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ПЃ в€€ О¦. 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 ОЈ are open polyhedral conesвЂ”connected components of V Hв€€ОЈ H.
The closures of these cones are called closed chambers. Facets of chambers
(i.e. faces of maximal dimension) are panels and mirrors in ОЈ 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 ПЃв€€О  VПЃв€’ for some simple
system О . Every panel of C belongs to one of the walls HПЃ for a root ПЃ в€€ О .
Vice versa, if ПЃ в€€ О  then HПЃ в€© C is a panel of C.

Proof. Take any vector Оі in the chamber C and consider the linear func-
tional f (О») = в€’(Оі, О»). Since Оі does not belong to any mirror HО± in ОЈ,
the functional f does not vanish on roots in О¦. Therefore the condition
f (О±) > 0 determines a positive system О¦+ and the corresponding simple
system О . Now consider the cone C deп¬Ѓned by the inequalities (О», ПЃ) < 0
for all ПЃ в€€ О . Obviously Оі в€€ C and therefore C вЉ† C . If C = C
then some hyperplane HО± , О± в€€ О¦, bounds C and intersects C nontriv-
ially. But О± = cПЃ ПЃ where all cПЃ are all non-negative or all non-positive,
and (Оі, О±) = cПЃ (Оі, ПЃ) cannot be equal 0. This contradiction shows that
C = C . The closure C of C is deп¬Ѓned by the inequalities (О», ПЃ) 0 for
ПЃ в€€ О  which is equivalent to (О», О±) 0 for О± в€€ О“. Therefore the cone

63
64

Positive cone О“

e
u e0
e
e
e e The fundamental chamber C is
e
e e deп¬Ѓned as the interior of the cone
e ee
'e e e e e e e e e E dual to the positive cone О“, i.e.
ВЃВЃ
eВЁ
as the set of vectors О» such that
 ВЁВЁВЁ e ВЁВЃ
 ВЁe ВЁВЃВЃ (О», Оі) < 0 for all Оі в€€ О“.
ВЁВЁ ВЁ ВЁ
eВЁ
ВЁВЁВЁВЁВЁВЃ

ВЁВЁe
ВЁВЁ

) В…
ВЁ
Fundamental
chamber C

Figure 3.1: The fundamental chamber.

\$
\$
r
В  rrВ  \$\$В ВЈ В
\$
В  \$В
\$\$\$ rrВ  ВЈ The Coxeter complex of type BC3 is
\$
В  В
В e ВЈ
d formed by all the mirrors of symmetry of
ВЈ
d В  eВ  В
the cube; here they are shown by their
ВЈ
dВ  e
В  ВЈe
В
d lines of intersection with the faces of the
ВЈe
В d cube.
В
d ВЈВ
В
В  dВ
ВЈ

Figure 3.2: The Coxeter complex BC3 .

C is dual to the positive cone О“ (see Figure 3.1) and every facet of C is
perpendicular to some edge of О“, and vice versa. In particular, every panel
of C belongs to the wall HПЃ for some simple root ПЃ в€€ О .
The same argument works in the reverse direction: if О  is any simple
system then, since О  is linearly independent, we can п¬Ѓnd a vector Оі such
that (Оі, ПЃ) < 0 for all ПЃ в€€ О . Then (Оі, О±) = 0 for all roots О± в€€ О¦ and the
chamber C containing Оі has the form C = ПЃв€€О  VПЃв€’ .

The set of all chambers associated with the root system О¦ is called the
Coxeter complex and will be denoted by C. See, for example, Figures 3.2
and 3.3. If О  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 вЂў Mirrors and Reп¬‚ections вЂў Version 01 вЂў 25.02.00 65



 d

d
ВЈ In the case that ОЈ is the system
d
ВЈ d
ВЂВЂ of mirrors of symmetry of a regu-
ВЈ 7 ВЂВЂ
7
ВЈ lar polytope в€†, the Coxeter complex
В€В€ 7
ВЈВ€
is basically the c subdivision of the
В€7
i
Вў
e
i faces of в€† by the mirrors of sym-
Вў
e
ii metries of faces (here shown only on
2Вў
e 222
d e2 В’ one face of the dodecahedron в€†).
3Вў
В’
d D 3
3
dD 3

Figure 3.3: Chambers and baricentric subdivision

1
Вђ
The symmetry group of the tetra- В„Вђ
В„Вђ
hedron acts on its 4 vertices as
В„Вђ
the symmetric group Sym4 . The Вђ
В„
(23)
В„В–В–ВђВђ
(34)
reп¬‚ections in the walls of the fun-
В„ В–В–Вђ 4В–
damental chamber are the trans- (12)
ВЎ
В„
positions (12), (23) and (34). ВЎ
В„
В„ВЎ
Therefore they generate Sym4 .
В„ВЎ
3 2

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

vex.

3.2 Generation by simple reп¬‚ections
Simple reп¬‚ections. Let О  = { ПЃ1 , . . . , ПЃn } be a simple system of roots.
The corresponding reп¬‚ections ri = sПЃi are called simple reп¬‚ections.

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

Proof. Set W = r1 , . . . , rn . We shall prove п¬Ѓrst 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 reп¬‚ections r1 , . . . , rn . Therefore the neighbour-
ing chambers (i.e. the chambers sharing a common mirror with C) can be
obtained from C by reп¬‚ections in these mirrors and equal r1 C, . . . , rn C.
66

Let now w в€€ W , then the panels of the chamber wC belong to the mirrors
of reп¬‚ections wr1 wв€’1 , . . . , wrn wв€’1 . If D is a chamber adjacent to wC then
it can be obtained from wC by reп¬‚ecting wC in the common mirror, hence
D = wri wв€’1 В· wC = wri C for some i = 1, . . . , n. Notice that wri в€€ 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 в€€ W .
We can now complete the proof. If О± в€€ О¦ is any root and sО± the
corresponding reп¬‚ection then the wall HО± bounds some chamber D. We
know that D = wC for some w в€€ W . The fundamental chamber C is
bounded by the walls HПЃi for simple roots ПЃi (Theorem 3.1.1) and therefore
the wall HО± equals wHПЃi for some simple root ПЃi . Thus sО± = wri wв€’1 belongs
to W . Since the group W is generated by reп¬‚ections sО± 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 в€’ 1, n) (see Fig-
ure 3.4).

3.2.2 Prove that the reп¬‚ections

r1 = (12)(1в€— 2в€— ), . . . , rnв€’1 = (n в€’ 1, n)(n в€’ 1в€— , nв€— ), rn = (n, nв€— )

generate the hyperoctahedral group BCn .

3.3 Foldings
Given a non-zero vector О± в€€ V , the hyperplane HО± = { Оі в€€ V | (Оі, О±) = 0 }
cuts V in two subspaces
+ в€’
VО± = { Оі | (Оі, О±) 0 } and VО± = { Оі | (Оі, О±) 0}

intersecting along the common hyperplane HО± . The folding in the direction
of О± is the map fО± deп¬Ѓned by the formula

ОІ if (ОІ, О±) 0
fО± (ОІ) = .
sО± ОІ if (ОІ, О±) < 0
A. & A. Borovik вЂў Mirrors and Reп¬‚ections вЂў Version 01 вЂў 25.02.00 67

fО±
C
ВЎe
ВЎe
In the 2-dimensional case, a ВЎ e
ВЎ
R2 e
folding is exactly what its
ВЎ ВЎ
name suggests: the plane is ВЎ
e О±e
e 'e ВЎ
being folded on itself like a
e V+ e ВЎ в€’
VО±
sheet of paper. eО± eВЎ
HО±

Figure 3.5: Folding

+ в€’ +
Thus fО± п¬Ѓxes all points in VО± and maps VО± onto VО± symmetrically (see
Figure 3.5). Notice that fО± is an idempotent map, i.e. fО± fО± = fО± . The
folding fв€’О± is called the opposite to fО± . The reп¬‚ection sО± is made up of
two foldings fО± and fв€’О± :

sО± = fО± |VО± в€Є fв€’О± |VО± .
+ в€’

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