character

Let $\rho:G\longrightarrow\operatorname{GL}(V)$ 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 $\chi_{V}:G\longrightarrow K$ defined by

\chi_{V}(g):=\operatorname{Tr}(\rho(g))

where $\operatorname{Tr}$ is the trace function.

Properties:

•

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

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).
•

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	2013-03-22 12:17:54
Last modified on	2013-03-22 12:17:54
Owner	djao (24)
Last modified by	djao (24)
Numerical id	7
Author	djao (24)
Entry type	Definition
Classification	msc 20C99