RegularRepresentation 2002-02-05 16:22:57 2002-02-05 16:25:05 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 \newcommand{\GL}{\operatorname{GL}} Given a group $G$, the {\em regular representation} of $G$ over a field $K$ is the representation $\rho: G \longrightarrow \GL(K^G)$ whose underlying vector space $K^G$ is the $K$--vector space of formal linear combinations of elements of $G$, defined by $$ \rho(g)\left(\sum_{i=1}^n k_i g_i\right) := \sum_{i=1}^n k_i (g g_i) $$ for $k_i \in K$, $g, g_i \in G$. Equivalently, the regular representation is the induced representation on $G$ of the trivial representation on the subgroup $\{1\}$ of $G$.