**
An Action by the Coxeter Group E _{8}**

(i.e. Another view of Gosset's figure 4

**Introduction.**

This unusual arrangement of 21 cubes was extracted from the graph of an action of the Coxeter group
E_{8}, the symmetry group of Gosset's 8-dimensional semiregular polytope
4_{21},
on a set of 120 elements. This action is quite easy to describe, assuming one is
familiar with the root system **D** for E_{8}. The group E_{8}
obviously acts on this root system, but, moreover, it contains an "antipodal"
involution which maps every vector to its negative. This involution lies in
the center of E_{8}, and so one may quotient to obtain
an action on P**D**, a "projectivized" root system in which each root is identified
with its negative. This projective action of E_{8} is not effective since
the antipodal involution acts as an identity element, but, since the quotient
of E_{8} by the 2-element group generated by this involution is a simple group,
the action is effective if considered an action of the quotient PE_{8}.

**The Graph.**

One can see the entire graph of this action in the following JavaSketchpad applet:

Figure. An Action by the Coxeter Group E_{8}.

One should observe three distinct elements in this figure. First is obviously the
Coxeter graph for E_{8}, with each node assigned a distingishing color.
Next is the graph of the action on the projectivized root system. There are 120 vertices
in this graph, each corresponding to a pair of antipodal roots. The colors of the edges
correspond to the colors of the 8 generating reflections. The generating reflections, of
course, each have order 2, and each edge designates that the endpoints are interchanged
by the reflection corresponding to the color of that edge.
For example, notice that the colored vertex on the far left is translated by the black
reflection and fixed by all the non-black reflections.

The third element of this figure is a "frame". By moving any of the four vertices of this frame, one can alter the perspective of the graph. Notice also that one may re-position the graph of the action by moving the colored point on the far left.

**Commutative Cubes.**

As one can probably see by playing with the applet, there appears to be a fairly natural
way to imbed this graph in **R**^{3} in such a way that (a) the 120 elements of the set
are placed at integral lattice points (i.e. in **Z**^{3}),
and (b) the edges lie on the unit segments of this lattice.
(If not, one may also convince oneself by attempting to draw the graph by hand. It isn't that
hard, really, as long as one can faithfully follow the relations among the generators.)
This imedding is "natural" in the particular sense that the 21 cubes which appear in the graph
must have a unique arrangement subject to the above conditions.

These 21 cubes appearing in this graph have some significance. It is fairly clear that one obtains such a cube exactly when one has three generating reflections, say {a,b,c}, which mutually commute. These correspond to triples {a,b,c} in the Coxeter graph for which no edges join any of the pairs of elements of the triple. One can quickly determine that there are precisely 21 such "commutative triples", and that each appears in this graph exactly once as such a cube.

**Conclusion.**

What about commutative hypercubes? There are 7 of these....

This page uses **JavaSketchpad**, a World-Wide-Web component of *The Geometer's Sketchpad.* Copyright © 1990-2001 by KCP Technologies, Inc. Licensed only for non-commercial use.