# Frobenius reciprocity

Let $V$ be a finite-dimensional representation of a finite group^{} $G$, and let $W$ be a representation of a subgroup^{} $H\subset G$. Then the characters^{} of $V$ and $W$ satisfy the inner product relation^{}

$$({\chi}_{\mathrm{Ind}(W)},{\chi}_{V})=({\chi}_{W},{\chi}_{\mathrm{Res}(V)})$$ |

where $\mathrm{Ind}$ and $\mathrm{Res}$ denote the induced representation^{} ${\mathrm{Ind}}_{H}^{G}$ and the restriction representation ${\mathrm{Res}}_{H}^{G}$.

The Frobenius reciprocity theorem is often given in the stronger form which states that $\mathrm{Res}$ and $\mathrm{Ind}$ are adjoint functors^{} between the category^{} of $G$–modules and the category of $H$–modules:

$${\mathrm{Hom}}_{H}(W,\mathrm{Res}(V))={\mathrm{Hom}}_{G}(\mathrm{Ind}(W),V),$$ |

or, equivalently

$$V\otimes \mathrm{Ind}(W)=\mathrm{Ind}(\mathrm{Res}(V)\otimes W).$$ |

Title | Frobenius reciprocity |
---|---|

Canonical name | FrobeniusReciprocity |

Date of creation | 2013-03-22 12:17:51 |

Last modified on | 2013-03-22 12:17:51 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 7 |

Author | djao (24) |

Entry type | Theorem |

Classification | msc 20C99 |