Unformatted text preview: the angle φ = ∠(H1 , H2 ) = ∠(H1 , H2 ) is deﬁned by
cos φ = −(e1 , e2 ), 0 ≤ φ ≤ π. If X n = H n we use the same deﬁnition if (e1 , e2 ) ≤ 1; otherwise we say that
the angle is divergent (in this case (e1 , e2 ) is equal to the hyperbolic cosine of the
distance between the hyperplanes). 8 IGOR V. DOLGACHEV EE "
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Figure 6. A convex polytope in X n is a nonempty intersection of a locally ﬁnite1 set of
halfspaces
−
P = ∩i∈I Hi .
−
The normal vectors ei deﬁning the halfspaces Hi all look inside the polytope. The
hyperplanes Hi ’s are called faces of the polytope. In the case X n = H n we also add
to P the points of the intersection lying on the absolute. We will always assume
that no ei is a positive linear combination of others or, equivalently, none of the
halfspaces contains the intersection of the rest of the halfspaces. In this case the
set of bounding hyperplanes can be reconstructed from P .
A convex polytope has a ﬁnite volume if and only if it is equal to a convex hull of
ﬁnitely many points (vertices) from X n (or from the absolute if X n = H n ). Such
a polytope has ﬁnitely many faces. If X n = E n or S n , it is a compact polytope.
A polytope of ﬁnite volume in H n is compact only if its vertices do not lie on the
absolute.
2.3. Reﬂection groups. A reﬂection group in a space of constant curvature is a
discrete group of motions of X n generated by reﬂections.
Theorem 2.1. Let Γ be a reﬂection group in X n . There exists a convex polytope
−
P (Γ) = ∩i∈I Hi such that
(i) P is a fundamental domain for the action of Γ in X n ;
−
−
(ii) the angle between any two halfspaces Hi , Hj is equal to zero or π/mij for
some positive integer mij unless the angle is divergent;
(iii) Γ is generated by reﬂections rHi , i ∈ I .
Conversely, for every convex polytope P satisfying property (ii) the group Γ(P )
generated by the reﬂections into its facets is a reﬂection group and P satisﬁes (i).
Proof. Consider the set H of mirror hyperplanes of all reﬂections contained in
Γ. For any mirror hyperplane H and g ∈ Γ, the hyperplane g (H ) is the mirror
hyperplane for the reﬂection grH g −1 . Thus the set H is invariant with respect to
Γ. Let K be a compact subset of X n . For any hyperplane H ∈ H meeting K , we
have rH (K ) ∩ K = ∅. Since Γ is a discrete group, the set {g ∈ G : g (K ) ∩ K = ∅}
1 Locally ﬁnite means that each compact subset of X n is intersected by only ﬁnitely many
hyperplanes. REFLECTION GROUPS IN ALGEBRAIC GEOMETRY 9 is ﬁnite. This shows that the set H is locally ﬁnite. The closure of a connected
component of
H
Xn \
H ∈H is a convex polytope called a cell (or Γcell or a fundamental polyhedron...
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 Fall '04
 IgorDolgachev
 Algebra, Geometry, Vector Space, The Land, Igor V. Dolgachev, Reflection group

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