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ρ ∈ ¦. Of course, this a hyperplane arrangement in the sense of Section 1.2,
and we shall freely use the relevant terminology. In particular, chambers
of Σ are open polyhedral cones”connected components of V H∈Σ H.
The closures of these cones are called closed chambers. Facets of chambers
(i.e. faces of maximal dimension) are panels and mirrors in Σ as walls.
Notice that every panel belongs to a unique wall. To fully appreciate this
architectural terminology (introduced by J. Tits), imagine a building built
out of walls of double-sided mirrors. Two chambers are adjacent if they
have a panel in common. Notice that every chamber is adjacent to itself.

Theorem 3.1.1 Every chamber C has the form ρ∈Π Vρ’ for some simple
system Π. Every panel of C belongs to one of the walls Hρ for a root ρ ∈ Π.
Vice versa, if ρ ∈ Π then Hρ © C is a panel of C.


Proof. Take any vector γ in the chamber C and consider the linear func-
tional f (») = ’(γ, »). Since γ does not belong to any mirror H± in Σ,
the functional f does not vanish on roots in ¦. Therefore the condition
f (±) > 0 determines a positive system ¦+ and the corresponding simple
system Π. Now consider the cone C de¬ned by the inequalities (», ρ) < 0
for all ρ ∈ Π. Obviously γ ∈ C and therefore C ⊆ C . If C = C
then some hyperplane H± , ± ∈ ¦, bounds C and intersects C nontriv-
ially. But ± = cρ ρ where all cρ are all non-negative or all non-positive,
and (γ, ±) = cρ (γ, ρ) cannot be equal 0. This contradiction shows that
C = C . The closure C of C is de¬ned by the inequalities (», ρ) 0 for
ρ ∈ Π which is equivalent to (», ±) 0 for ± ∈ “. Therefore the cone

63
64



Positive cone “

e
u e0
e
e
e e The fundamental chamber C is
e
e e de¬ned as the interior of the cone
e ee
'e e e e e e e e e E dual to the positive cone “, i.e.


as the set of vectors » such that
 ¨¨¨ e ¨
 ¨e ¨ (», γ) < 0 for all γ ∈ “.
¨¨ ¨ ¨

¨¨¨¨¨

¨¨e
¨¨

) …
¨
Fundamental
chamber C


Figure 3.1: The fundamental chamber.



$
$
r
  rr  $$ £  
$
  $ 
$$$ rr  £ The Coxeter complex of type BC3 is
$
   
 e £
d formed by all the mirrors of symmetry of
£
d   e   
the cube; here they are shown by their
£
d  e
  £e
 
d lines of intersection with the faces of the
£e
 d cube.
 
d £ 
 
  d 
£


Figure 3.2: The Coxeter complex BC3 .

C is dual to the positive cone “ (see Figure 3.1) and every facet of C is
perpendicular to some edge of “, and vice versa. In particular, every panel
of C belongs to the wall Hρ for some simple root ρ ∈ Π.
The same argument works in the reverse direction: if Π is any simple
system then, since Π is linearly independent, we can ¬nd a vector γ such
that (γ, ρ) < 0 for all ρ ∈ Π. Then (γ, ±) = 0 for all roots ± ∈ ¦ and the
chamber C containing γ has the form C = ρ∈Π Vρ’ .

The set of all chambers associated with the root system ¦ is called the
Coxeter complex and will be denoted by C. See, for example, Figures 3.2
and 3.3. If Π is a simple system then the corresponding chamber is called
the fundamental chamber of C.
The following lemma is an immediate consequence of Lemma 1.2.3
Lemma 3.1.2 The union of two distinct adjacent closed chambers is con-
A. & A. Borovik • Mirrors and Re¬‚ections • Version 01 • 25.02.00 65



 d

d
£ In the case that Σ is the system
d
£ d
€€ of mirrors of symmetry of a regu-
£ 7 €€
7
£ lar polytope ∆, the Coxeter complex
ˆˆ 7
£ˆ
is basically the c subdivision of the
ˆ7
i
¢
e
i faces of ∆ by the mirrors of sym-
¢
e
ii metries of faces (here shown only on

e 222
d e2 ’ one face of the dodecahedron ∆).


d D 3
3
dD 3


Figure 3.3: Chambers and baricentric subdivision

1

The symmetry group of the tetra- „
„
hedron acts on its 4 vertices as
„
the symmetric group Sym4 . The 

(23)
„––
(34)
re¬‚ections in the walls of the fun-
„ –– 4–
damental chamber are the trans- (12)
¡

positions (12), (23) and (34). ¡

„¡
Therefore they generate Sym4 .
„¡
3 2

Figure 3.4: Generation by simple re¬‚ections (Theorem 3.2.1).

vex.


3.2 Generation by simple re¬‚ections
Simple re¬‚ections. Let Π = { ρ1 , . . . , ρn } be a simple system of roots.
The corresponding re¬‚ections ri = sρi are called simple re¬‚ections.

Theorem 3.2.1 The group W is generated by simple re¬‚ections.

Proof. Set W = r1 , . . . , rn . We shall prove ¬rst that

the group W is transitive in its action on C.

Proof of the claim. The fundamental chamber C is bounded by panels lying
on the mirrors of the simple re¬‚ections r1 , . . . , rn . Therefore the neighbour-
ing chambers (i.e. the chambers sharing a common mirror with C) can be
obtained from C by re¬‚ections in these mirrors and equal r1 C, . . . , rn C.
66

Let now w ∈ W , then the panels of the chamber wC belong to the mirrors
of re¬‚ections wr1 w’1 , . . . , wrn w’1 . If D is a chamber adjacent to wC then
it can be obtained from wC by re¬‚ecting wC in the common mirror, hence
D = wri w’1 · wC = wri C for some i = 1, . . . , n. Notice that wri ∈ W . We
can proceed to move from a chamber to an adjacent one until we present
all chambers in C in the form wC for appropriate elements w ∈ W .
We can now complete the proof. If ± ∈ ¦ is any root and s± the
corresponding re¬‚ection then the wall H± bounds some chamber D. We
know that D = wC for some w ∈ W . The fundamental chamber C is
bounded by the walls Hρi for simple roots ρi (Theorem 3.1.1) and therefore
the wall H± equals wHρi for some simple root ρi . Thus s± = wri w’1 belongs
to W . Since the group W is generated by re¬‚ections s± we have W = W .


In the course of the proof we have obtained one more important result:
Corollary 3.2.2 The action of W on C is transitive.
This observation will be later incorporated into Theorem 3.5.1.

Exercises

3.2.1 Use Theorem 3.2.1 to prove the (well-known) fact that the symmetric
group Symn is generated by transpositions (12), (23), . . . , (n ’ 1, n) (see Fig-
ure 3.4).

3.2.2 Prove that the re¬‚ections

r1 = (12)(1— 2— ), . . . , rn’1 = (n ’ 1, n)(n ’ 1— , n— ), rn = (n, n— )

generate the hyperoctahedral group BCn .


3.3 Foldings
Given a non-zero vector ± ∈ V , the hyperplane H± = { γ ∈ V | (γ, ±) = 0 }
cuts V in two subspaces
+ ’
V± = { γ | (γ, ±) 0 } and V± = { γ | (γ, ±) 0}

intersecting along the common hyperplane H± . The folding in the direction
of ± is the map f± de¬ned by the formula

β if (β, ±) 0
f± (β) = .
s± β if (β, ±) < 0
A. & A. Borovik • Mirrors and Re¬‚ections • Version 01 • 25.02.00 67



C
¡e
¡e
In the 2-dimensional case, a ¡ e
¡
R2 e
folding is exactly what its
¡ ¡
name suggests: the plane is ¡
e ±e
e 'e ¡
being folded on itself like a
e V+ e ¡ ’

sheet of paper. e± e¡


Figure 3.5: Folding

+ ’ +
Thus f± ¬xes all points in V± and maps V± onto V± symmetrically (see
Figure 3.5). Notice that f± is an idempotent map, i.e. f± f± = f± . The
folding f’± is called the opposite to f± . The re¬‚ection s± is made up of
two foldings f± and f’± :

s± = f± |V± ∪ f’± |V± .
+ ’



We say that a folding f covers a subset X ‚ V if X ⊆ f (V ).
By de¬nition, a folding of the chamber complex C is a folding along one
of its walls.

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