82 Reflections, and reflection groups

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 one of its subfields. A complex reflection is an element s∈ GL(V) whose fixed point set is an hyperplane and which is of finite order (we will in the following just call it a reflection to abbreviate, which is an abuse since the term reflection is usually only employed when the order is 2). Thus a reflection has a unique eigenvalue not equal to 1. If K is a subfield of the 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 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 in CHEVIE are always distinguished reflections.

Reflection groups in CHEVIE are groups W with the following fields (in the group record) defined

.nbGeneratingReflections:

the number of reflections which generate W

.reflections:

a list of distinguished reflections, given as elements of W, such that a list of reflections which generate W is W.reflections{[1..W.nbGeneratingReflections]}.

.OrdersGeneratingReflections:

a list (of length at least 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.

Subsections

  1. Reflection
  2. AsReflection
  3. CartanMat
  4. Rank
  5. SemisimpleRank

82.1 Reflection

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 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

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 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 Weyl groups and positive roots that the i-reflection corresponds to the i-th root since two roots corresponding to the same reflection may have been listed before all the reflections have been obtained.

    gap> Reflection(CoxeterGroup("A",3),6);
    ( 1,11)( 3,10)( 4, 9)( 5, 7)( 6,12)

82.2 AsReflection

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:

the root of the reflection s (equal to r if given)

.coroot:

the coroot of s

.eigenvalue:

the non-trivial eigenvalue of s

.isOrthogonal:

a boolean which is 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 )

82.3 CartanMat

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

82.4 Rank

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

82.5 SemisimpleRank

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

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gap3-jm
19 Feb 2018