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Unformatted text preview: itting an
ordered set of generators S of order 2 with deﬁning relations
(ss )m(s,s ) = 1, s, s ∈ S,
where m(s, s ) is the order of the product ss (the symbol ∞ if the order is inﬁnite).
The pair (W, S ) is called a Coxeter system.
The Coxeter graph of (W, S ) is the graph whose vertices correspond to S and
any two diﬀerent vertices are connected by m(s, s ) − 2 edges or by an edge labeled
with m(s, s ) − 2 or a thick edge if m(s, s ) = ∞. We say that (W, S ) is irreducible
if the Coxeter graph is connected.
One proves the following theorem (see [112]).
Theorem 2.5. Let P be a nondegenerate Coxeter polytope of ﬁnite volume in X n
and Γ(P ) be the corresponding reﬂection group. The pair (Γ(P ), S ), where S is
the set of reﬂections with respect to the set of faces of P is a Coxeter system. Its
Coxeter graph is equal to the Coxeter diagram of P , and the Gram matrix of P is
equal to the matrix
π
)(s,s )∈S ×S .
(2.2)
(− cos
m(s, s )
The converse is partially true. The following facts can be found in [14]. Let
(W, S ) be an irreducible Coxeter system with no m(s, s ) equal to ∞. One considers
the linear space V = RS and equips it with a symmetric bilinear form B deﬁned by
π
.
(2.3)
B (es , es ) = − cos
m(s, s )
Assume that B is positive deﬁnite ((W, S ) is elliptic ). Then W is ﬁnite and
isomorphic to a reﬂection group Γ in the spherical space S n , where n + 1 = #S − 1.
The corresponding Γcell can be taken as the intersection of the sphere with the
simplex in Rn+1 with facets orthogonal to the vectors es .
Assume that B is degenerate and semipositive deﬁnite ((W, S ) is parabolic ).
Then its radical V0 is onedimensional and is spanned by a unique vector v0 =
as = 1. The group W acts naturally
as es satisfying as > 0 for all s and
as a reﬂection group Γ in the aﬃne subspace E n = {φ ∈ V ∗ : φ(v0 ) = 1} with
the associated linear space (V /V0 )∗ . The corresponding Γcell has n + 1 facets
orthogonal to the vectors es and is a simplex in aﬃne space.
Assume that B is nondegenerate, indeﬁnite and W is of coﬁnite volume in the
orthogonal group of B ((W, S ) is hyperbolic ). In this case the signature of B is
equal to (n − 1, 1) and C = s∈S R+ es is contained in one of the two connected
components of the set {x ∈ E : B (x, x) < 0}. Let H n be the hyperbolic space equal REFLECTION GROUPS IN ALGEBRAIC GEOMETRY 13 to the image of this component in P(E ). Then the action of W in H n is isomorphic
¯
to a reﬂection group Γ. A Γcell can be chosen to be the image of the closure C of
C in P(E ).
In all cases the Coxeter diagram of a Γcell coincides with the Coxeter graph of
(W, S ). We will call the matrix given by (2.3) the Gram matrix of (W, S ).
Remark 2.6. Irreducible reﬂection groups in S n and E n correspond to elliptic or
parabolic irreducible Coxeter systems. Hyperbolic Coxeter systems deﬁne Coxeter
simplices in H n of ﬁnite volume. Their Coxeter diagrams are called quasiLanner
[120] or hyperbolic [14]. They are characterized by the condition that each of its
proper subdiagrams is either elliptic or...
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This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of MichiganDearborn.
 Fall '04
 IgorDolgachev
 Algebra, Geometry, The Land

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