In
mathematics, a complex reflection group is a
finite group acting on a
finite-dimensional complex vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
that is generated by complex reflections: non-trivial elements that fix a complex
hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its '' ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hype ...
pointwise.
Complex reflection groups arise in the study of the
invariant theory of
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variable ...
s. In the mid-20th century, they were completely classified in work of Shephard and Todd. Special cases include the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
of permutations, the
dihedral groups, and more generally all finite real reflection groups (the
Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...
s or
Weyl groups, including the symmetry groups of
regular polyhedra).
Definition
A (complex) reflection ''r'' (sometimes also called ''pseudo reflection'' or ''unitary reflection'') of a finite-dimensional complex vector space ''V'' is an element
of finite order that fixes a complex hyperplane pointwise, that is, the ''fixed-space''
has codimension 1.
A (finite) complex reflection group
is a finite subgroup of
that is generated by reflections.
Properties
Any real reflection group becomes a complex reflection group if we extend the scalars from
R to C. In particular, all finite
Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...
s or
Weyl groups give examples of complex reflection groups.
A complex reflection group ''W'' is irreducible if the only ''W''-invariant proper subspace of the corresponding vector space is the origin. In this case, the dimension of the vector space is called the rank of ''W''.
The Coxeter number
of an irreducible complex reflection group ''W'' of rank
is defined as
where
denotes the set of reflections and
denotes the set of reflecting hyperplanes.
In the case of real reflection groups, this definition reduces to the usual definition of the Coxeter number for finite Coxeter systems.
Classification
Any complex reflection group is a product of irreducible complex reflection groups, acting on the sum of the corresponding vector spaces. So it is sufficient to classify the irreducible complex reflection groups.
The irreducible complex reflection groups were classified by . They proved that every irreducible belonged to an infinite family ''G''(''m'', ''p'', ''n'') depending on 3 positive integer parameters (with ''p'' dividing ''m'') or was one of 34 exceptional cases, which they numbered from 4 to 37. The group ''G''(''m'', 1, ''n'') is the
generalized symmetric group; equivalently, it is the
wreath product of the symmetric group Sym(''n'') by a cyclic group of order ''m''. As a matrix group, its elements may be realized as
monomial matrices whose nonzero elements are ''m''th
roots of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important i ...
.
The group ''G''(''m'', ''p'', ''n'') is an
index-''p'' subgroup of ''G''(''m'', 1, ''n''). ''G''(''m'', ''p'', ''n'') is of order ''m''
''n''''n''!/''p''. As matrices, it may be realized as the subset in which the product of the nonzero entries is an (''m''/''p'')th root of unity (rather than just an ''m''th root). Algebraically, ''G''(''m'', ''p'', ''n'') is a
semidirect product of an abelian group of order ''m''
''n''/''p'' by the symmetric group Sym(''n''); the elements of the abelian group are of the form (''θ''
''a''1, ''θ''
''a''2, ..., ''θ''
''a''''n''), where ''θ'' is a
primitive
Primitive may refer to:
Mathematics
* Primitive element (field theory)
* Primitive element (finite field)
* Primitive cell (crystallography)
* Primitive notion, axiomatic systems
* Primitive polynomial (disambiguation), one of two concepts
* Pr ...
''m''th root of unity and Σ''a''
''i'' ≡ 0 mod ''p'', and Sym(''n'') acts by permutations of the coordinates.
The group ''G''(''m'',''p'',''n'') acts irreducibly on C
''n'' except in the cases ''m'' = 1, ''n'' > 1 (the symmetric group) and ''G''(2, 2, 2) (the
Klein four-group). In these cases, C
''n'' splits as a sum of irreducible representations of dimensions 1 and ''n'' − 1.
Special cases of ''G''(''m'', ''p'', ''n'')
Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...
s
When ''m'' = 2, the representation described in the previous section consists of matrices with real entries, and hence in these cases ''G''(''m'',''p'',''n'') is a finite Coxeter group. In particular:
* ''G''(1, 1, ''n'') has type ''A''
''n''−1 =
,3,...,3,3= ...; the symmetric group of order ''n''!
* ''G''(2, 1, ''n'') has type ''B''
''n'' =
,3,...,3,4= ...; the
hyperoctahedral group of order 2
''n''''n''!
* ''G''(2, 2, ''n'') has type ''D''
''n'' =
1,1">,3,...,31,1= ..., order 2
''n''''n''!/2.
In addition, when ''m'' = ''p'' and ''n'' = 2, the group ''G''(''p'', ''p'', 2) is the
dihedral group of order 2''p''; as a Coxeter group, type ''I''
''2''(''p'') =
'p''= (and the Weyl group ''G''
2 when ''p'' = 6).
Other special cases and coincidences
The only cases when two groups ''G''(''m'', ''p'', ''n'') are isomorphic as complex reflection groups are that ''G''(''ma'', ''pa'', 1) is isomorphic to ''G''(''mb'', ''pb'', 1) for any positive integers ''a'', ''b'' (and both are isomorphic to the
cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bi ...
of order ''m''/''p''). However, there are other cases when two such groups are isomorphic as abstract groups.
The groups ''G''(3, 3, 2) and ''G''(1, 1, 3) are isomorphic to the symmetric group Sym(3). The groups ''G''(2, 2, 3) and ''G''(1, 1, 4) are isomorphic to the symmetric group Sym(4). Both ''G''(2, 1, 2) and ''G''(4, 4, 2) are isomorphic to the
dihedral group of order 8. And the groups ''G''(2''p'', ''p'', 1) are cyclic of order 2, as is ''G''(1, 1, 2).
List of irreducible complex reflection groups
There are a few duplicates in the first 3 lines of this list; see the previous section for details.
*ST is the Shephard–Todd number of the reflection group.
*Rank is the dimension of the complex vector space the group acts on.
*Structure describes the structure of the group. The symbol * stands for a
central product of two groups. For rank 2, the quotient by the (cyclic) center is the group of rotations of a tetrahedron, octahedron, or icosahedron (''T'' = Alt(4), ''O'' = Sym(4), ''I'' = Alt(5), of orders 12, 24, 60), as stated in the table. For the notation 2
1+4, see
extra special group.
*Order is the number of elements of the group.
*Reflections describes the number of reflections: 2
64
12 means that there are 6 reflections of order 2 and 12 of order 4.
*Degrees gives the degrees of the fundamental invariants of the ring of polynomial invariants. For example, the invariants of group number 4 form a polynomial ring with 2 generators of degrees 4 and 6.
For more information, including diagrams, presentations, and codegrees of complex reflection groups, see the tables in .
Degrees
Shephard and Todd proved that a finite group acting on a complex vector space is a complex reflection group if and only if its ring of invariants is a polynomial ring (
Chevalley–Shephard–Todd theorem). For
being the ''rank'' of the reflection group, the degrees
of the generators of the ring of invariants are called ''degrees of W'' and are listed in the column above headed "degrees". They also showed that many other invariants of the group are determined by the degrees as follows:
*The center of an irreducible reflection group is cyclic of order equal to the greatest common divisor of the degrees.
*The order of a complex reflection group is the product of its degrees.
*The number of reflections is the sum of the degrees minus the rank.
*An irreducible complex reflection group comes from a real reflection group if and only if it has an invariant of degree 2.
*The degrees ''d''
i satisfy the formula
Codegrees
For
being the ''rank'' of the reflection group, the codegrees
of W can be defined by
*For a real reflection group, the codegrees are the degrees minus 2.
*The number of reflection hyperplanes is the sum of the codegrees plus the rank.
Well-generated complex reflection groups
By definition, every complex reflection group is generated by its reflections. The set of reflections is not a minimal generating set, however, and every irreducible complex reflection groups of rank has a minimal generating set consisting of either or reflections. In the former case, the group is said to be ''well-generated''.
The property of being well-generated is equivalent to the condition
for all
. Thus, for example, one can read off from the classification that the group is well-generated if and only if ''p'' = 1 or ''m''.
For irreducible well-generated complex reflection groups, the ''
Coxeter number
In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter.
Definitions
Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there a ...
'' defined above equals the largest degree,
. A reducible complex reflection group is said to be well-generated if it is a product of irreducible well-generated complex reflection groups. Every finite real reflection group is well-generated.
Shephard groups
The well-generated complex reflection groups include a subset called the ''Shephard groups''. These groups are the symmetry groups of
regular complex polytopes. In particular, they include the symmetry groups of regular real polyhedra. The Shephard groups may be characterized as the complex reflection groups that admit a "Coxeter-like" presentation with a linear diagram. That is, a Shephard group has associated positive integers and such that there is a generating set satisfying the relations
:
for ,
:
if
,
and
:
where the products on both sides have terms, for .
This information is sometimes collected in the Coxeter-type symbol , as seen in the table above.
Among groups in the infinite family , the Shephard groups are those in which . There are also 18 exceptional Shephard groups, of which three are real.
Cartan matrices
An extended
Cartan matrix defines the unitary group. Shephard groups of rank ''n'' group have ''n'' generators.
Ordinary Cartan matrices have diagonal elements 2, while unitary reflections do not have this restriction.
[Unitary Reflection Groups, pp.91-93]
For example, the rank 1 group of order ''p'' (with symbols p[], ) is defined by the matrix
.
Given:
.
References
*
*
*
*Hiller, Howard ''Geometry of Coxeter groups.'' Research Notes in Mathematics, 54. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. iv+213 pp. *
*
*{{Citation , last1=Shephard , first1=G. C. , last2=Todd , first2=J. A. , title=Finite unitary reflection groups , url=https://books.google.com/books?id=Bi7EKLHppuYC , mr=0059914 , year=1954 , journal=Canadian Journal of Mathematics , issn=0008-414X , volume=6 , pages=274–304 , publisher=Canadian Mathematical Society , doi=10.4153/CJM-1954-028-3, s2cid=3342221
*
Coxeter, ''Finite Groups Generated by Unitary Reflections'', 1966, 4. ''The Graphical Notation'', Table of n-dimensional groups generated by n Unitary Reflections. pp. 422–423
External links
''MAGMA Computational Algebra System'' page
Reflection groups
Geometry
Group theory