Coxeter Groups
2.2 Examples
2.2.3 The Group PGL 2 (Z)
s(f) =f−2B(es, f)es, t(f) =f−2B(et, f)et.
Note that s(es) = −es and s fixes the hyperplane B(es,−) = 0, which is spanned byes+et; sosis indeed a reflection. Similarly,tis a reflection, with the same fixed hyperplane.
It turns out thats andt are the reflectionss∗ andt∗ onV∗ induced by s and t. To check this, one can simply computes∗ andt∗ on es and et. For example,
s∗(es),(x, y)=es, s(x, y)=es,(−x, y)=−x=−es,(x, y), so s∗(es) = −es =s(es). The remaining computations are equally easy and are left to the reader.
In summary, our reflection representation ofD∞ on V could have been obtained as follows: Start with an abstract vector space Res⊕Ret [which is our V∗] and define a linear action of D∞ on it by copying the formulas from the finite case, using the Coxeter matrix. Now pass to the dual space (Res⊕Ret)∗[which is ourV] to obtain an action in which we have the familiar sort of chamber geometry.
Remark 2.6.It is natural to ask whether we had to pass to the dual space in order to obtain the chamber geometry. The answer is yes—our two fundamen- tal reflections acting on Res⊕Ret have the same fixed hyperplane, so they do not determine a chamber in that space. See Exercise 2.7 below for further insight into the difference between theD∞-action onV and its action onV∗. In the finite case, on the other hand, the duality was hidden because, in the presence of aW-invariant inner product, there is a canonical identification of V with its dual. We will return to this circle of ideas in Section 2.5 below.
Exercise 2.7.
(a) Ifs is a linear reflection on a 2-dimensional vector space V, show that the onlys-invariant affine lines not passing through the origin are those parallel to the (−1)-eigenspace.
(b) Deduce that two linear reflections s, t of V have a common invariant line not passing through the origin if and only if they have the same (−1)-eigenspace.
(c) Suppose sand thave the same (+1)-eigenspace. Show that the induced reflectionss∗ andt∗ ofV∗ have the same (−1)-eigenspace.
72 2 Coxeter Groups
obtained from GL2(Z) by identifying a matrix with its negative. We denote a typical element of GL2(Z) by
a b c d
and its image in PGL2(Z) by a b c d
.
It is easy to check that W := PGL2(Z) is generated by the set S = {s1, s2, s3} of elements of order 2 defined by
s1= 0 1
1 0
, s2=
−1 1 0 1
, s3=
−1 0 0 1
.
(One can see this by thinking about elementary row operations.) We now show that condition (A) is satisfied. We will use three different methods, analogous to those used for D∞. In each of the three cases, however, we will have to use one or more nontrivial facts that will be stated without proof. Readers who are not familiar with these facts are advised to just read the discussion casually for the main ideas.
(i) Combinatorial group theory
A simple computation shows that the products s1s2, s1s3, and s2s3 have orders 3, 2, and ∞, respectively. It is also true (but not obvious) that W admits a presentation in which the defining relations simply specify the orders of these pairwise products:
W =
s1, s2, s3;s21=s22=s23= (s1s2)3= (s1s3)2= 1 .
[Some readers will be familiar with the fact thatW has a subgroup PSL2(Z) of index 2 that admits a presentation
u, v;u3=v2= 1
; see Serre [217, Sec- tion I.4.2] or Lehner [152, Sections IV.5H and VII.2F]. It is not too hard to de- duce the presentation forW stated above from this presentation for PSL2(Z).]
To verify (A), now, we need only check that the involutionsρi :=ρsi that occur in the statement of (A) satisfy the defining relations forW. Consider, for instance, the relation (ρ1ρ2)3 = 1. LetS := {s1, s2} and letW be the dihedral group of order 6 generated byS. The setTof reflections inW(i.e., theW-conjugates ofs1ands2), form a subset of the setT of reflections inW (theW-conjugates of the elements ofS).
Suppose, now, that we apply (ρ1ρ2)3to (t, )∈T×{±1}. Since (s1s2)3= 1 in W, we will get (t,±). Clearly the only thing we have to worry about is the possibility of sign changes in the second factor as we successively apply theρi. But no sign changes will ever occur unless there is an elementw∈W with wtw−1 ∈ S, in which case we have t ∈ T. Thus we are reduced to showing that (ρ1ρ2)3is the identity onT× {±1}, which follows from the fact
thatW is a finite reflection group and hence is already known to satisfy (A).
[Alternatively, we could complete the proof by doing an easy computation in the dihedral group D6.]
Remark 2.8.Note, for future reference, that this proof works wheneverW admits a presentation of the form
W =
S; (st)m(s,t)= 1
,
wherem(s, t) is the order ofstand there is one relation for each pairs, tsuch that m(s, t)<∞. We will return to this in Section 2.4.
(ii) Hyperbolic geometry
There is a famous tessellation of the hyperbolic plane by ideal hyperbolic tri- angles (i.e., triangles having their vertices on the circle at infinity). Figure 2.3 shows this tessellation in the unit disk model of the hyperbolic plane.∗
Fig. 2.3.A tessellation of the hyperbolic plane.
The (hyperbolic) lines of symmetry of this tessellation barycentrically sub- divide it; see Figure 2.4. The group of symmetries of the original tessellation is the group of hyperbolic isometries generated by the reflections with respect
∗Figures 2.3 and 2.4 first appeared in Klein–Fricke [145, pp. 111 and 112]. Fig- ure 2.5 is based on a picture in [145, p. 106]. We are grateful to Cornell Univer- sity Library’s Historic Monograph Collection for providing digital images of these pictures.
74 2 Coxeter Groups
Fig. 2.4.The same tessellation, subdivided by the lines of symmetry.
to the lines of symmetry, and it is, in fact, precisely the groupW. In order to explain this in slightly more detail, we switch to the upper-half-plane model of the hyperbolic plane. Figure 2.5 shows the barycentric subdivision in this model. To relate the two models of the hyperbolic plane, one should think of the vertices of the big triangle in Figure 2.3 as corresponding to the points 0, 1, and∞ in Figure 2.5. The barycenter of this big triangle is shown as a heavy dot in Figure 2.5. The action ofW on the upper half-plane is given by
a b c d
·z=
⎧⎪
⎪⎨
⎪⎪
⎩ az+b
cz+d ifad−bc= 1, a¯z+b
c¯z+d ifad−bc=−1,
where ¯zis the complex conjugate ofz. Many readers will be familiar with this action restricted to PSL2(Z), where, of course, complex conjugation does not arise. Complex conjugation is necessary for the full groupW, however, because elements of negative determinant acting by linear fractional transformations interchange the upper and lower half-planes.
Now under this action, the generating setSofW is the set of reflections in the three sides of one of the “chambers” C, as indicated in Figure 2.5. More- over, it is known thatC is a strict fundamental domain for the action ofW.
C s3
s1
s2
0 1/2 1
Fig. 2.5.The barycentric subdivision in the upper-half-plane model.
A proof of this can be found in almost any book that discusses modular forms.
[Actually it is more likely that the analogous fact about PSL2(Z) is proved:
C∪s3Cis a fundamental domain (but not a strict fundamental domain) for this group. See, for instance, Serre [215, Section VII.1.2] or Lehner [152, Sec- tion IV.5H].]
Readers who have followed all of this can probably complete the geometric proof that (W, S) satisfies condition (A). Just identifyT× {±1} with the set of hyperbolic half-planes determined by the hyperbolic lines in Figure 2.5, and use the action of W on these half-planes.
(iii) Linear algebra
As was the case with the group D∞, the linear algebra approach will take the longest to explain. But it is quite instructive and worth at least reading through, without necessarily checking all the details. It is based on a 3-dimen- sional linear representation of W that has been studied extensively, starting with Gauss.
The vector spaceV on whichW acts is the space of real quadratic formsq in two variables, i.e., the space of functionsq: R2→Rgiven byq(x) =ax21+ 2bx1x2+cx22, wherex= (x1, x2). Note that we can also writeq(x) =β(x, x), whereβ is the bilinear form onR2 with matrix
A:=
a b b c
.
76 2 Coxeter Groups
Thus we can, when it is convenient, identify V with the space of symmetric bilinear forms onR2, or, equivalently, with the space of real, symmetric 2×2 matrices.
The groupG= GL2(R) acts onV by (g·q)(x) :=q(xg)
forg∈G,q∈V, andx∈R2, where xis viewed as a row vector on the right side of the equation. This action is said to be bychange of variable, sinceg·qis obtained fromqby replacingx1andx2 by linear functions ofx1andx2 (with coefficients given by the columns of g). In terms of the symmetric matrixA corresponding to q, the action of g is given by A → gAgt, where gt is the transpose ofg.
The elements q ∈ V fall into exactly six orbits under the action of G.
First, there are three types of nondegenerate forms: positive definite (G-equiv- alent tox21+x22); negative definite (G-equivalent to−x21−x22); and indefinite (G-equivalent tox21−x22). Next, there are the nonzero degenerate forms, which are either positive semidefinite (G-equivalent tox21) or negative semidefinite (G-equivalent to−x21). And finally, there is the zero form.
It is easy to visualize this partition ofV intoG-orbits. LetQ:V →Rbe given by
Q(q) :=−detA=b2−ac ,
where A is the matrix corresponding to q as above. (Thus Qis a quadratic form on the 3-dimensional spaceV of quadratic forms.) Then the degenerate formsqare the points of the quadric surfaceQ= 0 inV. If we introduce new coordinates x, y, zinV by setting
b=x , a=z+y ,
c=z−y ,
thenQbecomesx2+y2−z2, so the quadric surface of degenerate forms is the double cone z2=x2+y2. [Draw a picture!] The exterior of the cone is given byQ >0 and consists of the indefinite forms. And the interiorQ <0 has two components, the upper half (z >0), consisting of the positive definite forms, and the lower half, consisting of the negative definite forms.
The action of G = GL2(R) on V is really an action of the quotient G/{±1} = PGL2(R), so we may restrict the action to W = PGL2(Z) ≤ PGL2(R). This is the desired 3-dimensional representation ofW. Here are the basic facts about this representation:
First, theW-action leaves the form Q invariant, i.e., Q(wq) = Q(q) for w ∈ W and q ∈ V. This follows from the fact that every g ∈ GL2(Z) has detg=±1, so that
detgAgt= (detg)2detA= detA
for any symmetric 2×2 matrixA. SoW also leaves invariant the symmetric bilinear form B onV such that Q(q) = B(q, q). One can easily compute B explicitly; in terms of symmetric matrices, we have
B(A, A) =bb−1
2(ac+ac), where
A= a b
b c
and A=
a b b c
.
The next observation is that the generators si of W act on V as linear reflections. In fact, computing the (±1)-eigenspaces of si, one finds that si
has a 1-dimensional (−1)-eigenspace Rei and that si fixes the hyperplane Hi :=ei⊥, where ei⊥ is defined with respect to our bilinear form B(−,−).
One can take theei, which are determined up to scalar multiplication, to be the following symmetric matrices:
e1=
1 0 0 −1
, e2=
−1 −1
−1 0
, e3= 0 1
1 0
.
And the fixed hyperplanes Hi are given, respectively, by a = c, c = 2b, andb= 0.
We chose the eigenvectorsei above so that they would satisfyQ(ei) = 1;
this determines them up to sign. It then follows as a formal consequence that the reflections si are given by the usual formula:
siq=q−2B(ei, q)ei;
for the map defined by this formula is the identity onei⊥and sendseito−ei. We now focus on the action of W on the cone P of positive definite forms, and we look for a fundamental domain for this action. Concretely, this means that we are looking for canonical representatives for the positive definite formsqunder integral change of variable. Gauss found the following fundamental domain. LetCbe the simplicial cone inV defined by the inequal- ities a > c > 2b > 0. Then C ⊆ P, and C is (more or less) a fundamental domain for the action ofW onP.
The qualifier “more or less” here refers to the fact that C touches the boundary ofP. For if one computes the vertices ofC (i.e., the rays that are 1-dimensional faces of C), one finds that they are represented by the forms x21,x21+x22, andx21+x1x2+x22, the first of which is degenerate. So the correct statement is the following: Let X be the convex cone in V consisting of the positive definite forms together with the formsλ(ax1+bx2)2withλ≥0 and a, b∈Z. ThenX =
w∈WwC, andC is a strict fundamental domain for the action ofW onX; moreover, the open simplicial coneswC are disjoint from one another.
Note that the walls of C are precisely the fixed hyperplanes Hi of the reflections si. So we have, once again, the usual sort of chamber geometry,
78 2 Coxeter Groups
and it is possible to verify condition (A) by identifying T× {±1} with the half-spaces in V determined by the walls of the chambers wC. Details are omitted.
One final comment: We normalized the ei above so that we would have B(ei,−) > 0 on C. In view of Chapter 1 and the infinite dihedral group example, it is therefore to be expected that
B(ei, ej) =−cos π mij ,
wheremij is the order ofsisj. This is indeed the case, as direct computation shows. Thus our representation of W acting onV is what we should now be ready to call the “canonical linear representation” ofW. Note also, for future reference, that the bilinear formBin this example is nondegenerate, although not positive definite. Indeed, we showed above that Q could be written as x2+y2−z2 after a change of coordinates in V, so B has signature (2,1) [there are 2 plus signs and 1 minus sign].
Exercise 2.9.What is the connection between the points of view in (ii) and (iii)?