Unformatted text preview: ections in Γ contain x0 . Therefore each Γcell is a
polyhedral cone with vertex at x0 . In the case when X n = E n , we can use x0
to identify E n with its linear space V and the group Γ with a reﬂection group in
S n−1 . The same is true if X n = S n . If X n = H n and x0 ∈ H n (resp. x0 lies on the 10 IGOR V. DOLGACHEV absolute), then by considering the orthogonal subspace in Rn+1 to the line deﬁned
by x0 we ﬁnd an isomorphism from Γ to a reﬂection group in S n−1 (resp. E n−1 ).
Any convex polytope of ﬁnite volume in E n or H n is nondegenerate in the sense
that its faces do not have a common point and the unit norm vectors of the faces
span the vector space V . A spherical convex polytope is nondegenerate if it does
not contain opposite vertices.
Theorem 2.2. Let Γ be an irreducible reﬂection group in X n with nondegenerate
Γcell P . If X n = S n , then Γ is ﬁnite and P is equal to the intersection of S n with
a simplicial cone in Rn+1 . If X n = E n , then Γ is inﬁnite and P is a simplex in
E n.
This follows from the following simple lemma ([14], Chapter V, §3, Lemma 5):
Lemma 2.3. Let V be a real vector space with positive deﬁnite symmetric bilinear
form (v, w) and let (vi )i∈I be vectors in V with (vi , vj ) ≤ 0 for i = j . Assume
that the set I cannot be nontrivially split into the union of two subsets I1 and I2
such that (vi , vj ) = 0 for i ∈ I1 , j ∈ I2 . Then the vectors vi are either linearly
independent or span a hyperplane and a linear dependence can be chosen of the
form
ai vi with all ai positive.
The classiﬁcation of irreducible nondegenerate Coxeter polytopes, and hence
irreducible reﬂection groups Γ in S n and Rn with nondegenerate Γcell, was given
by Coxeter [27]. The corresponding list of Coxeter diagrams is given in Table 1.
Here the number of nodes in the spherical (resp. euclidean) diagram is equal to
the subscript n (resp. n + 1) in the notation. The number n is equal to the rank of
the corresponding Coxeter polytope. We will refer to diagrams from the ﬁrst (resp.
second) column as elliptic Coxeter diagrams (resp. parabolic Coxeter diagrams ) of
rank n.
Our classiﬁcation of plane reﬂection groups in section 1.1 ﬁts in this classiﬁcation:
˜
˜
˜
˜
˜
(2, 4, 4) ←→ C2 , (2, 3, 6) ←→ G2 , (3, 3, 3) ←→ A2 , (2, 2, 2, 2) ←→ A1 × A1 .
The list of ﬁnite reﬂection groups not of type H3 , H4 , I2 (m), m = 6 (I2 (6) is often
denoted by G2 ) coincides with the list of Weyl groups of simple Lie algebras of the
corresponding type An , Bn or Cn , G2 , F4 , E6 , E7 , E8 . The corresponding Coxeter
diagrams coincide with the Dynkin diagrams only in the cases A, D, and E . The
second column corresponds to aﬃne Weyl groups.
The group of type An is the symmetric group Σn+1 . It acts in the space
(2.1) V = {(a1 , . . . , an+1 ) ∈ Rn+1 : a1 + . . . an+1 = 0} with the standard inner product as the group generated by reﬂections in vectors
ei − ei+1 , i = 1, . . . n.
The group of type Bn is isomorphic...
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 Fall '04
 IgorDolgachev
 Algebra, Geometry, Vector Space, The Land, Igor V. Dolgachev, Reflection group

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