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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 r_i for
each conjugacy class of reflections in the reflection group G
and if we choose a root alpha_i of r_i, then the union of the
orbits of the alpha_i form a suitable set on which G acts as
a group of permutations. For any set {r_1, r_2, ..., r_n} of
reflections that generate G, every reflection in G is
conjugate to a power of some r_i.
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 in his thesis on the classification of the groups. (There
is a different type of diagram used by Brou'e, Malle and others.)
In this package we use Cohen's naming scheme for the diagrams.
This extends the standard notation A_n, B_n, ..., H_3,
H_4 used for Coxeter groups.
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
alpha_i and if this is -1, the node is shown as a circle,
otherwise it is represented by alpha_i itself.
This construction includes all finite irreducible Coxeter
groups.
beginschema{A_n}
1 2 3 n
o---o---o- ... -o
endschema
beginschema{B_n = C_n}
1 2 3 n
o===o---o- ... -o
sqrt2
endschema
beginschema{D_n}
1 o
3 4 n
o---o- ... -o
/
2 o
endschema
beginschema{E_6}
2 3 4 5 6
o---o---o---o---o
|
1 o
endschema
beginschema{E_7}
2 3 4 5 6 7
o---o---o---o---o---o
|
1 o
endschema
beginschema{E_8}
2 3 4 5 6 7 8
o---o---o---o---o---o---o
|
1 o
endschema
beginschema{F_4}
1 2 3 4
o---o===o---o
sqrt2
endschema
beginschema{G_2}
1 2
o===o
sqrt3
endschema
beginschema{H_3}
1 2 3
o===o---o tau^2 = tau + 1
tau
endschema
beginschema{H_4}
1 2 3 4
o===o---o---o
tau
endschema
beginschema{J_3(4)}
2
o
/ c^2 = c - 2
1 o===o 3
-c
endschema
beginschema{J_3(5)}
2
o
/ omega^2 + omega+ 1 = 0
1 o===o 3
omegatau
endschema
beginschema{K_4}
3
o
/ \
o---o===o W(K_4) = G(3, 3, 4)
1 2 omega4
endschema
beginschema{K_5}
3
o
/ \
o---o===o---o
1 2 omega4 5
endschema
beginschema{K_6}
3
o
/ \
o---o===o---o---o
1 2 omega4 5 6
endschema
beginschema{L_3}
1 2 3
omega=== omega=== omega -omega^2 omega^2
endschema
beginschema{L_4}
1 2 3 4
omega=== omega=== omega=== omega -omega^2 omega^2 -omega^2
endschema
beginschema{M_3}
1 2 3
o === omega=== omega sqrt2 -omega^2
endschema
beginschema{N_4}
2
o
/ 3 4
1 o===o---o
i - 1
endschema
beginschema{O_4}
3
2 o --- o --- o 4 |W(O_4) : W(N_4)| = 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
C_m of order m and represent the elements of B by
diagonal matrices diag(theta_1, theta_2, ..., theta_n). 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} C_m wreath Sym(n).
For each divisor p of m define
A(m, p, n) := { (theta_1, theta_2, ..., theta_n) in B | (theta_1theta_2.stheta_n)^(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 unitary 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), where
p divides m. The field 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]
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, J_3(4), J_3(5),
K, L, M, N, and O. (The function accepts the
abbreviations J4 and J5 for the types J_3(4) and J_3(5).)
The (i, j)-th entry of the root system matrix for the roots
a_1, a_2, ..., a_k is delta_(ij) + (alpha_j -
1)(a_i, a_j), where alpha_j is an m-th root of unity, for
some m. The effect of the reflection r_j with root a_j on
the root a_i is given by
a_i r_j = a_i + (alpha_j - 1)(a_i, a_j) a_j.
Given a root system matrix M the function returns the corresponding
unitary 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 G_n 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 unitary reflection groups other than the
symmetric groups Sym(n) for n >= 5. The groups are listed
below.
There are nineteen 2-dimensional primitive unitary
reflection groups:
Tetrahedral family: G_4, ..., G_7
Octahedral family: G_8, ..., G_(15)
Icosahedral family: G_(16), ..., G_(22)
There are five 3-dimensional unitary reflection groups:
G_(23): W(H_3) = Z_2 x PSL(2, 5), order 120
G_(24): W(J_3(4)) = Z_2 x PSL(2, 7), order 336
G_(25): W(L_3) = W(P_3) = 3^(1 + 2).SL(2, 3), order 648; Hessian group
G_(26): W(M_3) = Z_2 x 3^(1 + 2).SL(2, 3), order 1296; Hessian group
G_(27): W(J_3(5)) = Z_2 x (Z_3.Alt(6)), order 2160 (non-split)
There are five 4-dimensional unitary reflection groups in addition to
Sym(5):
G_(28): W(F_4) = (SL(2, 3)
SL(2, 3)).(Z_2 x Z_2), order 1152
G_(29): W(N_4) = (Z_4 2^(1 + 4)).Sym(5), order 7680 (splits)
G_(30): W(H_4) = (SL(2, 5) SL(2, 5)).Z_2, order 14400
G_(31): W(O_4) = (Z_4 2^(1 + 4)).Sp(4, 2), order 46080 (non-split) 5
generators
G_(32): W(L_4) = Z_3 x Sp(4, 3), order 155520 = 2^7 x 3^5 x 5
There is one 5-dimensional unitary reflection group in addition to Sym(6):
G_(33): W(K_5) = Z_2 x Omega(5, 3) = Z_2 x PSp(4, 3)
= Z_2 x PSU(4, 2), order 51840 = 2^7 x 3^4 x 5.
There are two 6-dimensional unitary reflection groups in addition to Sym(7):
G_(34): W(K_6) = Z_3.SO^ - (6, 3), order 39191040 =
2^9 x 3^7 x 5 x 7
G_(35): W(E_6) = SO(5, 3) = O^ - (6, 2) = PSp(4, 3).Z_2 = PSU(4, 2).Z_2,
order 51840 = 2^7 x 3^4 x 5
There is one 7-dimensional unitary reflection group in addition to
Sym(8):
G_(36): W(E_7) = Z_2 x Sp(6, 2), order 2903040 =
2^(10) x 3^4 x 5 x 7.
There is one 8-dimensional unitary reflection group in addition to
Sym(9):
G_(37): W(E_8) = Z_2.O^ + (8, 2), order 696729600 =
2^(14) x 3^5 x 5^2 x 7
> W := ComplexReflectionGroup( "O", 4 );
> G := ShephardTodd( 31 );
> W eq G;
true
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