An Action by the Coxeter Group E8
(i.e. Another view of Gosset's figure 421)

Introduction.

This unusual arrangement of 21 cubes was extracted from the graph of an action of the Coxeter group E8, the symmetry group of Gosset's 8-dimensional semiregular polytope 421, on a set of 120 elements. This action is quite easy to describe, assuming one is familiar with the root system D for E8. The group E8 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 E8, and so one may quotient to obtain an action on PD, a "projectivized" root system in which each root is identified with its negative. This projective action of E8 is not effective since the antipodal involution acts as an identity element, but, since the quotient of E8 by the 2-element group generated by this involution is a simple group, the action is effective if considered an action of the quotient PE8.

The Graph.

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

#### Sorry, this page requires a Java-compatible web browser.Figure. An Action by the Coxeter Group E8.

One should observe three distinct elements in this figure. First is obviously the Coxeter graph for E8, 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 R3 in such a way that (a) the 120 elements of the set are placed at integral lattice points (i.e. in Z3), 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....

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