# Weyl group

The Weyl group^{} ${W}_{R}$ of a root system^{} $R\subset E$, where $E$ is a Euclidean vector space,
is the subgroup^{} of $\mathrm{GL}(E)$ generated by reflection^{} in the hyperplanes^{} perpendicular^{}
to the roots. The map of reflection in a root $\alpha $ is given by

$${r}_{\alpha}(v)=v-2\frac{(\alpha ,v)}{(\alpha ,\alpha )}\alpha .$$ |

The Weyl group is generated by reflections in the simple roots for any choice of a set of positive roots. There is a well-defined length function $\mathrm{\ell}:{W}_{R}\to \mathbb{Z}$, where $\mathrm{\ell}(w)$ is the minimal number of reflections in simple roots that $w$ can be written as. This is also the number of positive roots that $w$ takes to negative roots.

Title | Weyl group |
---|---|

Canonical name | WeylGroup |

Date of creation | 2013-03-22 13:11:52 |

Last modified on | 2013-03-22 13:11:52 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 6 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 17B20 |