# Weyl chamber

Let $E$ be a Euclidean vector space, $R\beta \x8a\x82E$ a root system^{}, and
${R}^{+}\beta \x8a\x82R$ a choice of positive roots. We define the positive Weyl chamber (relative to ${R}^{+}$) to be the closed set

$$\mathrm{\pi \x9d\x92\x9e}=\{u\beta \x88\x88E\beta \x88\pounds (u,\mathrm{\Xi \pm})\beta \x89\u20af0\beta \x81\u2019\text{\Beta for all\Beta}\beta \x81\u2019\mathrm{\Xi \pm}\beta \x88\x88{R}^{+}\}.$$ |

A weight which lies inside the positive Weyl chamber is called dominant.

The interior of $\mathrm{\pi \x9d\x92\x9e}$ is a fundamental domain for the action
of the Weyl group^{} on $E$. The image $w\beta \x81\u2019(\mathrm{\pi \x9d\x92\x9e})$ of $\mathrm{\pi \x9d\x92\x9e}$
under the any element $w$ of the Weyl group is called a Weyl
chamber. The Weyl group $W$ acts simply transitively on the set of
Weyl chambers.

Title | Weyl chamber |
---|---|

Canonical name | WeylChamber |

Date of creation | 2013-03-22 13:12:00 |

Last modified on | 2013-03-22 13:12:00 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 8 |

Author | rmilson (146) |

Entry type | Definition |

Classification | msc 17B20 |

Defines | positive Weyl chamber |

Defines | dominant weight |