# 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 induced representation^{} of $\rho $ on $G$, denoted ${\mathrm{Ind}}_{H}^{G}(V)$, is the $\mathbb{Z}[G]$–module whose underlying vector space^{} is the direct sum^{}

$$\underset{\sigma \in G/H}{\oplus}\sigma V$$ |

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 ${\mathrm{Ind}}_{H}^{G}(V)$ is independent of the choice of coset representatives $\{{g}_{\sigma}\}$.

Title | induced representation |
---|---|

Canonical name | InducedRepresentation |

Date of creation | 2013-03-22 12:17:33 |

Last modified on | 2013-03-22 12:17:33 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 4 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 20C99 |