Difference between revisions of "Reflection group"
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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 | + | {{TEX|auto}} |
+ | {{TEX|done}} | ||
+ | |||
+ | 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 | + | 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 | + | 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 | + | 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 | + | 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 | + | 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. | ||
− | and, thereby, of reflection groups as well. For | + | 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 | + | 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 | ||
− | + | $$ | |
+ | ( 1+ m _ {1} t) \dots ( 1+ m _ {n} t) = \ | ||
+ | c _ {0} + c _ {1} t + \dots + c _ {n} t ^ {n} | ||
+ | $$ | ||
− | holds, where | + | 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: | ||
− | + | $$ | |
+ | h = \max \{ m _ {k} \} + 1. | ||
+ | $$ | ||
− | 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 | + | 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 | + | 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 |
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[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 |
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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 |
Reflection group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Reflection_group&oldid=24123