[ 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

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 
