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  or
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
... • ⎧•
⎩ • r
... • • •
• 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. Pseudo-reﬂections. 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 pseudo-reﬂection 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 pseudo-reﬂection (reﬂection) subgroup of GL(E ) is a subgroup generated by
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 pseudo-reﬂection subgroup in GL(E ) is a
unitary pseudo-reﬂection 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).
Finite reﬂection groups are characterized by the following property of its algebra
of invariant polynomials (, 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 , and in
the case of arbitrary characteristic but for groups generated by reﬂectio...
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