# 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

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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].