The Infinite Dihedral Group

Coxeter Groups

2.2 Examples

2.2.2 The Infinite Dihedral Group

Let W be the infinite dihedral group D_∞. By definition, this is the group defined by the presentation

W :=

s, t;s²=t²= 1 .

For readers not familiar with this notation for group presentations, it simply means that we start with the free groupF :=F(s, t) on two generatorss, tand

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then divide out by the smallest normal subgroup containing s² and t². Note that the finite dihedral groups D_2m are quotients of W. It follows that the generatorss, tofF map to distinct nontrivial elements ofW, so no confusion will result if we use the same letterss, tto denote those elements ofW. It also follows that sthas infinite order inW and hence thatW is infinite.

LetS :={s, t} ⊆W. We will explain from three different points of view why (W, S) satisfies (A).

(i) Combinatorial group theory

The definition ofW via the presentation above makes it easy to define homo- morphisms fromW to another group. One need only specify two elements of the target group whose squares are trivial, and there is then a homomorphism takingsand tto these elements. In particular, if we want W to act on some set, it suffices to specify involutions ρs and ρt of that set, and then we can makesandt act asρsandρt, respectively. Condition (A) is now evident.

(ii) Euclidean geometry

We makeW act as a group of isometries of the real lineLby lettingsact as the reflection about 0 (x→ −x) andtact as the reflection about 1 (x→2−x).

Note, then, thatW acts as a group ofaffinetransformationsx→ax+b. This action has an associated “chamber geometry,” entirely analogous to what we saw in Chapter 1 for finite (linear) reflection groups. It is illustrated in Figure 2.1, whereCdenotes the open unit interval. The vertices in the picture are the integers. The two colors, black and white, indicate the two W-orbits of vertices.

C tsC

stC sC tC · · ·

· · ·

Fig. 2.1.The chambers forD_∞; affine version.

One can now check that the set T in the statement of (A) is the set of elements of W that act as reflections about integers, and one can identify T × {±1} with the set of half-lines whose endpoint is an integer. The action ofW onLinduces an action ofW on this set of half-lines, and condition (A) follows.

(iii) Linear algebra

There is a standard method for “linearizing” affine objects by embedding the affine space in question as an affine hyperplane (i.e., a translate of a linear hyperplane) in a vector space of one higher dimension. In the present case, we do this by identifying the lineLabove with the affine liney= 1 in the plane V =R². The affine action ofW onL extends to a linear action ofW onV.

Explicitly, since we want s(x,1) = (−x,1), we can set s(x, y) = (−x, y); in other words, we can makesact via the matrix

−1 0

0 1

.

Similarly, to maket(x,1) = (2−x,1), we can sett(x, y) := (2y−x, y); thust

acts via the matrix

−1 2

0 1

.

The picture of W acting on V is shown in Figure 2.2. It is simply the cone over the picture of W acting on L. (C now denotes thecone over the unit interval in the line y = 1.) The set T is now the set of reflections with respect to the walls of the chambers shown in the picture, and we may identify T×{±1}with the set of half-planes determined by these walls. Condition (A) now follows easily from the action of W on these half-planes.

· · · stC sC C tC tsC · · ·

Fig. 2.2.The chambers forD_∞; linear version.

Let’s compare this situation with that of Chapter 1. As in that context,s andt act as linear reflections onV, provided we interpret this term suitably:

Definition 2.4.IfV is a real vector space, not necessarily endowed with an inner product, then alinear reflection onV is a linear map that is the identity on a (linear) hyperplaneH and is multiplication by−1 on some complement ofH, i.e., a 1-dimensional subspaceHsuch thatV =H⊕H. The reflections considered in Chapter 1, where V has an inner product andH = H^⊥, will be called orthogonal reflections from now on to distinguish them from the more general linear reflections that we have just defined. Note that a linear reflection isnot uniquely determined by its hyperplane H of fixed points.

In the present example it is still true, as in Chapter 1, that W is gener- ated by linear reflections whose associated hyperplanes are the two walls of a

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“fundamental chamber” C. And it is still true thatC is a strict fundamental domain for the action of W on

w∈WwC. But this union is not the whole vector space V. It is, rather, the convex cone consisting of the open upper half-plane together with the origin. This is a very general phenomenon, as we will see in Section 2.6.

Note that the chamber geometry for W acting on V is very similar to the chamber geometry for finite reflection groups. For example, the chamber graph can be identified with the Cayley graph of (W, S), and the analogue of Proposition 1.118 remains valid (with the same proof). We record this explicitly for future reference. Let Hs and Ht be the fixed hyperplanes of s and t, respectively, and let U_±(s) and U_±(t) be the corresponding open half-spaces, where the positive half-space is the one containingC.

Lemma 2.5.For all w∈W, wC ⊆U+(s) if and only if l(sw)> l(w), and

wC⊆U+(t) if and only ifl(tw)> l(w).

Finally, we will show that our representation ofW as a “linear reflection group” admits a description resembling the description of a finite reflection group in terms of its Coxeter matrix (Section 1.5.5). Since we have no natural inner product on V, we introduce the dual space V^∗ and use inner-product notation for the canonical pairingV^∗×V →R, i.e.,

f, v:=f(v) forf ∈V^∗ andv∈V. Define es, et∈V^∗ by

es,(x, y):=x , et,(x, y):=y−x .

With these definitions, the fixed hyperplane forsis given byes,−= 0, the fixed hyperplane fortis given byet,−= 0, and the fundamental chamberC is given byes,−>0 andet,−>0.

We still have a Coxeter matrix M specifying the orders of the pairwise products of the generators. It is given by

M =

1 ∞

∞ 1

.

The corresponding Coxeter diagram is ∞ . Imitating equation (1.22), we now put a symmetric bilinear form onV^∗ by setting

B(e_s, e_s) =B(e_t, e_t) =−cosπ 1 = 1 and

B(es, et) =B(et, es) =−cos π

∞ =−1.

Next, define linear reflections s, t on V^∗ by using this bilinear form as in equation (1.23):

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^∗(e_s),(x, y)=e_s, s(x, y)=e_s,(−x, y)=−x=−e_s,(x, y), so s^∗(e_s) = −e_s =s(e_s). 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.