Central in CHEVIE is the notion of reflection groups.
Let V be a vector space over a subfield K of the complex numbers; in
GAP3 this usually means the Rationals, the Cyclotomics, or a
subfield. A complex reflection is an element s∈ GL(V) of finite order
whose fixed point set is an hyperplane (we will in the following just call
it a reflection to abbreviate; in some literature the term reflection is
only employed when the order is 2 and the more general case is called a
pseudo-reflection). Thus a reflection has a unique eigenvalue not equal
to 1. If K is a subfield of the real numbers, we get a real reflection
which is necessarily of order 2 and the non-trivial eigenvalue is equal to
-1.
A reflection group W is a group generated by a finite number of complex reflections.
Since when W contains a reflection s it contains its powers, W 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 reflection groups in CHEVIE are
always distinguished reflections. In a real reflection group all
reflections are distinguished.
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)
A (complex) reflection s acting on the vector space V (over some subfield of the complex numbers), is a linear map of finite order whose fixed points are an hyperplane H (called the reflecting hyperplane of s); an eigenvector r for the non-trivial eigenvalue ζ (a root of unity) is called a root of s. We may chose a linear form r∨ (called a coroot of s) defining H such that r∨(r)=1-ζ and then as a linear map s is given by x→ x-r∨(x)r.
A first way of specifying a reflection is by giving a root and a coroot,
which 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 gives the matrix of the corresponding reflection in the
standard basis of V, where the root and the coroot are vectors given
in the standard bases of V and V∨ (thus in GAP3 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 GAP3 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 GAP3 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 |
so s is specified by just root and eigenvalue. When eigenvalue is
omitted it is assumed to be equal to -1. The function Reflection gives
again the matrix of the reflection.
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 reflection with respect to the i-th root in the
finite reflection group W (this works only for groups represented as
permutation groups of the roots, see Finite Reflection Groups). Note that
one would not get the same result with W.reflections[i] since this entry
might not yet be bound (not yet have been computed), and also it is not
guaranteed apart from the generating roots (and the positive roots of Weyl
groups) that the i-th reflection corresponds to the i-th root, since
two roots corresponding to the same reflection may have been obtained
before all the reflections have been obtained.
gap> Reflection(CoxeterGroup("A",3),6);
( 1,11)( 3,10)( 4, 9)( 5, 7)( 6,12)
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 root and coroot, which otherwise are
determined only up to a scalar and its inverse). 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 )
CartanMat( W )
Let s1,...,sn be a list of reflections with associated root vectors ri and coroots ri∨. Then the matrix Ci,j of the ri∨(rj) is called the Cartan matrix of the list of reflections. It is uniquely determined by the reflections up to conjugating by diagonal matrices.
If s1,...,sn are the generators of a reflection group W, the
matrix C up to conjugation by diagonal matrices is an 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
Rank( W )
Let W be a reflection group in the vector space V. This function returns the dimension of V, if known. If reflections of W are generated by a root and a coroot, it is the length of the root as a list. If W is a matrix group it is the dimension of the matrices.
gap> W:=ReflectionSubgroup(CoxeterGroup("A",3),[1,3]);
ReflectionSubgroup(CoxeterGroup("A",3), [ 1, 3 ])
gap> Rank(W);
3
SemisimpleRank( W )
Let W be a reflection group in the vector space V. This function
returns the dimension of the subspace V' of V where W effectively
acts, which is the subspace generated by the roots of the reflections of
W. The space V' is W-stable and has a W-stable complement on which
W acts trivially. The SemisimpleRank is independent of the reflection
representation. W is called essential if V'=V.
gap> W:=ReflectionSubgroup(CoxeterGroup("A",3),[1,3]);
ReflectionSubgroup(CoxeterGroup("A",3), [ 1, 3 ])
gap> SemisimpleRank(W);
2
gap3-jm