InducedRepresentation 2002-02-05 12:04:55 2002-02-05 12:04:55 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}} \newcommand{\lra}{\longrightarrow} Let $G$ be a group, $H \subset G$ a subgroup, and $V$ a representation of $H$, considered as a $\mathbb{Z}[H]$--module. The {\em induced representation} of $\rho$ on $G$, denoted $\operatorname{Ind}_H^G(V)$, is the $\mathbb{Z}[G]$--module whose underlying vector space is the direct sum $$ \bigoplus_{\sigma \in G/H} \sigma V $$ of formal translates of $V$ by left cosets $\sigma$ in $G/H$, and whose multiplication operation is defined by choosing a set $\{g_\sigma\}_{\sigma \in G/H}$ of coset representatives and setting $$ g(\sigma v) := \tau (h v) $$ where $\tau$ is the unique left coset of $G/H$ containing $g \cdot g_\sigma$ (i.e., such that $g \cdot g_\sigma = g_\tau \cdot h$ for some $h \in H$). One easily verifies that the representation $\operatorname{Ind}_H^G(V)$ is independent of the choice of coset representatives $\{g_\sigma\}$.