Unformatted text preview: parabolic. If no parabolic subdiagram is
present, then the simplex is compact and the diagram is Lanner or compact hyperbolic. The complete list of hyperbolic Coxeter diagrams can be found in [120] or
[58].
Example 2.7. Let W (p, q, r ), 1 ≤ p ≤ q ≤ r, be the Coxeter group with Coxeter
graph of type Tp,q,r given in Figure 7.
• • q
... • ⎧•
⎪
⎪
⎪
⎪
⎪•
⎪
⎪
⎨
p.
.
⎪.
⎪
⎪
⎪
⎪•
⎪
⎪
⎩ • r
... • • •
Figure 7.
Then
• W (p, q, r ) is of elliptic type if and only if
(p, q, r ) = (1, q, r ), (2, 2, r ), (2, 3, 3), (2, 3, 4), (2, 3, 5);
• W (p, q, r ) is of parabolic type if and only if
(p, q, r ) = (2, 4, 4), (2, 3, 6), (3, 3, 3);
• W (p, q, r ) is of hyperbolic type if and only if
(p, q, r ) = (3, 4, 4), (2, 4, 5), (2, 3, 7).
3. Linear reflection groups
3.1. Pseudoreﬂections. Let E be a vector space over any ﬁeld K . A pseudoreﬂection in E is a linear invertible transformation s : E → E of ﬁnite order greater
than 1 which ﬁxes pointwise a hyperplane. A reﬂection is a diagonalizable pseudoreﬂection. A pseudoreﬂection is a reﬂection if and only if its order is coprime to
the characteristic of K .
Let v be an eigenvector of a reﬂection s of order d. Its eigenvalue η diﬀerent
from 1 is a dth root of unity η = 1. We can write s in the form
(3.1) s ( x) = x − ( x) v 14 IGOR V. DOLGACHEV for some linear function : E → K . Its zeroes deﬁne the hyperplane of ﬁxed
points of s, the reﬂection hyperplane. Taking x = v we obtain (v ) = 1 − η . This
determines uniquely when v is ﬁxed; we denote it by v .
A pseudoreﬂection (reﬂection) subgroup of GL(E ) is a subgroup generated by
pseudoreﬂections (reﬂections).
Assume that we are given an automorphism σ of K whose square is the identity.
¯
We denote its value on an element λ ∈ K by λ. Let B : E × E → K be a σ hermitian
form on E ; i.e. B is K linear in the ﬁrst variable and satisﬁes B (x, y ) = B (y, x).
Let U(E, B, σ ) be the unitary group of B , i.e. the subgroup of K linear transformations of E which preserve B . A pseudoreﬂection subgroup in GL(E ) is a
unitary pseudoreﬂection group if it is contained in a unitary group U(E, B, σ ) for
some σ, B and for any reﬂection (3.1) one can choose a vector v with B (v, v ) = 0.
The additional condition implies that
(3.2) v ( x) = (1 − η )B (x, v )
for all x ∈ V.
B (v, v ) In particular, the vector v is orthogonal to the hyperplane −1 (0).
v
Finite reﬂection groups are characterized by the following property of its algebra
of invariant polynomials ([14], Chapter V, §5, Theorem 4).
Theorem 3.1. A ﬁnite subgroup G of GL(E ) of order prime to char(K ) is a
reﬂection group if and only if the algebra S (E )G of invariants in the symmetric
algebra of E is isomorphic to a polynomial algebra.
In the case K = C this theorem was proven by Shephard and Todd [102], and in
the case of arbitrary characteristic but for groups generated by reﬂectio...
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 Fall '04
 IgorDolgachev
 Algebra, Geometry, Vector Space, The Land, Igor V. Dolgachev, Reflection group

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