If you want a more down-to-earth explanation, read What's the E8 Picture?

More Coxeter plane projections are available.

Let

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]:

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

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

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

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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This page last modified Tue Nov 27 10:51:13 EST 2007