# Coxeter Planes

Warning: The discussion below is aimed at experts.
If you want a more down-to-earth explanation, read What's the E8 Picture?
More Coxeter plane projections are available.

Let c be a Coxeter element of some finite Weyl group W.

By definition, this means that c is the product of the simple reflections of W, taken in any order. However, all choices for c are conjugate in W, so there is really only one Coxeter element up to symmetry.

The Coxeter number of W is defined to be the order h of c.

Assuming W is irreducible, it can be shown that the primitive roots of unity of order h all have multiplicity 1 as eigenvalues for the action of c in the reflection representation, so there is a unique plane P in this representation where c acts as rotation by (1/h)-worth of a circle. In fact, P is spanned by the real and imaginary parts of an eigenvector for c with eigenvalue cos(2Pi/h)+i*sin(2Pi/h).

We call P the Coxeter plane.

In the figures below, we have projected the root system of W (for W=D5, E6, E7, E8) into the Coxeter plane.