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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r0805201.png" />, a sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r0805202.png" /> or a hyperbolic (Lobachevskii) space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r0805203.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r0805204.png" />-dimensional case, and, in connection with problems of the theory of functions of a complex variable, the hyperbolic plane; regular partitions of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r0805205.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r0805206.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r0805207.png" /> were enumerated (those in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r0805208.png" /> can be considered as a particular case of reflection groups in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r0805209.png" />). 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052010.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052011.png" /> with a bounded fundamental polyhedron (see [[Discrete group of transformations|Discrete group of transformations]]).
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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.==
 
==Basic results of the theory of reflection groups.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052013.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052014.png" />. Every reflection group in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052015.png" /> is generated by reflections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052016.png" /> in hyperplanes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052018.png" />, which bound a fundamental polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052019.png" />. Relative to this system of generators, the reflection group is a [[Coxeter group|Coxeter group]] with defining relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052020.png" />, where the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052021.png" /> are obtained as follows: If the faces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052023.png" /> are adjacent and the angle between them is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052024.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052025.png" />; if they are not adjacent, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052026.png" /> (and the hyperplanes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052028.png" /> do not intersect). On the other hand, any convex polyhedron in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052029.png" /> all dihedral angles of which are submultiples of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052030.png" /> is the fundamental polyhedron of the group generated by the reflections in its bounding hyperplanes.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052031.png" /> (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:
+
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
 
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
Line 12: Line 54:
 
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
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052032.png" /> with a bounded fundamental polyhedron (as a group of motions) is the direct product of groups of type (II).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052033.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052034.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052035.png" /> is defined by a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052036.png" />-gon with angles
+
A reflection group in $  \Lambda  ^ {2} $
 +
is defined by a $  k $-
 +
gon with angles
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052037.png" /></td> </tr></table>
+
$$
  
(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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052038.png" /> parameters.
+
\frac \pi {n _ {1} }
 +
\dots
 +
\frac \pi {n _ {k} }
 +
,\  \textrm{ where }
 +
 +
\frac{1}{n _ {1} }
 +
+ \dots +
 +
\frac{1}{n _ {k} }
 +
  < k - 2
 +
$$
  
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052039.png" />, the fundamental polyhedron of a reflection group in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052040.png" /> is uniquely defined by its combinatorial structure and its dihedral angles. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052041.png" />, an exhaustive description of these polyhedra has been obtained
+
(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.
  
and, thereby, of reflection groups as well. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052042.png" />, only examples and a few general methods of construction for reflection groups in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052043.png" /> are known (see [[#References|[6]]], ). It is not known (1990) whether there exists a reflection group in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052044.png" /> with a bounded fundamental polyhedron when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052045.png" /> and with a fundamental polyhedron of finite volume when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052046.png" />.
+
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]]]).
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052047.png" /> be the degrees of the generators of the invariants of a finite reflection group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052048.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052049.png" /> is the dimension of the space); the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052050.png" /> are called the exponents of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052051.png" />. The formula
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052052.png" /></td> </tr></table>
+
$$
 +
( 1+ m _ {1} t) \dots ( 1+ m _ {n} t)  = \
 +
c _ {0} + c _ {1} t + \dots + c _ {n} t  ^ {n}
 +
$$
  
holds, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052053.png" /> is the number of elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052054.png" /> for which the space of fixed points has dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052055.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052056.png" /> is equal to the number of reflections in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052057.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052058.png" /> is equal to the order of the group. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052059.png" /> is irreducible, then the eigenvalues of its Killing–Coxeter element (see [[Coxeter group|Coxeter group]]) are equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052060.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052061.png" /> is the Coxeter number:
+
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} + \dots + m _ {n} $
 +
is equal to the number of reflections in $  G $;  
 +
$  ( m _ {1} + 1) \dots ( 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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052062.png" /></td> </tr></table>
+
$$
 +
= \max \{ m _ {k} \} + 1.
 +
$$
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052063.png" />. 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]]].
+
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====
 
====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>
 
<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====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080520/r08052064.png" />, see [[#References|[a2]]]–[[#References|[a4]]].
+
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====
 
====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>
 
<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>

Revision as of 08:10, 6 June 2020


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

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

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} + \dots + m _ {n} $ is equal to the number of reflections in $ G $; $ ( m _ {1} + 1) \dots ( 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
How to Cite This Entry:
Reflection group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Reflection_group&oldid=48469
This article was adapted from an original article by E.B. Vinberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article