Editing Character theory (section)

==Characters of Lie groups and Lie algebras==
{{See also|Weyl character formula|Algebraic character}}

If <math>G</math> is a [[Lie group]] and <math>\rho</math> a finite-dimensional representation of <math>G</math>, the character <math>\chi_\rho</math> of <math>\rho</math> is defined precisely as for any group as
:<math>\chi_\rho(g)=\operatorname{Tr}(\rho(g))</math>.
Meanwhile, if <math>\mathfrak g</math> is a [[Lie algebra]] and <math>\rho</math> a finite-dimensional representation of <math>\mathfrak g</math>, we can define the character <math>\chi_\rho</math> by
:<math>\chi_\rho(X)=\operatorname{Tr}(e^{\rho(X)})</math>.
The character will satisfy <math>\chi_\rho(\operatorname{Ad}_g(X))=\chi_\rho(X)</math> for all <math>g</math> in the associated Lie group <math> G</math> and all <math>X\in\mathfrak g</math>. If we have a Lie group representation and an associated Lie algebra representation, the character <math>\chi_\rho</math> of the Lie algebra representation is related to the character <math>\Chi_\rho</math> of the group representation by the formula
:<math>\chi_\rho(X)=\Chi_\rho(e^X)</math>.

Suppose now that <math>\mathfrak g</math> is a complex [[semisimple Lie algebra]] with Cartan subalgebra <math>\mathfrak h</math>. The value of the character <math>\chi_\rho</math> of an irreducible representation <math>\rho</math> of <math>\mathfrak g</math> is determined by its values on <math>\mathfrak h</math>. The restriction of the character to <math>\mathfrak h</math> can easily be computed in terms of the [[weight space (representation theory)|weight space]]s, as follows:
:<math>\chi_\rho(H) = \sum_\lambda m_\lambda e^{\lambda(H)},\quad H\in\mathfrak h</math>,
where the sum is over all [[weight (representation theory)|weights]] <math>\lambda</math> of <math>\rho</math> and where <math>m_\lambda</math> is the multiplicity of <math>\lambda</math>.<ref>{{harvnb|Hall|2015}} Proposition 10.12</ref>

The (restriction to <math>\mathfrak h</math> of the) character can be computed more explicitly by the Weyl character formula.