# regular representation

Given a group $G$, the regular representation^{} of $G$ over a field $K$ is the representation^{} $\rho :G\u27f6\mathrm{GL}({K}^{G})$ 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 $G$.

Title | regular representation |
---|---|

Canonical name | RegularRepresentation |

Date of creation | 2013-03-22 12:17:40 |

Last modified on | 2013-03-22 12:17:40 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 5 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 20C99 |