Examples

Finite Reflection Groups

1.2 Examples

Definition 1.5.A setΦsatisfying the hypotheses of Lemma 1.4 will be called a generalized root system. The elements of Φ will be called roots. We will always assume (without loss of generality) that our generalized root systems arereduced, in the sense that±α(forα∈Φ) are the only scalar multiples ofα that are again roots. Thus there is exactly one pair ±αfor each generating reflection in the statement of Lemma 1.4.

To emphasize the distinction between generalized root systems and the classi- cal ones that leave a lattice invariant, we will sometimes refer to the classical ones ascrystallographic root systems.

It is also convenient to have some terminology for the sort of decomposition of V that arose in the proof of Lemma 1.4. Let W be a group generated by reflectionss_H (H ∈ H), whereHis a set of hyperplanes. LetV₀ be the fixed- point set

V^W =

H∈H

H .

Definition 1.6.We callV0theinessential part ofV, and we call its orthogo- nal complementV1theessential part ofV. The pair (W, V) is calledessential ifV1 =V, or, equivalently, if V0 = 0. The dimension ofV1 is called therank of the finite reflection groupW.

The study of a general (W, V) is easily reduced to the essential case. Indeed, V1isW-invariant sinceV0is, and clearly (V1)^W = 0; so we have an orthogonal decomposition V = V0⊕V1, where the action of W is trivial on the first summand and essential on the second. We may therefore identify W with a group acting on V1, and as such, W is essential (and still generated by reflections). IfW is the groupWΦassociated with a generalized root system, thenW is essential if and only ifΦspansV.

Exercise 1.7.Show that every finite reflection groupW has the formWΦfor some generalized root system Φ.

1.2 Examples

There are two classical families of examples of finite reflection groups. The first, as we have already indicated, consists of Weyl groups of (crystallo- graphic) root systems. The second consists of symmetry groups of regular solids. We will not assume that the reader knows anything about either of these two subjects. But it will be convenient to use the language of root sys- tems or regular solids informally as we discuss examples. It is a fact that all finite reflection groups can be explained in terms of one or both of these theories; we will return to this in the next section.

Example 1.8.The groupW of order 2 generated by a single reflectionsαis a finite reflection group of rank 1. After passing to the essential part ofV, we may identifyW with the group{±1} acting onRby multiplication. It is the group of symmetries of the regular solid [−1,1] inR. It is also the Weyl group of the root system Φ:={±α}, which is called the root system of type A₁. Example 1.9.Let V be 2-dimensional, and choose two hyperplanes (lines) that intersect at an angle ofπ/mfor some integerm≥2. Let sandtbe the corresponding reflections and let W be the group s, t they generate. [Here and throughout this book we use angle brackets to denote the group generated by a given set.] Then the product ρ:= st is a rotation through an angle of 2π/mand hence is of orderm. Moreover,sconjugatesρtos(st)s=ts=ρ⁻¹ and similarly fort, so the cyclic subgroupC:=ρof ordermis normal inW. Finally, the quotient W/C is easily seen to be of order 2; henceW is indeed a finite reflection group, of order 2m.

This groupW is called thedihedral group of order 2m, and we will denote it by D2m. If m ≥ 3, W is the group of symmetries of a regular m-gon. If m = 3, 4, or 6, then W can also be described as the Weyl group of a root systemΦ, said to be of type A2, B2, or G2, respectively. The root system of type A2 (m= 3) consists of 6 equally spaced vectors of the same length, as shown in Figure 1.1, which also shows the three reflecting hyperplanes (lines).

There are two oppositely oriented root vectors for each hyperplane. To get

Fig. 1.1.The root system of type A2 and the reflecting hyperplanes.

B2 and G2 (m = 4 and m= 6), we can take m equally spaced unit vectors together with the sum of any two cyclically consecutive ones, as shown in Figure 1.2

Of course, we can always getD2m from the generalized root system con- sisting of 2mequally spaced unit vectors; but this is not crystallographic for m >3.

Example 1.10.LetW be the group of linear transformations ofRⁿ (n≥2) that permute the standard basis vectors e1, e2, . . . , en. ThusW is isomorphic to the symmetric group Sn onnletters and can be identified with the group

1.2 Examples 13

Fig. 1.2.The root systems of type B2 and G2.

of n×n permutation matrices. It is generated by the_n

2

transpositions s_ij (i < j), where s_ij interchanges the ith and jth coordinates, so it is a finite reflection group (see Example 1.2). Note that (W,Rⁿ) is not essential. In fact V^W is the linex₁ =x₂ =· · · =x_n spanned by the vectore:= (1,1, . . . ,1).

So the subspace V₁of Rⁿ on whichW is essential is the (n−1)-dimensional subspace e^⊥ defined by_n

i=1xi= 0, whenceW has rankn−1.

The interested reader can verify that W is the group of symmetries of a regular (n−1)-simplex in V1. [Hint: The convex hull σ of e1, . . . , en is a regular (n−1)-simplex in the affine hyperplane

xi = 1, which is parallel to V1. The desired regular simplex in V1 is now obtained from σ via the translation x→ x−b, where b is the barycenter of σ.] W is also the Weyl group of a root system inV1, called the root system of type An−1. It consists of then(n−1) vectorsei−ej (i=j).

Whenn= 2, this example reduces to Example 1.8; whenn= 3, it reduces to Example 1.9 with m = 3 (after we pass to the essential part), i.e., W is dihedral of order 6. For n = 4, Figure 1.3 shows the unit sphere in the

Fig. 1.3.The hyperplanes for the reflection group of type A3.

3-dimensional space V1 on which W is essential; the 6 =₄

2

planes xi =xj

(corresponding to the reflectionss_ij) cut the sphere in the solid great circles.

The dotted great circle represents an equator and does not correspond to a reflecting hyperplane. [Note: Figure 1.3 is a schematic picture; it accurately shows the combinatorics, but it distorts the geometry. See Figure 0.1 for a more accurate picture.]

Note that the hyperplanes induce a triangulation of the sphere as the barycentric subdivision of the boundary of a tetrahedron. The black vertices are the vertices of the original tetrahedron (only 3 of which are visible in the hemisphere shown in the picture); the gray vertices (of which 3 are visible) are the barycenters of the edges of the tetrahedron; and the white vertices (of which one is visible) are the barycenters of the 2-dimensional faces of the tetrahedron. We will see later that this generalizes to arbitraryn: the reflecting hyperplanes triangulate the sphere inV1 as the barycentric subdivision of the boundary of an (n−1)-simplex. See Exercise 1.112.

Example 1.11.LetW be the group of linear transformations ofRⁿ (n≥1) leaving invariant the set{±ei}of standard basis vectors and their negatives.

In terms of matrices, W can be viewed as the group of n×n monomial matrices whose nonzero entries are±1. [Recall that amonomial matrix is one with exactly one nonzero element in every row and every column.] Elements ofW are sometimes called “signed permutations.” The groupW is generated by transpositions sij as above, together with reflections t1, . . . , tn, where ti

changes the sign of theith coordinate (i.e.,t_iis the reflection in the hyperplane x_i= 0). HenceW is a finite reflection group of order 2ⁿn!, and this time it is essential.

Once again, the interested reader is invited to verify thatW is the group of symmetries of a regular solid in Rⁿ, which one can take to be then-cube [−1,1]ⁿ. Alternatively, take the solid to be the convex hull of the 2n vectors {±ei}; this is a “hyperoctahedron.” [The hyperoctahedron is thedual of the cube, which means that it is the convex hull of the barycenters of the faces of the cube. Since a solid and its dual have the same symmetry group, it makes no difference which one we choose. We had no reason to mention this in our previous examples because the dual of a regular m-gon is again a regular m-gon, and the dual of a regular simplex is again a regular simplex.]

And once again, W is the Weyl group of a root system, called the root system of type Bn, consisting of the vectors ±ei±ej (i = j) together with the vectors±ei. Alternatively,W can be described as the Weyl group of the root system of type C_n, consisting of the vectors ±e_i±e_j (i = j) together with the vectors ±2e_i; this is dual to type B_n. [Every root system Φ has a

“dual,” as we explain in Appendix B. A root system and its dual have the same Weyl group. The root systems mentioned in Examples 1.8–1.10, like the regular solids, are self-dual, so this issue did not arise.]

Whenn= 1 this example reduces to Example 1.8; whenn= 2 it reduces to Example 1.9 with m= 4, i.e.,W is dihedral of order 8.