Central in CHEVIE is the notion of reflection groups, which are by definition groups generated by a finite number of complex reflections.
Let V be a vector space over a subfield K of the complex numbers (in
GAP this usually means the Rationals
, a cyclotomic field CF(n)
or
the Cyclotomics
). A complex reflection (that we will in the following
just call a reflection to abbreviate) is an element s∈ GL(V) whose
fixed point set is an hyperplane and which is of finite order --- here we
use the slightly non-standard terminology complex reflection since the
usual term pseudo-reflection is usually not meant to imply finite order;
and we abuse the term reflection which is usually only employed for
pseudo-reflections of order 2. Thus a reflection has a unique eigenvalue
not equal to 1. If K is the field of real numbers, then a complex
reflection is necessarily of order 2 and the non-trivial eigenvalue equal
to -1.
Since a group containing s contains its powers, a group generated by
reflections is always generated by reflections s with eigenvalue E(d)
where d is the order of s; we may in addition assume that s is not a
power of another reflection with eigenvalue E(d')
with d'>d
. Such a
reflection is called distinguished; we take it as the canonical generator
of the cyclic subgroup it generates. The generators of the reflection
groups we construct in CHEVIE are always distinguished reflections.
Reflection groups in CHEVIE are groups W with the following fields (in
the group record) defined
.nbGeneratingReflections
:
.reflections
:W.reflections{[1..W.nbGeneratingReflections]}
.
.OrdersGeneratingReflections
:W.nbGeneratingReflections
) such that its i-th element is
the order of W.reflections[i]
. By the above conventions
W.reflections[i]
thus has E(W.OrdersGeneratingReflections[i])
as
its nontrivial eigenvalue.
Note that W does not need to be a matrix group. The meaning of the
above fields is just that W has a representation (called the
reflection representation of W) where the elements W.reflections
operate as reflections. It is much more efficient to compute with
permutation groups which have such fields defined, than with matrix
groups, when possible. Information sufficient to determine a particular
reflection representation is stored for such groups (see CartanMat
).
Also note that, although .reflections
is usually just initialized to the
generating reflections, it is usually augmented by adding other reflections
to it as computations require. For instance, when W is finite, the set of
all reflections in W is finite (they are just the elements of the
conjugacy classes of the generating reflections and their powers), and all
the distinguished reflections in W are added to .reflections
when
required, for instance when calling Reflections(W)
which returns the
list of all (distinguished) reflections. Note that when W is finite, the
distinguished reflections are in bijection with the reflecting hyperplanes.
There are very few functions in CHEVIE which deal with reflections groups in full generality. Usually the groups one wants to deal with is in a more restricted class (Coxeter groups, finite reflection groups) which are described in the following chapters.
Reflection( root, coroot)
If s is a (complex) reflection acting on the vector space V, with fixed hyperplane H (called the reflecting hyperplane of s) and with eigenvector r (called a root of s) for its non-trivial value ζ, then as a linear map s is given by x→ x-r∨(x)r, where r∨ (called the coroot of s associated to r) is a linear form of kernel H such that r∨(r)=1-ζ.
Thus a first way of specifying a reflection is by giving a root and a
coroot. The root and coroot are uniquely determined by the reflection up to
multiplication of the root by a scalar and of the coroot by the inverse
scalar. The function Reflection
in this form gives the matrix of the
corresponding reflection, where the root and the coroot are vectors
given in the standard bases of V and V∨ (thus in GAP r∨(r)
is obtained as root*coroot
).
gap> r:=Reflection([1,0,0],[2,-1,0]); [ [ -1, 0, 0 ], [ 1, 1, 0 ], [ 0, 0, 1 ] ] gap> r=CoxeterGroup("A",3).matgens[1]; true gap> [1,0,0]*r; [ -1, 0, 0 ]
As we see in the last line, in GAP the matrices operate from the right on the vector space.
Reflection( root [, eigenvalue] )
We may give slightly less information if we assume that the
standard hermitian scalar product (x,y) on V (given in GAP by
x*ComplexConjugate(y)
) is s-invariant. Then, identifying V and
V∨ via this scalar product, s is given by the formula
x→ x-(1-ζ)(x,r)/(r,r)r |
where ζ is the non-trivial eigenvalue of s, so s is specified
by just root and eigenvalue. When eigenvalue is omitted it is
assumed to be equal to -1. The function Reflection
in this form gives
the matrix of the reflection given such data.
gap> Reflection([0,0,1],E(3)); [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, E(3) ] ] gap> last=ComplexReflectionGroup(25).matgens[1]; true
Reflection( W, i )
This form returns the i-th reflection in the finite reflection group W
(this works only for groups represented as finite permutation groups). The
point of using this function rather than W.reflections[i]
is that this
entry might not yet be bound (not yet have been computed). The above
function then computes it and returns it.
gap> Reflection(CoxeterGroup("A",3),6); ( 1,11)( 3,10)( 4, 9)( 5, 7)( 6,12)
This function requires the package "chevie" (see RequirePackage).
AsReflection( s [,r])
Here s is a square matrix with entries cyclotomic numbers, and if
given r is a vector of the same length as s of cyclotomic numbers.
The function determines if s is the matrix of a reflection (resp. if
r is given if it is the matrix of a reflection of root r; the point
of giving r is to specify exactly the desired coroot, which otherwise
is determined only up to a scalar). The returned result is false
if
s is not a reflection (resp. not a reflection with root r), and
otherwise is a record with four fields :
.root
:
.coroot
:
.eigenvalue
:
.isOrthogonal
:true
if and only if s
is orthogonal with respect to the usual scalar product (then the root and
eigenvalue are sufficient to determine s)
gap> AsReflection([[-1,0,0],[1,1,0],[0,0,1]]); rec( root := [ 2, 0, 0 ], coroot := [ 1, -1/2, 0 ], eigenvalue := -1, isOrthogonal := false ) gap> AsReflection([[-1,0,0],[1,1,0],[0,0,1]],[1,0,0]); rec( root := [ 1, 0, 0 ], coroot := [ 2, -1, 0 ], eigenvalue := -1, isOrthogonal := false )
This function requires the package "chevie" (see RequirePackage).
CartanMat( W )
Let s1,...,sn be a set of reflections which generate a group W
with associated root vectors ri and coroots r∨i. Then the
matrix C
of the ri∨(rj) is called the Cartan matrix of the
set of reflections. It is uniquely determined by the reflections up to
conjugating by diagonal matrices, and up to such conjugation it is a
conjugation invariant of the reflection representation of W. It
actually completely determines this representation if the ri are
linearly independent (which is e.g. the case if C
is invertible),
since in the ri basis the matrix for the si differs from the
identity only on the i-th line, where the corresponding line of C
has been subtracted.
gap> W:=CoxeterGroup("A",3);; gap> CartanMat(W); [ [ 2, -1, 0 ], [ -1, 2, -1 ], [ 0, -1, 2 ] ]
CartanMat( W, l )
Returns the Cartan matrix of the roots of W specified by the list of integers l (for a finite reflection group represented as a group of permutation of root vectors, these integers are indices in the list of roots of the parent reflection group).
CartanMat( type )
This form returns the Cartan matrix of some standard reflection representations for Coxeter groups, taking a symbolic description of the Coxeter group given by the arguments. See CartanMat for Dynkin types
This function requires the package "chevie" (see RequirePackage).
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GAP 3.4.4