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 , a sphere or a hyperbolic (Lobachevskii) space .

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 -dimensional case, and, in connection with problems of the theory of functions of a complex variable, the hyperbolic plane; regular partitions of the space 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 and were enumerated (those in can be considered as a particular case of reflection groups in ). 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 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 with a bounded fundamental polyhedron (see Discrete group of transformations).

Basic results of the theory of reflection groups.

Let , or . Every reflection group in is generated by reflections in hyperplanes , , which bound a fundamental polyhedron . Relative to this system of generators, the reflection group is a Coxeter group with defining relations , where the numbers are obtained as follows: If the faces and are adjacent and the angle between them is equal to , then ; if they are not adjacent, then (and the hyperplanes and do not intersect). On the other hand, any convex polyhedron in all dihedral angles of which are submultiples of is the fundamental polyhedron of the group generated by the reflections in its bounding hyperplanes.

Every reflection group in (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 with a bounded fundamental polyhedron (as a group of motions) is the direct product of groups of type (II).

Reflection groups in 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 .

A reflection group in is defined by a -gon with angles

(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 parameters.

When , the fundamental polyhedron of a reflection group in is uniquely defined by its combinatorial structure and its dihedral angles. For , an exhaustive description of these polyhedra has been obtained

and, thereby, of reflection groups as well. For , only examples and a few general methods of construction for reflection groups in are known (see [6], ). It is not known (1990) whether there exists a reflection group in with a bounded fundamental polyhedron when and with a fundamental polyhedron of finite volume when .

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 be the degrees of the generators of the invariants of a finite reflection group ( is the dimension of the space); the numbers are called the exponents of the group . The formula

holds, where is the number of elements in for which the space of fixed points has dimension . In particular, is equal to the number of reflections in ; is equal to the order of the group. If is irreducible, then the eigenvalues of its Killing–Coxeter element (see Coxeter group) are equal to , where is the Coxeter number:

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 . 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 , 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