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3.2 Coxeter groups – The general theory

We have just shown that the billiard group in the case of pure gravity in four spacetime dimensions is the group P GL (2,ℤ ). This group is generated by reflections and is a particular example of a Coxeter group. Furthermore, as we shall explain below, this Coxeter group turns out to be the Weyl group of the (hyperbolic) Kac–Moody algebra A++ 1. Our first encounter with Lorentzian Kac–Moody algebras in more general gravitational theories will also be through their Weyl groups, which are, exactly as in the four-dimensional case just described, particular instances of (non-Euclidean) Coxeter groups, and which arise as the groups of billiard reflections.

For this reason, we start by developing here some aspects of the theory of Coxeter groups. An excellent reference on the subject is [107Jump To The Next Citation Point], to which we refer for more details and information. We consider Kac–Moody algebras in Section 4.

3.2.1 Examples

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.

The dihedral group I2(3 ) ≡ A2

Consider the dihedral group I2(3) of order 6 of symmetries of the equilateral triangle in the Euclidean plane.

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Figure 3: The equilateral triangle with its 3 axes of symmetries. The reflections s1 and s2 generate the entire symmetry group. We have pictured the vectors α1 and α2 orthogonal to the axes of reflection and chosen to make an obtuse angle. The shaded region {w |(w|α1) ≥ 0} ∩ {w |(w |α2) ≥ 0} is a fundamental domain for the action of the group on the triangle. Note that the fundamental domain for the action of the group on the entire Euclidean plane extends indefinitely beyond the triangle but is, of course, still bounded by the two walls orthogonal to α1 and α2.

This group contains the identity, three reflections s1, s2 and s3 about the three medians, the rotation R1 of 2π∕3 about the origin and the rotation R2 of 4π∕3 about the origin (see Figure 3View Image),

I (3) = {1,s ,s ,s ,R ,R }. (3.19 ) 2 1 2 3 1 2
The reflections act as follows7,
(γ|αi) si(γ) = γ − 2 (α-|α-)αi, (3.20 ) i i
where ( | ) is here the Euclidean scalar product and where αi is a vector orthogonal to the hyperplane (here, line) of reflection.

Now, all elements of the dihedral group I (3) 2 can be written as products of the two reflections s 1 and s2:

1 = s0, s = s , s = s , R = s s , R = s s , s = s s s . (3.21 ) 1 1 1 2 2 1 1 2 2 2 1 3 12 1
Hence, the dihedral group I2(3) is generated by s1 and s2. The writing Equation (3.21View Equation) is not unique because s1 and s2 are subject to the following relations,
s21 = 1, s22 = 1, (s1s2)3 = 1. (3.22 )
The first two relations merely follow from the fact that s1 and s2 are reflections, while the third relation is a consequence of the property that the product s1s2 is a rotation by an angle of 2π ∕3. This follows, in turn, from the fact that the hyperplanes (lines) of reflection make an angle of π∕3. There is no other relation between the generators s1 and s2 because any product of them can be reduced, using the relations Equation (3.22View Equation), to one of the 6 elements in Equation (3.21View Equation), and these are independent.

The dihedral group I2(3) is also denoted A2 because it is the Weyl group of the simple Lie algebra A2 (see Section 4). It is isomorphic to the permutation group S3 of three objects.

The infinite dihedral group I2(∞ ) ≡ A+ 1

Consider now the group of isometries of the Euclidean line containing the symmetries about the points with integer or half-integer values of x (x 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 s1 about the origin and s2 about the point with coordinate 1 ∕2,

s (x) = − x, s (x ) = − (x − 1). (3.23 ) 1 2
The product s2s1 is a translation by +1 while the product s1s2 is a translation by − 1, so no power of s1s2 or s2s1 gives the identity. All the powers (s2s1)k and (s1s2)j are distinct (translations by +k and − j, respectively). The only relations between the generators are
s2 = 1 = s2. (3.24 ) 1 2
This infinite dihedral group I2(∞ ) is also denoted by A+1 because it is the Weyl group of the affine Kac–Moody algebra A+1.

3.2.2 Definition

A Coxeter group ℭ is a group generated by a finite number of elements si (i = 1,⋅⋅⋅ ,n) subject to relations that take the form

s2i = 1 (3.25 )
mij (sisj) = 1, (3.26 )
where the integers mij associated with the pairs (i,j) fulfill
mij = mji, mij ≥ 2 (i ⁄= j). (3.27 )
Note that Equation (3.25View Equation) is a particular case of Equation (3.26View Equation) with mii = 1. If there is no power of sisj that gives the identity, as in our second example, we set, by convention, mij = ∞. The generators si are called “reflections” because of Equation (3.25View Equation), even though we have not developed yet a geometric realisation of the group. This will be done in Section 3.2.4 below.

The number n of generators is called the rank of the Coxeter group. The Coxeter group is completely specified by the integers mij. It is useful to draw the set {mij} pictorially in a diagram Γ, called a Coxeter graph. With each reflection si, one associates a node. Thus there are n nodes in the diagram. If m > 2 ij, one draws a line between the node i and the node j and writes m ij over the line, except if mij is equal to 3, in which case one writes nothing. The default value is thus “3”. When there is no line between i and j (i ⁄= j), the exponent mij is equal to 2. We have drawn the Coxeter graphs for the Coxeter groups I2(3), I2(m ) and for the Coxeter group H3 of symmetries of the icosahedron.

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Figure 4: The Coxeter graph of the symmetry group I (3) ≡ A 2 2 of the equilateral triangle.
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Figure 5: The Coxeter graph of the dihedral group I2(m ).
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Figure 6: The Coxeter graph of the symmetry group H3 of the regular icosahedron.

Note that if mij = 2, the generators si and sj commute, sisj = sjsi. Thus, a Coxeter group ℭ 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.

Another example: + C 2

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 x-axis (s1), (ii) the straight line joining the points (1, 0) and (0,1) (s 2), and (iii) the y-axis (s 3). The Coxeter exponents are finite and equal to 4 (m12 = m21 = m23 = m32 = 4) and 2 (m13 = m31 = 2). The Coxeter graph is given in Figure 7View Image. The Coxeter group is the symmetry group of the regular paving of the plane by squares and contains translations. Indeed, the product s2s1s2 is a reflection in the line parallel to the y-axis going through (1,0) and thus the product t = s2s1s2s3 is a translation by +2 in the x-direction. All powers of t are distinct; the group is infinite. This Coxeter group is of affine type and is called C+ 2 (which coincides with + B 2).

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Figure 7: The Coxeter graph of the affine Coxeter group C+2 corresponding to the group of isometries of the Euclidean plane.

The isomorphism problem

The Coxeter presentation of a given Coxeter group may not be unique. Consider for instance the group I2(6) of order 12 of symmetries of the regular hexagon, generated by two reflections s1 and s2 with

s21 = s22 = 1, (s1s2)6 = 1.

This group is isomorphic with the rank 3 (reducible) Coxeter group I2(3) × ℤ2, with presentation

r2 = r2 = r2 = 1, (r1r2)3 = 1, (r1r3)2 = 1, (r2r3)2 = 1, 1 2 3

the isomorphism being given by f(r1) = s1, f(r2) = s1s2s1s2s1, 3 f(r3) = (s1s2). 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].

3.2.3 The length function

An important concept in the theory of Coxeter groups is that of the length of an element. The length of w ∈ ℭ is by definition the number of generators that appear in a minimal representation of w as a product of generators. Thus, if w = si1si2 ⋅⋅⋅sil and if there is no way to write w as a product of less than l generators, one says that w has length l.

For instance, for the dihedral group I2(3), the identity has length zero, the generators s1 and s2 have length one, the two non-trivial rotations have length two, and the third reflection s3 has length three. Note that the rotations have representations involving two and four (and even a higher number of) generators since for instance s1s2 = s2s1s2s1, 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 s3 = s1s2s1 = s2s1s2. Both involve three generators and define the length of s3 to be three.

Let w be an element of length l. The length of wsi (where si is one of the generators) differs from the length of w by an odd (positive or negative) integer since the relations among the generators always involve an even number of reflections. In fact, l(wsi) is equal to l + 1 or l − 1 since l(wsi) ≤ l(w) + 1 and l(w ≡ wsisi) ≤ l(wsi ) + 1. Thus, in wsi, there can be at most one simplification (i.e., at most two elements that can be removed using the relations).

3.2.4 Geometric realization

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 n, equipped with a scalar product preserved by the group.

To each generator si, associate a vector αi of a basis {α1,⋅⋅⋅ ,αn} of an n-dimensional vector space V. Introduce a scalar product defined as follows,

( ) B (α ,α ) = − cos -π-- , (3.28 ) i j mij
on the basis vectors and extend it to V by linearity. Note that for i = j, mii = 1 implies B (αi,αi) = 1 for all i. In the case of the dihedral group A2, this scalar product is just the Euclidean scalar product in the two-dimensional plane where the equilateral triangle lies, as can be seen by taking the two vectors α1 and α2 respectively orthogonal to the first and second lines of reflection in Figure 3View Image and oriented as indicated. But in general, the scalar product (3.28View Equation) might not be of Euclidean signature and might even be degenerate. This is the case for the infinite dihedral group I2(∞ ), for which the matrix B reads
( 1 − 1) B = (3.29 ) − 1 1
and has zero determinant. We shall occasionally use matrix notations for the scalar product, B (α,γ ) ≡ αTB γ.

However, the basis vectors are always all spacelike since they have norm squared equal to 1. For each i, the vector space V splits then as a direct sum

V = ℝ αi ⊕ Hi, (3.30 )
where Hi is the hyperplane orthogonal to αi (δ ∈ Hi iff B (δ,αi) = 0). One defines the geometric reflection σi as
σi(γ) = γ − 2B (γ,αi)αi. (3.31 )
It is clear that σi fixes Hi pointwise and reverses αi. It is also clear that σ2 = 1 i and that σi preserves B,
′ ′ B (σi(γ),σi(γ )) = B (γ,γ ). (3.32 )
Note that in the particular case of A2, we recover in this way the reflections s1 and s2.

We now verify that the σ i’s also fulfill the relations (σ σ )mij = 1 i j. To that end we consider the plane Π spanned by αi and αj. This plane is left invariant under σi and σj. Two possibilities may occur:

  1. The induced scalar product on Π is nondegenerate and in fact positive definite, or
  2. the induced scalar product is positive semi-definite, i.e., there is a null direction orthogonal to any other direction.

The second case occurs only when mij = ∞. The null direction is given by λ = αi + αj.

As the defining relations are preserved, we can conclude that the map f from the Coxeter group generated by the si’s to the geometric group generated by the σi’s defined on the generators by f (si) = σi 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 Bij is indecomposable. A matrix A ij 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 i,j in two sets J and Λ such that Aij = 0 whenever i ∈ J,j ∈ Λ or i ∈ Λ, j ∈ J. The indecomposability of B 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 Λ to a point in the set J.

3.2.5 Positive and negative roots

A root is any vector in the space V of the geometric realisation that can be obtained from one of the basis vectors αi by acting with an element w of the Coxeter group (more precisely, with its image f (w ) under the above homomorphism, but we shall drop “f” for notational simplicity). Any root α can be expanded in terms of the αi’s,

∑ α = c α . (3.33 ) i i i
If the coefficients ci are all non-negative, we say that the root α is positive and we write α > 0. If the coefficients ci are all non-positive, we say that the root α is negative and we write α < 0. Note that we use strict inequalities here because if c = 0 i for all i, then α is not a root. In particular, the α i’s themselves are positive roots, called also “simple” roots. (Note that the simple roots considered here differ by normalization factors from the simple roots of Kac–Moody algebras, as we shall discuss below.) We claim that roots are either positive or negative (there is no root with some ci’s in Equation (3.33View Equation) > 0 and some other ci’s < 0). The claim follows from the fact that the image of a simple root by an arbitrary element w of the Coxeter group is necessarily either positive or negative.

This, in turn, is the result of the following theorem, which provides a useful criterion to tell whether the length l(wsi) of wsi is equal to l(w ) + 1 or l(w ) − 1.

Theorem: l(wsi ) = l(w ) + 1 if and only if w (αi ) > 0.

The proof is given in [107Jump To The Next Citation Point], page 111.

It easily follows from this theorem that l(wsi ) = l(w ) − 1 if and only if w (αi) < 0. Indeed, l(wsi) = l(w) − 1 is equivalent to l(w ) = l(wsi ) + 1, i.e., l((wsi)si) = l(wsi) + 1 and thus, by the theorem, wsi(αi) > 0. But since si(αi) = − αi, this is equivalent to w (αi) < 0.

We have seen in Section 3.2.3 that there are only two possibilities for the length l(ws ) i. It is either equal to l(w ) + 1 or to l(w) − 1. From the theorem just seen, the root w(αi) is positive in the first case and negative in the second. Since any root is the Coxeter image of one of the simple roots αi, i.e., can be written as w (αi ) for some w and αi, 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 [107Jump To The Next Citation Point] that l(w ) is equal to the number of positive roots sent by w to negative roots. In particular, the fundamental reflection s associated with the simple root αs maps αs 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 f (w ) = 1 where w is an element of the Coxeter group that is not the identity. It is clear that there exists one group generator si such that l(wsi) = l(w) − 1. Take for instance the last generator occurring in a reduced expression of w. For this generator, one has w (αi) < 0, which is in contradiction with the assumption f (w ) = 1.

Because f is an isomorphism, we shall from now on identify the Coxeter group with its geometric realisation and make no distinction between si and σi.

3.2.6 Fundamental domain

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 A2 of the equilateral triangle. The shaded region ℱ in Figure 4View Image contains the vectors γ such that B (α1, γ) ≥ 0 and B (α2,γ ) ≥ 0. It has the following important property: Any orbit of the group A2 intersects ℱ 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 B 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 V, but rather to a fundamental domain of the action of the Coxeter group on the so-called Tits cone 𝒳, which is such that the inequalities B (αi,γ) ≥ 0 continue to play the central role.

We assume that the scalar product is nondegenerate. Define for each simple root αi the open half-space

Ai = {γ ∈ V |B (αi,γ) > 0}. (3.34 )
We define ℰ to be the intersection of all Ai,
⋂ ℰ = A . (3.35 ) i i
This is a convex open cone, which is non-empty because the metric is nondegenerate. Indeed, as B is nondegenerate, one can, by a change of basis, assume for simplicity that the bounding hyperplanes B (α ,γ ) = 0 i are the coordinate hyperplanes x = 0 i. ℰ is then the region xi > 0 (with appropriate orientation of the coordinates) and ℱ is xi ≥ 0. The closure
¯ ⋂ ¯ ℱ = ℰ = iAi, A¯ = {γ ∈ V |B (α ,γ) ≥ 0} (3.36 ) i i
is then a closed convex cone8.

We next consider the union of the images of ℱ under the Coxeter group,

⋃ 𝒳 = w (ℱ). (3.37 ) w∈ℭ
One can show [107Jump To The Next Citation Point] that this is also a convex cone, called the Tits cone. Furthermore, ℱ is a fundamental domain for the action of the Coxeter group on the Tits cone; the orbit of any point in 𝒳 intersects ℱ once and only once [107Jump To The Next Citation Point]. The Tits cone does not coincide in general with the full space V and is discussed below in particular cases.
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