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In this section, we describe the classification of finite complex groups, and
functions for constructing these groups.
At present there is no satisfactory theory of root systems for
complex reflection groups comparable to the theory for finite
Coxeter groups. However, if we choose a representative ri for
each conjugacy class of reflections in the reflection group G
and if we choose a root αi of ri, then the union of the
orbits of the αi form a suitable set on which G acts as
a group of permutations. For any set {r1, r2, ..., rn} of
reflections that generate G, every reflection in G is
conjugate to a power of some ri.
Even though there is no satisfactory notion of "set of
fundamental roots", a reflection group can nevertheless be
described by specifying the set of roots corresponding to a set of
reflection generators together with a root of unity attached to
each root. Moreover, the inner products between the roots can be
described by means of a diagram similar to the Dynkin diagram of a
Coxeter group. This notation was suggested by Coxeter and used by
Cohen to classification the groups [Coh76]. (There
is a different type of diagram used by Brou'e, Malle and others.)
Cohen's naming scheme for the diagrams extends
the standard notation An, Bn, ..., H3,
H4 used for Coxeter groups.
An alternative numbering system for the primitive groups
is due to Shephard and Todd [ST54].
The ordering of the fundamental root vectors is given in the
following diagrams. A pair of nodes not joined by an edge
corresponds to a matrix entry of 0. A single bond corresponds
to 1 and all other bonds are labelled by the matrix entry
(reading from left to right, from lower numbered node to higher).
Thus an unlabelled edge between nodes of reflections of order 2
corresponds to an inner product of -1/2.
In the associated diagram (given below) there is a node for each
root. The (i, i)-th entry of the root system matrix is
αi and if this is -1, the node is shown as a circle,
otherwise it is represented by αi itself.
This construction includes all finite irreducible Coxeter
groups.
hrule
beginschema{An}
1 2 3 n
o---o---o- ... -o
endschema
beginschema{Bn = Cn}
1 2 3 n
o===o---o- ... -o
sqrt2
endschema
beginschema{Dn}
1 o
3 4 n
o---o- ... -o
/
2 o
endschema
beginschema{E6}
2 3 4 5 6
o---o---o---o---o
|
1 o
endschema
beginschema{E7}
2 3 4 5 6 7
o---o---o---o---o---o
|
1 o
endschema
beginschema{E8}
2 3 4 5 6 7 8
o---o---o---o---o---o---o
|
1 o
endschema
beginschema{F4}
1 2 3 4
o---o===o---o
sqrt2
endschema
beginschema{G2}
1 2
o===o
sqrt3
endschema
beginschema{H3}
1 2 3
o===o---o τ2 = τ+ 1
τ
endschema
beginschema{H4}
1 2 3 4
o===o---o---o
τ
endschema
beginschema{J3(4)}
2
o
/ c2 = c - 2
1 o===o 3
-c
endschema
beginschema{J3(5)}
2
o
/ ω2 + ω+ 1 = 0
1 o===o 3
ωτ
endschema
beginschema{K4}
3
o
/ \
o---o===o W(K4) = G(3, 3, 4)
1 2 ω4
endschema
beginschema{K5}
3
o
/ \
o---o===o---o
1 2 ω4 5
endschema
beginschema{K6}
3
o
/ \
o---o===o---o---o
1 2 ω4 5 6
endschema
beginschema{L3}
1 2 3
ω=== ω=== ω -ω2 ω2
endschema
beginschema{L4}
1 2 3 4
ω=== ω=== ω=== ω -ω2 ω2 -ω2
endschema
beginschema{M3}
1 2 3
o === ω=== ω sqrt2 -ω2
endschema
beginschema{N4}
2
o
/ 3 4
1 o===o---o
i - 1
endschema
beginschema{O4}
3
2 o --- o --- o 4 |W(O4) : W(N4)| = 6
// \ /
/ scriptstyle(i - 1)\
o === o
1 scriptstyle(i + 1) 5
endschema
hrule
Let B be the direct product of n copies of the cyclic group
Cm of order m and represent the elements of B by
diagonal matrices diag(θ1, θ2, ..., θn). The
elements of the symmetric group Sym(n) can be represented by
n x n permutation matrices and in this guise it acts on the
group B; the resulting semidirect product is also known as the
emph{wreath product} Cm wreath Sym(n).
For each divisor p of m define
A(m, p, n) := { diag(θ1, θ2, ..., θn)∈B | (θ1θ2 ... θn)m/p = 1 }.
It is immediately clear that A(m, p, n) is a subgroup of index p in
B that is invariant under the action of Sym(n). The semidirect
product of A(m, p, n) by the symmetric group Sym(n) is the
group G(m, p, n). Shephard and Todd proved that every irreducible
imprimitive complex reflection subgroup of GL(n, C) is conjugate
to G(m, p, n) for some m and p.
This function returns the Shephard-Todd group G(m, p, n) ⊂GL(n, F), where
p divides m. The field F of definition is returned as a second
value. In general, G(m, p, n) is irreducible but if m = p = 1,
the function returns Sym(n) in its natural permutation
representation, which is not irreducible.
> ImprimitiveReflectionGroup(6, 3, 3);
MatrixGroup(3, Cyclotomic Field of order 6 and degree 2)
Generators:
[0 1 0]
[1 0 0]
[0 0 1]
[1 0 0]
[0 0 1]
[0 1 0]
[ 0 z 0]
[-z + 1 0 0]
[ 0 0 1]
[-1 0 0]
[ 0 1 0]
[ 0 0 1]
Cyclotomic Field of order 6 and degree 2
Given a string X defining the type and an integer n specifying
the rank, this function returns the matrix of (modified) inner
products of roots corresponding to generating reflections of a
reflection group of type X and rank n. The rank is the
dimension of the space on which the group acts; it is not always
the number of generators.
The function constructs root system matrices for the types A,
B, C, D, E, F, G, H, J3(4), J3(5),
K, L, M, N, and O. (The function accepts the
abbreviations J4 and J5 for the types J3(4) and J3(5).)
The (i, j)-th entry of the root system matrix for the roots
a1, a2, ..., ak is δij + (αj -
1)(ai, aj), where αj is an m-th root of unity, for
some m. The effect of the reflection rj with root aj on
the root ai is given by
ai rj = ai + (αj - 1)(ai, aj) aj.
Given a root system matrix M the function returns the corresponding
complex reflection group. In addition, the field of definition is
returned.
We assume that M corresponds to a positive semidefinite
inner product and that the first n - 1 columns of M - I
are linearly independent.
The reflection generators are created as matrices with respect to
the standard basis of the reflection representation. The matrices
represent the action on row vectors.
The k-th reflection matrix is obtained from the identity
matrix by replacing its k-th column with the k-th column
of the root system matrix.
If the determinant of M - I is 0, the matrices can be
thought of as arising from transformations constructed as
just described, but acting on the quotient of the space
modulo the null space of M - I.
> M := RootSystemMatrix("O", 4);
> M;
[ -1 1 i - 1 0 i + 1]
[ 1 -1 1 0 0]
[-i - 1 1 -1 1 i - 1]
[ 0 0 1 -1 1]
[-i + 1 0 -i - 1 1 -1]
> #ReflectionGroup(M);
46080
Given a string X defining the type and an integer n specifying
the rank, this function returns the corresponding complex reflection group.
This function returns the primitive reflection group Gn in
GL(m, C), using the Shephard-Todd numbering. The field of
definition is returned as well.
The groups available via this function include all the finite
primitive irreducible complex reflection groups other than the
symmetric groups Sym(n) for n≥5. The groups are listed
below.
There are nineteen 2-dimensional primitive complex
reflection groups:
Tetrahedral family: G4, ..., G7
Octahedral family: G8, ..., G15
Icosahedral family: G16, ..., G22
There are five 3-dimensional complex reflection groups:
G23: W(H3) = Z2 x PSL(2, 5), order 120
G24: W(J3(4)) = Z2 x PSL(2, 7), order 336
G25: W(L3) = W(P3) = 31 + 2.SL(2, 3), order 648; Hessian group
G26: W(M3) = Z2 x 31 + 2.SL(2, 3), order 1296; Hessian group
G27: W(J3(5)) = Z2 x (Z3.Alt(6)), order 2160 (non-split)
There are five 4-dimensional complex reflection groups in addition to
Sym(5):
G28: W(F4) = (SL(2, 3) SL(2, 3)).(Z2 x Z2), order 1152
G29: W(N4) = (Z4 21 + 4).Sym(5), order 7680 (splits)
G30: W(H4) = (SL(2, 5) SL(2, 5)).Z2, order 14400
G31: W(O4) = (Z4 21 + 4).Sp(4, 2), order 46080 (non-split) 5
generators
G32: W(L4) = Z3 x Sp(4, 3), order 155520 = 27 x 35 x 5
There is one 5-dimensional complex reflection group in addition to Sym(6):
G33: W(K5) = Z2 x Ω(5, 3) = Z2 x PSp(4, 3)
= Z2 x PSU(4, 2), order 51840 = 27 x 34 x 5.
There are two 6-dimensional complex reflection groups in addition to Sym(7):
G34: W(K6) = Z3.SO^ - (6, 3), order 39191040 =
29 x 37 x 5 x 7
G35: W(E6) = SO(5, 3) = O^ - (6, 2) = PSp(4, 3).Z2 = PSU(4, 2).Z2,
order 51840 = 27 x 34 x 5
There is one 7-dimensional complex reflection group in addition to
Sym(8):
G36: W(E7) = Z2 x Sp(6, 2), order 2903040 =
210 x 34 x 5 x 7.
There is one 8-dimensional complex reflection group in addition to
Sym(9):
G37: W(E8) = Z2.O^ + (8, 2), order 696729600 =
214 x 35 x 52 x 7
> W := ComplexReflectionGroup("O", 4);
> G := ShephardTodd(31);
> W eq G;
true
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