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
Cyclotomics, or one of its
subfields. 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 abuse the
term reflection which is usually only employed when the order is 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. A reflection group is a group
generated by a finite number of complex reflections.
Since a group containing s contains its powers, a group generated by
reflections is always generated by reflections s with eigenvalue
where d is the order of s; we may in addition assume that s is not a
power of another reflection with eigenvalue
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 in CHEVIE are always distinguished reflections.
Reflection groups in CHEVIE are groups W with the following fields (in the group record) defined
W.nbGeneratingReflections) such that its i-th element is the order of
W.reflections[i]. By the above conventions
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
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
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
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 GAP3 r∨(r)
is obtained as
gap> r:=Reflection([1,0,0],[2,-1,0]); [ [ -1, 0, 0 ], [ 1, 1, 0 ], [ 0, 0, 1 ] ] gap> r=CoxeterGroup("A",3).matgens; 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
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; 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). 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)
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 :
trueif 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
C up to conjugation by diagonal matrices is an invariant 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 a reflection is 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 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
independent of the reflection representation.
gap> W:=ReflectionSubgroup(CoxeterGroup("A",3),[1,3]); ReflectionSubgroup(CoxeterGroup("A",3), [ 1, 3 ]) gap> SemisimpleRank(W); 2
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