character
Let $\rho :G\u27f6\mathrm{GL}(V)$ be a finite dimensional representation of a group $G$ (i.e., $V$ is a finite dimensional vector space^{} over its scalar field $K$). The character^{} of $\rho $ is the function ${\chi}_{V}:G\u27f6K$ defined by
$${\chi}_{V}(g):=\mathrm{Tr}(\rho (g))$$ 
where $\mathrm{Tr}$ is the trace function.
Properties:

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${\chi}_{V}(g)={\chi}_{V}(h)$ if $g$ is conjugate^{} to $h$ in $G$. (Equivalently, a character is a class function on $G$.)

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If $G$ is finite, the characters of the irreducible representations of $G$ over the complex numbers form a basis of the vector space of all class functions on $G$ (with pointwise addition and scalar multiplication).

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Over the complex numbers, the characters of the irreducible representations of $G$ are orthonormal under the inner product
$$({\chi}_{1},{\chi}_{2}):=\frac{1}{G}\sum _{g\in G}\overline{{\chi}_{1}(g)}{\chi}_{2}(g)$$
Title  character 

Canonical name  Character 
Date of creation  20130322 12:17:54 
Last modified on  20130322 12:17:54 
Owner  djao (24) 
Last modified by  djao (24) 
Numerical id  7 
Author  djao (24) 
Entry type  Definition 
Classification  msc 20C99 