# induced representation

Let $G$ be a group, $H\subset G$ a subgroup, and $V$ a representation of $H$, considered as a $\mathbb{Z}[H]$–module. The 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

 $g(\sigma v):=\tau(hv)$

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 $\{g_{\sigma}\}$.

Title induced representation InducedRepresentation 2013-03-22 12:17:33 2013-03-22 12:17:33 djao (24) djao (24) 4 djao (24) Definition msc 20C99