Longest Weyl-group element
= Longest Weyl-group element
{title2=$w_0$}
The longest element of a finite <Weyl group> is the element sending all positive <roots of a root system> to negative roots. It is the <longest element of a finite Coxeter group>. For the <G2 root system> it is $-I$: its twelve roots form two hexagons, and the dihedral <reflection group of a root system> contains the rotation through $\pi$. It determines the <highest weight of a dual representation>.