# 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 $\ell:W_{R}\to\mathbb{Z}$, where $\ell(w)$ 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 WeylGroup 2013-03-22 13:11:52 2013-03-22 13:11:52 mathcam (2727) mathcam (2727) 6 mathcam (2727) Definition msc 17B20