Reflection group

A discrete group of transformations generated by reflections in hyperplanes. The most frequently studied are those consisting of mappings of a simply-connected complete Riemannian manifold of constant curvature, i.e. of a Euclidean space $ E ^ {n} $, a sphere $ S ^ {n} $ or a hyperbolic (Lobachevskii) space $ \Lambda ^ {n} $.

The theory of reflection groups has its origin in research into regular polyhedra and regular partitions of the Euclidean plane and the sphere ( "ornamentornaments" ). In the second half of the 19th century, this research was extended to include both the $ n $-dimensional case, and, in connection with problems of the theory of functions of a complex variable, the hyperbolic plane; regular partitions of the space $ \Lambda ^ {n} $ into regular polyhedra were also described. The symmetry group of any regular polyhedron, as well as the symmetry group of a regular partition of space into regular polyhedra are reflection groups. In 1934 (see [1]), all reflection groups in $ E ^ {n} $ and $ S ^ {n} $ were enumerated (those in $ S ^ {n} $ can be considered as a particular case of reflection groups in $ E ^ {n+ 1} $). As early as 1925–1927, in the work of E. Cartan and H. Weyl, reflection groups appeared as Weyl groups (cf. Weyl group) of semi-simple Lie groups. It was subsequently established that Weyl groups are in fact those reflection groups in $ E ^ {n} $ that have a single fixed point and are written in a certain basis by integer matrices, while affine Weyl groups are all reflections groups in $ E ^ {n} $ with a bounded fundamental polyhedron (see Discrete group of transformations).

Basic results of the theory of reflection groups.

Let $ X ^ {n} = S ^ {n} $, $ E ^ {n} $ or $ \Lambda ^ {n} $. Every reflection group in $ X ^ {n} $ is generated by reflections $ r _ {i} $ in hyperplanes $ H _ {i} $, $ i \in I $, which bound a fundamental polyhedron $ P $. Relative to this system of generators, the reflection group is a Coxeter group with defining relations $ ( r _ {i} r _ {j} ) ^ {n _ {ij} } = 1 $, where the numbers $ n _ {ij} $ are obtained as follows: If the faces $ H _ {i} \cap P $ and $ H _ {j} \cap P $ are adjacent and the angle between them is equal to $ \alpha _ {ij} $, then $ \alpha _ {ij} = \pi /n _ {ij} $; if they are not adjacent, then $ n _ {ij} = \infty $ (and the hyperplanes $ H _ {i} $ and $ H _ {j} $ do not intersect). On the other hand, any convex polyhedron in $ X ^ {n} $ all dihedral angles of which are submultiples of $ \pi $ is the fundamental polyhedron of the group generated by the reflections in its bounding hyperplanes.

Every reflection group in $ E ^ {n} $ (as a group of motions) is the direct product of a trivial group operating in a Euclidean space of a certain dimension, and groups of motions of the following two types:

a finite reflection group whose fundamental polyhedron is a simplicial cone; and (II) an infinite reflection group whose fundamental polyhedron is a simplex. A group of type

can be seen as a reflection group on a sphere with its centre at the vertex of the fundamental cone; its fundamental polyhedron will then be a spherical simplex. A reflection group of type

is uniquely defined by its Coxeter matrix, for which reason the classification of these groups coincides with the classification of finite Coxeter groups. A reflection group of type (II) is defined by its Coxeter matrix up to a dilatation. The classification of these groups, up to a dilatation, coincides with the classification of indecomposable parabolic Coxeter groups. Every reflection group in $ E ^ {n} $ with a bounded fundamental polyhedron (as a group of motions) is the direct product of groups of type (II).

Reflection groups in $ \Lambda ^ {n} $ have been significantly less studied. For many reasons, it is natural to distinguish those whose fundamental polyhedron is bounded or tends to the absolute (the "sphere at infinity" ) only at a finite number of points (this is equivalent to finiteness of the volume). Only these groups are considered below. They are described more or less clearly only for $ n = 2, 3 $.

A reflection group in $ \Lambda ^ {2} $ is defined by a $ k $-gon with angles

$$ \frac \pi {n _ {1} }, \dots, \frac \pi {n _ {k} } ,\ \textrm{ where } \ \frac{1}{n _ {1} } + \dots + \frac{1}{n _ {k} } < k - 2 $$

(if a vertex is infinitely distant, then its angle is considered to be equal to zero). A polygon with such given angles always exists and depends on $ k- 3 $ parameters.

When $ n \geq 3 $, the fundamental polyhedron of a reflection group in $ \Lambda ^ {n} $ is uniquely defined by its combinatorial structure and its dihedral angles. For $ n= 3 $, an exhaustive description of these polyhedra has been obtained

and, thereby, of reflection groups as well. For $ n \geq 4 $, only examples and a few general methods of construction for reflection groups in $ \Lambda ^ {n} $ are known (see [6], ). It is not known (1990) whether there exists a reflection group in $ \Lambda ^ {n} $ with a bounded fundamental polyhedron when $ n \geq 9 $ and with a fundamental polyhedron of finite volume when $ n \geq 22 $.

Linear reflection groups, acting discretely in an open convex cone of a real vector space, are considered alongside reflection groups in spaces of constant curvature. This makes a geometric realization of all Coxeter groups with a finite number of generators possible (see [3], [4]).

Every finite reflection group can be seen as a linear group. Of all finite linear groups, finite reflection groups are characterized by the fact that the algebras of invariant polynomials of these groups possess algebraically independent systems of generators [4]. For example, for the group of all permutations of the basis vectors, these will be the elementary symmetric polynomials. Let $ m _ {1} + 1 \dots m _ {n} + 1 $ be the degrees of the generators of the invariants of a finite reflection group $ G $ ($ n $ is the dimension of the space); the numbers $ m _ {1}, \dots, m _ {n} $ are called the exponents of the group $ G $. The formula

$$ ( 1+ m _ {1} t) \cdots ( 1+ m _ {n} t) = \ c _ {0} + c _ {1} t + \dots + c _ {n} t ^ {n} $$

holds, where $ c _ {k} $ is the number of elements in $ G $ for which the space of fixed points has dimension $ n- k $. In particular, $ m _ {1} + \cdots + m _ {n} $ is equal to the number of reflections in $ G $; $ ( m _ {1} + 1) \cdots ( m _ {n} + 1) $ is equal to the order of the group. If $ G $ is irreducible, then the eigenvalues of its Killing–Coxeter element (see Coxeter group) are equal to $ \mathop{\rm exp} ( 2 \pi i m _ {k} /h) $, where $ h $ is the Coxeter number:

$$ h = \max \{ m _ {k} \} + 1. $$

The assertions of the previous paragraph, with the exception of the last, also apply to linear groups over an arbitrary field of characteristic zero (see [4]). In this case it is appropriate to understand a reflection to be a linear transformation with space of fixed points of dimension $ n- 1 $. All finite linear reflection groups over the field of complex numbers are listed in [8]. Finite linear reflection groups over fields of non-zero characteristic have been found [9].

References

[1]	H.S.M. Coxeter, "On complexes with transitive groups of automorphisms" Ann. of Math. , 35 (1934) pp. 588–621
[2]	H.S.M. Coxeter, W.O.J. Moser, "Generators and relations for discrete groups" , Springer (1984) MR0609520 MR0562913 MR0349820 MR0174618 MR0088489 Zbl 0487.20023 Zbl 0422.20001 Zbl 0239.20040 Zbl 0133.28002 Zbl 0077.02801
[3]	J. Tits, "Groupes simples et géométries associées" , Proc. Internat. Congress Mathematicians 1962, Dursholm , Mittag-Leffler Inst. (1963) pp. 197–221 MR0175903 Zbl 0131.26502
[4]	N. Bourbaki, "Groupes et algèbres de Lie" , Eléments de mathématiques , Hermann (1968) pp. Chapts. 4–6 MR0240238 Zbl 0186.33001
[5a]	E.M. Andreev, "On convex polyhedra in Lobačevskii spaces" Math. USSR-Sb. , 10 : 3 (1970) pp. 413–440 Mat. Sb. , 81 (1970) pp. 445–478
[5b]	E.M. Andreev, "On convex polyhedra of finite volume in Lobačevskii space" Math. USSR-Sb. , 12 : 2 (1970) pp. 255–259 Mat. Sb. , 83 (1970) pp. 256–260
[6]	V.S. Makarov, "On Fedorov groups of the four- and five-dimensional Lobachevskii spaces" , Studies in general algebra , 1 , Kishinev (1968) pp. 120–129 (In Russian)
[7a]	E.B. Vinberg, "Discrete groups generated by reflections in Lobačevskii spaces" Math. USSR-Sb. , 1 : 3 (1967) pp. 429–444 Mat. Sb. , 72 (1967) pp. 471–488 MR0207853 Zbl 0166.16303
[7b]	E.B. Vinberg, "On groups of unit elements of certain quadratic forms" Math. USSR-Sb. , 16 : 1 (1972) pp. 17–35 Mat. Sb. , 87 (1972) pp. 18–36 MR295193 Zbl 0252.20054
[8]	G.C. Shephard, J.A. Todd, "Finite unitary reflection groups" Canad. J. Math. , 6 (1954) pp. 274–304 MR0059914 Zbl 0055.14305
[9]	A.E. Zalesskii, V.N. Serezhkin, "Finite linear groups generated by reflections" Math. USSR-Izv. , 17 : 3 (1981) pp. 477–503 Izv. Akad. Nauk SSSR Ser. Mat. , 44 (1980) pp. 1279–1307 MR0603578 MR0554927

Comments

All finite linear reflection groups over the skew-field of real quaternions are listed in [a1]. For the determination of finite linear reflection groups over fields of characteristic $ \neq 2 $, see [a2]–[a4].

References

[a1]	A.M. Cohen, "Finite quaternionic reflection groups" J. of Algebra , 64 (1980) pp. 293–324 MR0579063 Zbl 0433.20035
[a2]	A. Wagner, "Determination of the finite primitive reflection groups over an arbitrary field of characteristic not 2, I" Geom. Ded. , 9 (1980) pp. 239–253 MR0608141 Zbl 0443.51009
[a3]	A. Wagner, "Determination of the finite primitive reflection groups over an arbitrary field of characteristic not 2, II" Geom. Ded. , 10 (1981) pp. 191–203 MR0608141
[a4]	A. Wagner, "Determination of the finite primitive reflection groups over an arbitrary field of characteristic not 2, III" Geom. Ded. , 10 (1981) pp. 475–523 MR0608141 Zbl 0471.51015