρ is called irreducible if ρ is an Irreducible Representation . It is called '''linear''' if the Dimension of ρ is 1.

The kernel of a character χρ is the set:
:\ker \chi_{ ho} := \left \lbrace g \in G \mid \chi_{ ho}(g) = \chi_{ ho}(1) ight brace where \chi_{ ho}(1) is the value of χρ on the group identity.

If ρ is a representation of ''G'' of dimension ''k'' and 1 is the identity of ''G'' then
:\chi_{ ho}(1) = \operatorname{Tr}( ho(1)) = \operatorname{Tr} \left ( \begin{vmatrix}1 & & 0\ & \ddots & \ 0 & & 1\end{vmatrix} ight ) = \sum_{i = 1}^k 1 = \dim ho = k

Unlike the situation with the Character Group , the characters ''of a'' group do not, in general, ''form'' a group themselves.


PROPERTIES




  • If a representation is the direct sum of subrepresentations, then the corresponding character is the sum of the subrepresentations' characters.


  • Every character \chi\ (g) is a sum of n ''m''th Roots Of Unity where n is the degree (ie, the dimension n of the vector space over which GL(n) acts) of the representation, and m is the Order of g.


  { Class "wikitable"
  :<math>\left \langle \chi I, \chi J Ight Angle : rac{1}{ \left G ight }\sum_{g \in G} \chi_i(g) \overline{\chi_j(g)}</math> where <math>\overline{\chi_j(g)}</math> means the complex conjugate of the value of <math>\chi_{j}</math> on ''g''
  :For <math>g, H \in G</math> The Sum <math>\sum {\chi I} \chi I(g) \overline{\chi I(h)} \begin{cases}\left C_G(g) ight , & \mbox{ if } g, h \mbox{ are conjugate } \ 0 & \mbox{ otherwise}\end{cases}</math>