[ Main page  Part II. Word processing  Part III. Multiplication  Part IV. Closures ] 
More precisely, the definition sets
Exercise 1.
Verify that if Exercise 2.
Verify that Exercise 3.
Verify in general that for all Exercise 4.
Verify that we could have stipulated
If a Coxeter diagram decomposes into two components
Exercise 5. Verify this last claim.
All words in an equivalence class have the same parity  either even or odd.
The sign of
A word is said to be reduced if it is of minimal
length in its equivalence class. The length Proposition 1. The length has these properties:
Exercise 6. Verify this claim. Exercise 7.
Suppose that
If

It is possible to develop the subject of Coxeter groups entirely
in combinatorial terms (this is done  well, at least thoroughly attempted  in
the book by Bourbaki), but certain geometric representations
of Coxeter groups, in which the group acts discretely
on a certain domain, and in which the generators are
represented by reflections, allow one to visualize nicely
what is going on.
Recall that a linear reflection is a linear transformation
fixing a hyperplane passing through the origin and acting
as multiplication by
There are other kinds of reflections, too, but they reduce to linear ones. They are discussed at the end of this section. Single reflectionsIf
The hyperplane fixed by the reflection is where
The function
Often a reflection will be orthogonal with respect to
some inner product , when the reflection takes
In this situation, the dot product induces
a linear map from
The group Pairs of reflectionsSuppose
Let
Let
for some nonzero constants
One way to understand conjugationinvariant phenomena
is to consider geometrical configurations  for example,
how the various hyperplanes and lines relate qualititatively to each other.
One possibility is that in which
the reflection hyperplanes of
Proposition 2. When Proof. Only the last claim remains to be proven. Exercise 8. Verify the last claim.
We are especially interested in the case where
The region on one side of that line and in between the two lines
Exercise 9.
In the situation described above,
explain how Exercise 10.
Show that in dimension
From now on, suppose that
There is one other exceptional collection of cases
to deal with  when
Otherwise, suppose that
Then the line
From now on, we assume
Proposition 3. If
Proof. Let
Proposition 4. Take
As one consequence,
Proposition 5.
Suppose
Proof.
Suppose that
Since The matrix of the quadratic form is
In order for the form to be definite, it is necessary
and sufficient that
Proposition 6.
In order for the group generated by
The picture accompanying the last assertion is this:
Exercise 11. Prove this in detail.
Suppose now that
Proposition 7.
The elements
Proof. The case
Exercise 12.
Finish the proof. (Hint: the reflection In summary:
Theorem 8.
Assume that
In the last two cases, Affine reflectionsAn affine reflection reflects points in an affine hyperplane
for some linear function
An affine reflection is a special case of a linear one, through
the familiar trick of embedding an affine space of dimension
NonEuclidean reflectionsA nonEuclidean reflection reflects points of a nonEuclidean space in a nonEuclidean hyperplane. This also can be explained in terms of linear reflections.
The nonEuclidean space
It is taken into
itself by the connected component of the orthogonal group of
For the Klein model, this ball
is identified with the intersection of the slice
The Poincaré model is a transformation of the Klein model.
A point
The point for us is that nonEuclidean reflections
are those induced on the hyperbolic
sphere, or either of its models, by linear reflections in Exercise 13. Design an algorithm to draw the geodesic between two points of the unit disc in the Poincaré model. There are two ways to approach this problem, depending on the tools available for drawing. (1) Suppose all you can do is produce line segments and circles, then you must first decide whether to draw a line or a circle; if it is a circle, you must find the centre, radius, and arc. This becomes delicate only for points nearly lying on a diameter of the unit disc, and suffers to some extent from stability. Discuss how serious this is, analyzing how continuously your drawing depends on the two points. (2) if you can draw Bezier curves, then you should probably use them to approximate circles. Use one or two bisections to allow a reasonable approximation; then use velocity vectors to find the control points. The best way to do all this is likely through stereographic projection, since the geodesics in the disc correspond to simple latitude circles on the sphere. 
In this section we shall look at some Coxeter groups defined geometrically.
Finite dihedral groupsSuppose
Rotations are not its only symmetries,
since any line through the center of any of its sides
and the origin, or any of its corners and the origin,
is an axis of mirror symmetry. Since any symmetry must take
a corner into some other corner, and can either preserve
or reverse orientation, there are
Proposition 9. The symmetry group of
a regular polygon of
This should be clear from the picture. The generators
They are orthogonal reflections, with the angle between their
lines of reflection equal to
The
We can see how these elements match up with transforms of
Affine dihedral groupsNow we look at the group generated by affine reflections in the points at the end of a line segment in one dimension.
It is an infinite Coxeter group with two generators, say
The elements of the group can be expressed as
Hyperbolic dihedral groupsNow we look at the group generated by two hyperbolic reflections in the ends of a segment on the hyperbola
An observation about groups of rank twoThe examples we have just examined exhaust the possible representations of rank two Coxeter groups with the property that the region
Proposition 10. In these circumstances,
if
In brief, the argument for this is that if Groups of rank 3Suppose now that
We know that the standard
representation preserves a metric in which
Proposition 11.
Let
Here is a table of cases of possible values of the
The regular solidsThe classification of the finite Coxeter groups in three dimensions is intimately related to the classification of the regular solids.Exercise 14.
Verify that the Coxeter subgroup with values of Exercise 15.
Verify that the Coxeter subgroup with values of Exercise 16.
Verify that the Coxeter subgroup with values of Exercise 17. Prove directly that the symmetry group of any regular polyhedron is a Coxeter group. Exercise 18.
Verify that the symmetry group of the regular simplex in
Exercise 19.
Verify that the symmetry group of the cube in
Affine Coxeter groups of rank 3
A hyperbolic Coxeter group of rank 3

A polyhedral realization of a Coxeter group is a linear representation in which
Every Coxeter group possesses at least
one realization, as we shall see
in a moment.
Geometric properties of realizations translate naturally to
combinatorial properties of the group. From the geometry
of the simplices neighbouring a fundamental
domain, for example, you can read off the Coxeter matrix.
This is because if
Given a realization, make a choice for each
The Cartan matrix associated to
a choice of functions Theorem 12. In any realization, the Cartan matrix satisfies these conditions:
Any Cartan matrix clearly gives rise to a representation of the associated Coxeter group. In fact: Theorem 13. The representation of a Coxeter group determined by any abstract Cartan matrix is a realization of the associated Coxeter group. This will be proven in the next section. One consequence is that every Coxeter group has at least one realization, since there exists always the standard Cartan matrix
Cartan matrices with integral matrices determine
KacMoody Lie algebras. In this case the representation
of its Weyl group on the lattice of roots is the one associated to this Cartan matrix.
Coxeter groups which occur as the Weyl groups
of KacMoody algebras are called crystallographic,
and are distinguished by the property
that for them the numbers
Two Cartan matrices
Proposition 14.
If each
Proof.
If all
Corollary. If
Proof. Because if the diagram were not a tree, the group would possess
a continuous family of nonisomorphic representations of dimension Distinct classes can give rise to realizations with very different geometric properties. We have seen this already in the case of the infinite dihedral group, and here are the pictures for two different realizations of the Coxeter group whose Coxeter diagram is :
The first of these is associated to the standard Cartan matrix, and the second to the integral matrix
which is that of a certain hyperbolic KacMoody Lie algebra. It is the second, therefore, which is likely to have intrinsic significance. Exercise 20.
Does the Coxeter group of rank Exercise 21.
(This is a research problem! ) Prove that the boundary
of the second is nonsmooth everywhere
(i.e. even though it does have tangent lines everywhere, it
will not likely be 
The chambers are parametrized by elements of
The following pictures illustrate how this works on the affine Weyl
group of
If 
Define
Proposition 26. A vector
Proof. It is to be shown that if the set of roots
Proposition 27. The region Proposition 28. The following are equivalent:
Exercise 22.
Prove that in every double coset
Proposition 29. The face 
These are the Coxeter diagrams for those irreducible
Coxeter groups which are finite:
This is justified in VI.4 of the book by Bourbaki.
The basic idea is to check when the standard
realization preserves a positive definite quadratic form.
These are the cases when the Tits `cone' is the whole
vector space. The starting point is that the Coxeter diagram
cannot contain any circuits.
Another easy remark is that the number of branches
from any point can be at most Exercise 23. How large are these groups? These are the Coxeter diagrams for those irreducible Coxeter groups which can be interpreted as affine reflections:
This also is in the book by Bourbaki. These are the cases when the Tits `cone' is a halfspace. Exercise 24.
The aim of this exercise and the next two is to explain how the regular polyhedra in
all dimensions of Euclidean space are classified in terms of Coxeter diagrams.
Recall that a regular polyhedron is a polyhedron in Euclidean space
whose symmetry group acts transitively on
the faces of any given dimension.
Prove that the symmetry group of any regular polyhedron in Exercise 25.
Suppose that Exercise 26.
Prove that the regular Euclidean polyhedra are classified by isomorphism classes
of (a) a Coxeter diagram associated to a finite Coxeter group together with
(b) a single node of the diagram on its boundary. List explicitly all
the ones occurring in dimension 
