By Theorem 3.7.1, the latter group is generated by those simple re¬‚ections

whose mirrors contain the both sets P J and P K , that is, by re¬‚ections ri

with i ∈ J © K. Therefore

WJ©K = WJ © WK .

In particular, this theorem means that

{ r1 , . . . , rn } © WJ = { ri | i ∈ J }.

We have an important geometric interpretation of this result.

Proposition 3.8.2 Let D and E be the chambers corresponding to the

elements u and v of a parabolic subgroup PJ . If D and E are rj -adjacent

then j ∈ J.

Proof. Since D and E are rj -adjacent, then, by Theorem 3.6.2, we have

urj = v and rj ∈ PJ . Therefore j ∈ J.

Exercises

3.8.1 Let W = An’1 and let us view W as the symmetric group Symn of the

set [n], so that the simple re¬‚ections in W are

r1 = (12), r2 = (23), . . . rn’1 = (n ’ 1, n).

Prove that the parabolic subgroup

P = r1 , . . . , rk’1 , rk+1 , . . . , rn’1

is the stabiliser in Symn of the set { 1, . . . , k } and thus is isomorphic to Symk —

Symn’k .

3.9 Residues

We retain the notation of the previous sections.

Let D be a chamber in C and J ‚ I. A J-residue of D is the set of all

chambers in C which can be connected to D by galleries in which types of

panels between consequent chambers are of type ri for i ∈ J.

Let C be the fundamental chamber of C and w the element of W which

canonically corresponds to D, that is, D = wC. Then Theorem 3.6.3

says that a chamber vC belongs to the J-residue of D if and only if v =

wri1 · · · rit with i1 , . . . , it ∈ J. Now we have to recall the de¬nition of a

parabolic subgroup PJ = ri | i ∈ J and conclude that the chamber vC

belongs to the J-residue of D = wC if and only if v ∈ wPJ . Therefore

J-residues are in one-two-one correspondence with the left cosets of the

parabolic subgroup PJ .

A. & A. Borovik • Mirrors and Re¬‚ections • Version 01 • 25.02.00 79

[1234]

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Figure 3.8: A permutahedron for the group A3 = Sym4 . Its vertices form

one orbit under the permutation action of Sym4 in R3 and can be labelled by

elements of Sym4 . Here [i1 i2 i3 i4 ] denotes the permutation 1 ’ i1 , . . . , 4 ’ i4 .

3.10 Generalised permutahedra

We say that a point ± ∈ V is in general position if ± does not belong to Σ.

Let now δ be any point in general position, W · δ its orbit under W and

∆ the convex hull of W · δ. We shall call ∆ a generalised permutahedron

and study it in some detail.

Theorem 3.10.1 In the notation above, the following statements hold.

(1) Vertices of ∆ are exactly all points in the orbit W · δ and each chamber

in C contains exactly one vertex of ∆.

(2) Every edge of ∆ is parallel to some vector in ¦ and intersects exactly

one wall of the Coxeter complex C.

(3) The edges emanating from the given vertex are directed along roots

forming a simple system.

(4) If ± is the vertex of ∆ contained in a chamber C then the vertices

adjacent to ± are exactly all the mirror images si ± of ± in walls of

C.

80

β

r

± The segment [±β] not normal to a

r

d mirror Hρ it crosses cannot be an

d Hρ

d edge of the permutahedron ∆; in-

d deed, if ± and β are re¬‚ections of

d ± and β in Hρ then ± and β are

r

d

± also vertices of ∆ and [±β] belongs

d

d to the convex hull of ±, β, ± , β .

r

d

β

Figure 3.9: For the proof of Theorem 3.10.1.

Proof. Notice, ¬rst of all, that, since all points in the orbit W · δ lie at

the same distance from the origin, they belong to some sphere centered at

the origin. Therefore points in W · δ are the vertices of the convex hull of

W · δ. Next, because of simple transitivity of W on the Coxeter complex

C, every chamber in C contains exactly one vertex of ∆. This proves (1).

Let now ± and β be two adjacent, i.e. connected by an edge, vertexes of

∆. Then β belongs to a chamber distinct from ± and, therefore, the edge

[±β] intersects some mirror Hρ . If the edge [±β] is not perpendicular to Hρ ,

we immediately have a contradiction with the following simple geometric

argument (see Figure 3.9).

In Figure 3.9, the points ± and β are symmetric to ±, β, respectively, and

the convex quadrangle ±± ββ lies in a 2-dimensional plane perpendicular

to the mirror Hρ of symmetry. Therefore the segment [±β] belongs to the

interior of the quadrangle and cannot be an edge of ∆.

Hence [±β] is perpendicular to Hρ , hence β ’ ± = cρ for some c and

the mirror Hρ is uniquely determined, which proves (2).

Now, select a linear functional f which attains its minimum on ∆ at

the point ± and does not vanish at roots in ¦. Let ¦+ and Π be the

positive and simple system in ¦ associated with f . If s±ρ = sρ = s’ρ

is the re¬‚ection in W for the roots ±ρ, then s±ρ ± is a vertex of ∆ and

f (s±ρ ± ’ ±) > 0. But s±ρ ± ’ ± = cρ for some c. After swapping notation

for +ρ and ’ρ we can assume without loss that f (ρ) > 0, i.e. ρ ∈ ¦+ and

c > 0. Let β1 , . . . , βm be all vertices of ∆ adjacent to ±. Then βi ’ ± = ci ρi

for some ρi ∈ ¦+ and ci > 0.

And here comes the punchline: notice that the convex polytope ∆ is

contained in the convex cone “ spanned by the edges emanating from ±

(Figure 3.10). Since every positive roots ρ ∈ ¦+ points from the vertex ±

to the vertex sρ ± of ∆, all positive roots lie in the convex cone spanned by

A. & A. Borovik • Mirrors and Re¬‚ections • Version 01 • 25.02.00 81

rr

r

rr

One of the simple principles of

@r

@r linear programming which is ex-

@ @

@

r@ @

r @ t tremely useful in the study of Cox-

d t eter groups: a convex polytope is

d

d

contained in the convex polyhedral

d

t tt

d@ @

tr @@ cone spanned by the edges emanat-

r

td

ing from the given vertex.

td

td

td

tr d

rr d

rd

d

r

Figure 3.10: For the proof of Theorem 3.10.1.

the roots ρi ∈ ¦+ pointing from ± to adjacent to ± vertices βi . But this

means exactly that ρi form the simple system Π in ¦+ , which proves (3).

Also, the fact that βi ’ ± = cρi for c > 0 means that ± ∈ Vρ’ . Since this

i

holds for all simple roots, ± belongs to the fundamental chamber C = Vρ’ i

(Theorem 3.1.1). But, by the same theorem, C is bounded by the mirrors

of simple re¬‚ections and βi = sρi ± is the mirror image of ± in the wall Hρi

containing a panel of C. (4) is also proven.

Exercises

3.10.1 Sketch permutahedra for the re¬‚ection groups

A1 + A1 , A2 , BC2 , A1 + A1 + A1 , A2 + A1 .

3.10.2 Label, in a way analogous to Figure 3.8, the vertices of a permutahedron

for the hyperoctahedral group BC3 (Figure 3.11) by elements of the group.

3.10.3 Let ∆ be a permutahedron for a re¬‚ection group W . Prove that there

is a one-to-one correspondence between faces of ∆ and residues in the Coxeter

complex C of W . Namely, the set of chambers containing vertices of a given face

is a residue.

82

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