For this reason, we start by developing here some aspects of the theory of Coxeter groups. An excellent
reference on the subject is [107], to which we refer for more details and information. We consider
Kac–Moody algebras in Section 4.
Coxeter groups generalize the familiar notion of reflection groups in Euclidean space. Before we present the basic definition, let us briefly discuss some more illuminating examples.
Consider the dihedral group of order 6 of symmetries of the equilateral triangle in the Euclidean
plane.
This group contains the identity, three reflections ,
and
about the three medians, the
rotation
of
about the origin and the rotation
of
about the origin (see Figure 3
),
Now, all elements of the dihedral group can be written as products of the two reflections
and
:
The dihedral group is also denoted
because it is the Weyl group of the simple Lie algebra
(see Section 4). It is isomorphic to the permutation group
of three objects.
Consider now the group of isometries of the Euclidean line containing the symmetries about the points with
integer or half-integer values of (
is a coordinate along the line) as well as the translations by an
integer. This is clearly an infinite group. It is generated by the two reflections
about the origin and
about the point with coordinate
,
A Coxeter group is a group generated by a finite number of elements
(
) subject to
relations that take the form
The number of generators is called the rank of the Coxeter group. The Coxeter group is completely
specified by the integers
. It is useful to draw the set
pictorially in a diagram
, called a
Coxeter graph. With each reflection
, one associates a node. Thus there are
nodes in the diagram. If
, one draws a line between the node
and the node
and writes
over the line, except if
is equal to 3, in which case one writes nothing. The default value is thus “3”. When there is no line
between
and
(
), the exponent
is equal to 2. We have drawn the Coxeter graphs for
the Coxeter groups
,
and for the Coxeter group
of symmetries of the
icosahedron.
Note that if , the generators
and
commute,
. Thus, a Coxeter group
is the direct product of the Coxeter subgroups associated with the connected components of its Coxeter
graph. For that reason, we can restrict the analysis to Coxeter groups associated with connected (also called
irreducible) Coxeter graphs.
The Coxeter group may be finite or infinite as the previous examples show.
It should be stressed that the Coxeter group can be infinite even if none of the Coxeter exponent is infinite.
Consider for instance the group of isometries of the Euclidean plane generated by reflections in the following
three straight lines: (i) the -axis (
), (ii) the straight line joining the points
and
(
), and (iii) the
-axis (
). The Coxeter exponents are finite and equal to
(
) and
(
). The Coxeter graph is given in Figure 7
. The
Coxeter group is the symmetry group of the regular paving of the plane by squares and contains
translations. Indeed, the product
is a reflection in the line parallel to the
-axis going through
and thus the product
is a translation by
in the
-direction. All powers of
are distinct; the group is infinite. This Coxeter group is of affine type and is called
(which coincides
with
).
The Coxeter presentation of a given Coxeter group may not be unique. Consider for instance the group
of order 12 of symmetries of the regular hexagon, generated by two reflections
and
with
This group is isomorphic with the rank 3 (reducible) Coxeter group , with presentation
the isomorphism being given by ,
,
. The question of
determining all such isomorphisms between Coxeter groups is known as the “isomorphism problem of
Coxeter groups”. This is a difficult problem whose general solution is not yet known [10].
An important concept in the theory of Coxeter groups is that of the length of an element. The length of
is by definition the number of generators that appear in a minimal representation of
as a
product of generators. Thus, if
and if there is no way to write
as a product of less
than
generators, one says that
has length
.
For instance, for the dihedral group , the identity has length zero, the generators
and
have length one, the two non-trivial rotations have length two, and the third reflection
has length
three. Note that the rotations have representations involving two and four (and even a higher number of)
generators since for instance
, but the length is associated with the representations
involving as few generators as possible. There might be more than one such representation as it occurs
for
. Both involve three generators and define the length of
to be
three.
Let be an element of length
. The length of
(where
is one of the generators) differs
from the length of
by an odd (positive or negative) integer since the relations among the
generators always involve an even number of reflections. In fact,
is equal to
or
since
and
. Thus, in
, there
can be at most one simplification (i.e., at most two elements that can be removed using the
relations).
We now construct a geometric realisation for any given Coxeter group. This enables one to view the Coxeter
group as a group of linear transformations acting in a vector space of dimension , equipped with a scalar
product preserved by the group.
To each generator , associate a vector
of a basis
of an
-dimensional vector
space
. Introduce a scalar product defined as follows,
However, the basis vectors are always all spacelike since they have norm squared equal to . For each
, the vector space
splits then as a direct sum
We now verify that the ’s also fulfill the relations
. To that end we consider the plane
spanned by
and
. This plane is left invariant under
and
. Two possibilities may
occur:
The second case occurs only when . The null direction is given by
.
As the defining relations are preserved, we can conclude that the map from the Coxeter group
generated by the
’s to the geometric group generated by the
’s defined on the generators by
is a group homomorphism. We will show below that its kernel is the identity so that it is in
fact an isomorphism.
Finally, we note that if the Coxeter graph is irreducible, as we assume, then the matrix is
indecomposable. A matrix
is called decomposable if after reordering of its indices, it decomposes as a
non-trivial direct sum, i.e., if one can slit the indices
in two sets
and
such that
whenever
or
. The indecomposability of
follows from the fact that if it were
decomposable, the corresponding Coxeter graph would be disconnected as no line would join a point in the
set
to a point in the set
.
A root is any vector in the space of the geometric realisation that can be obtained from one of the basis
vectors
by acting with an element
of the Coxeter group (more precisely, with its image
under the above homomorphism, but we shall drop “
” for notational simplicity). Any root
can be
expanded in terms of the
’s,
This, in turn, is the result of the following theorem, which provides a useful criterion to tell whether the
length of
is equal to
or
.
Theorem: if and only if
.
The proof is given in [107], page 111.
It easily follows from this theorem that if and only if
. Indeed,
is equivalent to
, i.e.,
and thus, by the
theorem,
. But since
, this is equivalent to
.
We have seen in Section 3.2.3 that there are only two possibilities for the length . It is either
equal to
or to
. From the theorem just seen, the root
is positive in the first
case and negative in the second. Since any root is the Coxeter image of one of the simple roots
, i.e.,
can be written as
for some
and
, we can conclude that the roots are either positive or
negative; there is no alternative.
The theorem can be used to provide a geometric interpretation of the length function. One can
show [107] that
is equal to the number of positive roots sent by
to negative roots. In particular,
the fundamental reflection
associated with the simple root
maps
to its negative and permutes
the remaining positive roots.
Note that the theorem implies also that the kernel of the homomorphism that appears in the geometric
realisation of the Coxeter group is trivial. Indeed, assume where
is an element of the
Coxeter group that is not the identity. It is clear that there exists one group generator
such that
. Take for instance the last generator occurring in a reduced expression of
. For this generator, one has
, which is in contradiction with the assumption
.
Because is an isomorphism, we shall from now on identify the Coxeter group with its geometric
realisation and make no distinction between
and
.
In order to describe the action of the Coxeter group, it is useful to introduce the concept of fundamental
domain. Consider first the case of the symmetry group of the equilateral triangle. The shaded region
in Figure 4
contains the vectors
such that
and
. It has the following
important property: Any orbit of the group
intersects
once and only once. It is called for this
reason a “fundamental domain”. We shall extend this concept to all Coxeter groups. However, when the
scalar product
is not positive definite, there are inequivalent types of vectors and the concept of
fundamental domain can be generalized a priori in different ways, depending on which region one wants to
cover. (The entire space? Only the timelike vectors? Another region?) The useful generalization turns out
not to lead to a fundamental domain of the action of the Coxeter group on the entire vector space
, but rather to a fundamental domain of the action of the Coxeter group on the so-called
Tits cone
, which is such that the inequalities
continue to play the central
role.
We assume that the scalar product is nondegenerate. Define for each simple root the open
half-space
We next consider the union of the images of under the Coxeter group,
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