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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|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 [[#References|[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|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|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|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 [[#References|[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 [[#References|[3]]], [[#References|[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 [[#References|[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|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 [[#References|[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 [[#References|[8]]]. Finite linear reflection groups over fields of non-zero characteristic have been found [[#References|[9]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H.S.M. Coxeter, "On complexes with transitive groups of automorphisms" ''Ann. of Math.'' , '''35''' (1934) pp. 588–621</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> H.S.M. Coxeter, W.O.J. Moser, "Generators and relations for discrete groups" , Springer (1984) {{MR|0609520}} {{MR|0562913}} {{MR|0349820}} {{MR|0174618}} {{MR|0088489}} {{ZBL|0487.20023}} {{ZBL|0422.20001}} {{ZBL|0239.20040}} {{ZBL|0133.28002}} {{ZBL|0077.02801}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J. Tits, "Groupes simples et géométries associées" , ''Proc. Internat. Congress Mathematicians 1962, Dursholm'' , Mittag-Leffler Inst. (1963) pp. 197–221 {{MR|0175903}} {{ZBL|0131.26502}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> N. Bourbaki, "Groupes et algèbres de Lie" , ''Eléments de mathématiques'' , Hermann (1968) pp. Chapts. 4–6 {{MR|0240238}} {{ZBL|0186.33001}} </TD></TR><TR><TD valign="top">[5a]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[5b]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> 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)</TD></TR><TR><TD valign="top">[7a]</TD> <TD valign="top"> 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 {{MR|0207853}} {{ZBL|0166.16303}} </TD></TR><TR><TD valign="top">[7b]</TD> <TD valign="top"> 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 {{MR|295193}} {{ZBL|0252.20054}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> G.C. Shephard, J.A. Todd, "Finite unitary reflection groups" ''Canad. J. Math.'' , '''6''' (1954) pp. 274–304 {{MR|0059914}} {{ZBL|0055.14305}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> 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 {{MR|0603578}} {{MR|0554927}} {{ZBL|}} </TD></TR></table>

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A.M. Cohen, "Finite quaternionic reflection groups" ''J. of Algebra'' , '''64''' (1980) pp. 293–324 {{MR|0579063}} {{ZBL|0433.20035}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> 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 {{MR|0608141}} {{ZBL|0443.51009}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> 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 {{MR|0608141}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> 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 {{MR|0608141}} {{ZBL|0471.51015}} </TD></TR></table>
[[Category:Group theory and generalizations]]