Character 2002-02-07 12:53:24 2003-02-10 13:50:23 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here 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 {\em 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: \begin{itemize} \item $\chi_V(g) = \chi_V(h)$ if $g$ is conjugate to $h$ in $G$. (Equivalently, a character is a class function on $G$.) \item 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). \item Over the complex numbers, the characters of the irreducible representations of $G$ are orthonormal under the inner product $$ (\chi_1, \chi_2) := \frac{1}{|G|} \sum_{g \in G} \overline{\chi_1(g)} \chi_2(g) $$ \end{itemize}