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2.3.7 Check that the complex numbers

e2kПЂi/n = cos 2kПЂ/n + i sin 2kПЂ/n, k = 0, 1, . . . , n в€’ 1

in the complex plane C are vertices of a regular n-gon в€†. Prove that the maps

r : z в†’ z В· e2ПЂi/n ,
t : z в†’ z,
ВЇ

where ВЇ denotes the complex conjugation, generate the group of symmetries of
в€†.

2.3.8 Use the idea of the proof of Theorem ?? to п¬Ѓnd the orders of the groups
of symmetries of the regular tetrahedron, cube, dodecahedron.
5
We use вЂ˜leftвЂ™ notation for action, so when we apply the composition st of two
transformations s and t to a point, we apply t п¬Ѓrst and then s: (st)A = s(tA).
44

2.4 Root systems
Mirrors and their normal vectors. Consider a reп¬‚ection s with the
mirror H. If we choose the orthogonal system of coordinates in V with the
origin O belonging to H then s п¬Ѓxes O and thus can be treated as a linear
orthogonal transformation of V . Let us take a nonzero vector О± perpen-
dicular to H then, obviously, RО± = H вЉҐ is the orthogonal complement of
H in V , s preserves H вЉҐ and therefore sends О± to в€’О±. Then we can easily
check that s can be written in the form
2(ОІ, О±)
sО± ОІ = ОІ в€’ О±,
(О±, О±)

where (О±, ОІ) denotes the scalar product of О± and ОІ. Indeed, a direct com-
putation shows that the formula holds when ОІ в€€ H and when ОІ = О±. By
the obvious linearity of the right side of the formula with respect to ОІ, it
is also true for all ОІ в€€ H + RО± = V .
Also we can check by a direct computation (left to the reader as an
exercise) that, given the nonzero vector О±, the linear transformation sО± is
orthogonal, i.e. (sО± ОІ, sО± Оі) = (ОІ, Оі) for all vectors ОІ and Оі. Finally, sО± = scО±
for any nonzero scalar c.
Notice that reп¬‚ections can be characterized as linear orthogonal trans-
formations of Rn with one eigenvalue в€’1 and (n в€’ 1) eigenvalues 1; the
vector О± in this case is an eigenvector corresponding to the eigenvalue в€’1.
Thus we have a one-to-one correspondence between the three classes of
objects:
вЂў hyperplanes (i.e. vector subspaces of codimension 1) in the Euclidean
vector space V ;

вЂў nonzero vectors deп¬Ѓned up to multiplication by a nonzero scalar;

вЂў reп¬‚ections in the group of orthogonal transformations of V .
The mirror H of the reп¬‚ection sО± will be denoted by HО± . Notice that
HО± = HcО± for any non-zero scalar c.
Notice, п¬Ѓnally, that orthogonal linear transformations of the Euclidean
vector space V (with the origin O п¬Ѓxed) preserve the relations between
mirrors, vectors and reп¬‚ections.

Root systems. Traditionally closed systems of reп¬‚ections were studied
in the disguise of root systems. By deп¬Ѓnition, a п¬Ѓnite set О¦ of vectors in V
is called a root system if it satisп¬Ѓes the following two conditions:
(1) О¦ в€© RПЃ = { ПЃ, в€’ПЃ } for all ПЃ в€€ О¦;
A. & A. Borovik вЂў Mirrors and Reп¬‚ections вЂў Version 01 вЂў 25.02.00 45

О¦ О¦
s
d T 
В  T
d В  s
d В

d В  d В
' dВ  E ' dВ  E
В d В d
В  d В  d
В  В‚
d
В  cd
В
В‚ c

Figure 2.8: If О¦ is a root system then the vectors ПЃ/|ПЃ| with ПЃ в€€ О¦ form the
root system О¦ with the same reп¬‚ection group. We are not much interested in
lengths of roots and in most cases can assume that all roots have length 1.

(2) sПЃ О¦ = О¦ for all ПЃ в€€ О¦.

The following lemma is an immediate corollary of Lemma 2.1.3.

Lemma 2.4.1 Let ОЈ be a п¬Ѓnite closed system of mirrors. For every mirror
H in ОЈ take two vectors В±ПЃ of length 1 perpendicular to H. Then the
collection О¦ of all these vectors is a root system. Vice versa, if О¦ is a root
system then { HПЃ | ПЃ в€€ О¦ } is a system of mirrors.

Proof. We need only to recall that a reп¬‚ection s, being an orthogonal
transformation, preserves orthogonality of vectors and hyperplanes: if ПЃ is
a vector and H is a hyperplane then ПЃ вЉҐ H if and only if sПЃ вЉҐ sH.
Also we can restate Lemma 2.2.1 in terms of root systems.

Lemma 2.4.2 Let О¦ be a root system. Then the group W generated by
reп¬‚ections sПЃ for ПЃ в€€ О¦ is п¬Ѓnite.

Exercises
2.4.1 Prove, by direct computation, that the linear transformation sО± given
by the formula
2(ОІ, О±)
sО± ОІ = ОІ в€’ О±,
(О±, О±)
is orthogonal, that is,
(sО± ОІ, sО± ОІ) = (ОІ, ОІ)
for all ОІ в€€ V .
46

2.4.2 Let О¦ be a root system in the Euclidean space V and U < V a vector
subspace of V . Prove that О¦ в€© U is a (possibly empty) root system in U .

2.4.3 Let V1 and V2 be two subspaces orthogonal to each other in the real
Euclidean vector space V and О¦i be a root system in Vi , i = 1, 2. Prove that
О¦ = О¦1 в€Є О¦2 is a root system in V1 вЉ• V2 ; it is called the direct sum of О¦1 and О¦2
and denoted
О¦ = О¦ 1 вЉ• О¦2 .

2.4.4 We say that a group W of orthogonal transformations of V is essential
if it acts on V without nonzero п¬Ѓxed points. Let О¦ be a root system in V , О¦
and W the corresponding system of mirrors and reп¬‚ection groups. Prove that
the following conditions are equivalent.

вЂў О¦ spans V .

вЂў The intersection of all mirrors in ОЈ consists of one point.

вЂў W is essential on V .

2.5 Planar root systems
We wish to begin the development of the theory of root systems with
referring to the readerвЂ™s geometric intuition.

Lemma 2.5.1 If О¦ is a root system in R2 then the angles formed by pairs
of neighbouring roots are all equal. (See Figure 2.9.)

Proof of this simple result becomes self-evident if we consider, instead of
roots, the corresponding system ОЈ of mirrors, see Figure 2.10. The mirrors
in ОЈ cut the plane into corners (later we shall call them chambers), and
adjacent corners, with the angles П† and П€, are congruent because they
are mirror images of each other. Therefore all corners are congruent. But
the angle between neighbouring mirrors is exactly the angle between the
corresponding roots.

Lemma 2.5.2 If a planar root system О¦ contains 2n vectors, n в‰Ґ 1, then
the reп¬‚ection group W (О¦) is the dihedral group D2n of order 2n.
A. & A. Borovik вЂў Mirrors and Reп¬‚ections вЂў Version 01 вЂў 25.02.00 47

T
The fundamental property of planar
s
d В

d В
d В  root systems: the angles П€ formed
d В  by pairs of neighbouring roots are
П€
d В
all equal. If the root system con-
' dВ  E
В d tains 2n vectors then П€ = ПЂ/n and
В  d
the reп¬‚ection group is the dihedral
В  d
group D2n of order 2n.
В  d
В  d
В  d
В‚
c

Figure 2.9: A planar root system (Lemma 2.5.1).

The fact that the angles formed by
d В
d В
d В  pairs of neighbouring roots are all
d В  equal becomes obvious if we con-
О±
d В
sider the corresponding system of
dВ
В d ОІ mirrors: О± = ОІ because the ad-
В  d
jacent angles are mirror images of
В  d
each other.
В  d
В  d
В  d

Figure 2.10: A planar mirror system (for the proof of Lemma 2.5.1).
48

Proof left to the reader as an exercise.

We see that a planar root system consisting of 2n vectors of equal length
is uniquely deп¬Ѓned, up to elation of R2 . We shall denote it I2 (n). Later we
shall introduced planar root systems A2 (which coincides with I2 (3)) as a
part of series of n-dimensional root systems An . In many applications of
the theory of reп¬‚ection groups the lengths of roots are of importance; in
particular, the root system I2 (4) associated with the system of mirrors of
symmetry of the square, comes in two versions, named B2 and C2 , which
contain 8 roots of two diп¬Ђerent lengths, see Figure 2.17 in Section 2.8.
Finally, the regular hexagon gives rise to the root system of type G2 , see
Figure 2.11.

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