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?
New! 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.

D5 [Download a high-resolution version in Encapsulated PostScript]:

D5 plane

E6 [Download a high-resolution version in Encapsulated PostScript]:

E6 plane

E7 [Download a high-resolution version in Encapsulated PostScript]:

E7 plane

E8 [Download a high-resolution version in Encapsulated PostScript]:

E8 plane

The edges in these figures join roots that are nearest neighbors. The fact that there are so many edges illustrates the point that the roots of a root system are tightly packed. Alternatively, one may describe each figure as the projection of the 1-skeleton of the convex hull of a root system.

It turns out that no root projects to 0, so every c-orbit of roots is in a circular arrangement of size h. In D5 and E6 there are rings that project on top of each other, so the number of projected roots in these cases is less than the number of true roots.

The colors of the edges were chosen so that all edges from outer ring i to inner ring j have a color that depends only on i.

April 9, 2007

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