84 Finite Reflection Groups

Let W be a finite reflection group on the vector space V over a subfield k of the complex numbers. An efficient representation that we use in CHEVIE for computing with such group is, is a permutation representation on a W-invariant set of root and coroot vectors for reflections of W; that is, a set R of pairs (r,r)∈ V× V* invariant by W and such each distinguished reflection in W is defined by some pair in R (see Reflection). There may be several pairs for each reflection, differing by roots of unity. This generalizes the usual construction for Coxeter groups (the case k=ℝ) where to each reflection of W is associated two roots, a positive and a negative one. For complex reflection groups, we need at least as many roots on a given line as the order of the center of W.

The finite irreducible complex reflection groups have been completely classified by Shepard and Todd. They contain one infinite family depending on 3 parameters, and 34 ``exceptional'' groups which have been given by Shephard and Todd names which range from G4 to G37. They cover the exceptional Coxeter groups, e.g., CoxeterGroup("E",8) is the same as G37.

CHEVIE provides functions to build any finite reflection group, either by giving a list of roots and corrots defining the generating reflections, or in terms of the classification. The output is a permutation group on set of roots (see ComplexReflectionGroup and PermRootGroup). In the context e.g. of Weyl groups, one wants to describe the particular root system chosen in term of the traditional classification of crystallographic root systems. This is done via calls to the function CoxeterGroup (see the chapter on finite Coxeter groups). There is not yet a general theory on how to construct a nice set of roots for a non-real reflection group; the roots chosen in CHEVIE where obtained case-by-case; however, they satisfy several important properties:

• The generating reflections satisfy braid relations which present the braid group associated to W (see PrintDiagram).

• The field of definition of W is the field k generated by the traces of the elements of W acting on V. It is a theorem that W may be realized as a reflection group over k. For almost all irreducible complex reflection groups, the generating matrices for W given by CHEVIE have coefficients in k. Further, the set of matrices for all elements of W is globally invariant under the Galois group of k/ℚ, thus the Galois action induces automorphisms of W. The exceptions are G22, G27 where the matrices are in a degree two extension of k (this is needed to have a globally invariant model, see MarinMichel10) and some dihedral groups as well as H3 and H4, where the matrices given (the usual Coxeter reflection representation over k) are not globally invariant.

It turns out that all representations of a complex reflection group W are defined over the field of definition of W (cf. Ben76 and D. Bessis thesis). This has been known for a long time in the case k=ℚ, the case of Weyl groups: their representations are defined over the rationals.

• The Cartan matrix (see CartanMat) for the generating roots (those which correspond to the generating reflections) has entries in the ring k of integers of k, and the roots (resp. coroots) are linear combination with coefficients in k of a linearly independent subset of them.

The finite reflection groups are reflection groups as described in the chapter "Reflections, and reflection groups", so in addition to the fields for permutation groups they have the fields .nbGeneratingReflections, .OrdersGeneratingReflections and .reflections. They also have the following additional fields:


a set of complex roots in V, given as a list of lists (vectors), on which W has a faithful permutation representation.


the coroots for the first .nbGeneratingReflections roots.

In this chapter we describe functions available for finite reflection groups W represented as permutation groups on a set of roots. These functions make use of the classification of W whenever it is known, but work even if it is not known.

Let SV be the symmetric algebra of V. The invariants of W in SV are called the polynomial invariants of W. They are generated as a polynomial ring by dim V homogeneous algebraically independent polynomials f1,...,fdim V. The polynomials fi are not uniquely determined but their degrees are. The fi are called the basic invariants of W, and their degrees the reflection degrees of W. Let I be the ideal generated by the homogeneous invariants of positive degree in SV. Then SV/I is isomorphic to the regular representation of W as a W-module. It is thus a graded (by the degree of elements of SV) version of the regular representation of W. The polynomial which gives the graded multiplicity of a character χ of W in the graded module SV/I is called the fake degree of χ.


  1. Functions for finite reflection groups
  2. PermRootGroup
  3. ReflectionType
  4. ReflectionName
  5. IsomorphismType
  6. ComplexReflectionGroup
  7. Reflections
  8. MatXPerm
  9. PermMatX
  10. MatYPerm
  11. InvariantForm for finite reflection groups
  12. ReflectionEigenvalues
  13. ReflectionLength
  14. ReflectionWord
  15. HyperplaneOrbits
  16. BraidRelations
  17. PrintDiagram
  18. ReflectionCharValue
  19. ReflectionCharacter
  20. ReflectionDegrees
  21. ReflectionCoDegrees
  22. GenericOrder
  23. TorusOrder
  24. ParabolicRepresentatives for reflection groups
  25. IsParabolic
  26. ParabolicClosure
  27. Invariants
  28. Discriminant
  29. Catalan

84.1 Functions for finite reflection groups

They are permutation groups, so all functions for permutation groups apply, although some are replaced by faster methods when available. A typical example is the function Size, which is obtained simply by the product of the reflection degrees, when they are known. Appropriate methods for String and Print are also defined.


Works like for Coxeter groups; in addition, as a special convention, if all .reflectionsLabels of W are positive integers, negative integers are accepted and represent the inverse of the corresponding generator.


A*B returns the product of the two reflection groups A and B as a reflection group.

84.2 PermRootGroup

PermRootGroupNC( roots [,eigenvalues]) PermRootGroup( roots [,eigenvalues])

PermRootGroupNC( roots, coroots) PermRootGroup( roots, coroots)

roots is a list of roots, that is of vectors in some vector space. PermRootGroup returns the reflection group generated by the reflections with respect to these roots (if this group is not finite, the function will never return). The precise way the reflections are constructed as matrices is specified by the second argument. In the second form the i-th reflection is computed as Reflection(roots[i], coroots[i]). In the first form eigenvalues represents non-trivial eigenvalues of the reflections to construct, represented as a list of fractions n/d, where such a fraction represents the eigenvalue E(d)^n (the reason for using such a representation instead of E(d)^n is that in GAP3 it is trivial to compute E(d)^n given d/n, but the converse is hard). In this form the i-th reflection is computed as Reflection(roots[i], E(d)^n) where eigenvalues[i]=n/d. If in the first form eigenvalues are omitted, they are all assumed to be 1/2 (which represents the number -1, i.e. all reflections are true reflections).

In the (faster) variant with NC, the group is not classified (thus for instance PrintDiagram will not work).

    gap> W:=PermRootGroupNC(IdentityMat(3),CartanMat("A",3));
    PermRootGroup([ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
      [ [ 2, -1, 0 ], [ -1, 2, -1 ], [ 0, -1, 2 ] ])
    gap> PrintDiagram(W);
    Error, PermRootGroup([ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
      [ [ 2, -1, 0 ], [ -1, 2, -1 ], [ 0, -1, 2 ] ]) has no method
    for PrintDiagram in
    Dispatcher( "PrintDiagram" )( W ) called from
    PrintDiagram( W ) called from
    main loop
    gap> ReflectionType(W);
    [ rec(rank    := 3,
          series  := "A",
          indices := [ 1, 2, 3 ]) ]
    gap> PrintDiagram(W);
    A3 1 - 2 - 3

In the above, the call to ReflectionType makes CHEVIE identify the classification of W, after which functions like PrintDiagram can work. Below is another way to build a group of type A3.

    gap> W:=PermRootGroup([[1,0,-1],[-1,1,0],[1,0,1]]);
    gap> ReflectionDegrees(W);
    [ 2, 3, 4 ]

84.3 ReflectionType

ReflectionType( W )

This function returns the type of W, which is a list each element of which describes an irreducible component of W; the elements of the list are objects of type Reflection Type, on which some functions can be called to obtain data on groups of that type, like ReflectionDegrees, etc...

Such an object is a record with a field series, the type ("A", "B", "D", etc...) of the component, a field indices, the indices in the list of generating reflections of W where it sits, a field rank (equal to Length(indices) for well-generated complex reflection groups such as Coxeter groups, and to Length(indices)+1 for the others).

For dihedral groups there is in addition a field bond giving the order of the braid relation between the two generators.

For complex reflection groups which are not real, the field series is equal to "ST", and there is an additional field ST, equal either to an integer n (for exceptional reflection groups Gn), or a triple (p,q,r) of integers (for imprimitive reflection groups G(p,q,r)).

This function is called automatically upon construction of a finite reflection group via PermRootGroup, or upon constructing a finite Coxeter group by CoxeterGroup. But since it is sometimes costly in time (it identifies the type of the group based on the order, the degree, the order of the generators and the Cartan matrix; sometimes it needs to search for another set of generators than the given one), a version PermRootGroupNC is given which does not call it.

This function is called automatically prior to calling any function depending on the classification, such as PrintDiagram, ReflectionName, ChevieClassInfo, BraidRelations, CharName, CharParams, Representations, Invariants.

    gap> W:=ComplexReflectionGroup(4)*CoxeterGroup("A",2);;
    gap> ReflectionType(W);
    [ rec(series  := "ST",
          ST      := 4,
          rank    := 2,
          indices := [ 1, 2 ]), rec(rank    := 2,
          series  := "A",
          indices := [ 3, 4 ]) ]

ReflectionType( C )

C should be a Cartan matrix. This function determines the type of each irreducible component of C which is the Cartan matrix of a finite Coxeter group; the result is a list of Reflection types. The corresponding field is set to false if the corresponding submatrix of C is not the Cartan matrix of a finite Coxeter group. Going from the above example:

    gap> C:=CartanMat(W);
    [ [ -2*E(3)-E(3)^2, E(3)^2, 0, 0 ], [ -E(3)^2, -2*E(3)-E(3)^2, 0, 0 ],
      [ 0, 0, 2, -1 ], [ 0, 0, -1, 2 ] ]
    gap> ReflectionType(C);
    [ false, rec(rank    := 2,
          series  := "A",
          indices := [ 3, 4 ]) ]

Note that a Cartan matrix for a finite Coxeter group is conjugate by a diagonal matrix of the matrices for the root systems given in the introduction of the chapter on root systems. This conjugation corresponds to changing the ratio of the length between long and short roots; for example one could construct a root system for type B where the quotient of the two root lengths is any cyclotomic number.

    gap> M:=[ [ 2, -E(7)^3-E(7)^5-E(7)^6 ], [ -E(7)-E(7)^2-E(7)^4, 2 ] ];;
    gap> ReflectionType(M);
    [ rec(rank       := 2,
          series     := "B",
          cartanType := E(7)^3+E(7)^5+E(7)^6,
          indices    := [ 1, 2 ]) ]

In the above example, the cartanType field shows that the two root lengths for B2 have a ratio which is (1+√-7)/(2).

84.4 ReflectionName

ReflectionName( type )

takes as argument a type type as returned by ReflectionType. Returns the name of the group system with that type, which is the concatenation of the names of its irreducible components, with x added in between. For reflection subgroups, it gives an indication about embedding in the parent

    gap> C := [ [ 2, 0, -1 ], [ 0, 2, 0 ], [ -1, 0, 2 ] ];;
    gap> ReflectionName( ReflectionType( C ) );
    gap> ReflectionName( ReflectionType( CartanMat( "I", 2, 7 ) ) );
    gap> ReflectionName(ReflectionSubgroup(CoxeterGroup("E",8),[2,3,6,7]));

ReflectionName( D )

The argument to ReflectionType can also be a record with a field operations.ReflectionType, and that function is then called with rec as argument --- this works for reflection groups and reflection cosets.

84.5 IsomorphismType

IsomorphismType( W )

takes as argument a reflection group or a reflection coset. Returns a description of the isomorphism type of the argument.

    gap> IsomorphismType(ReflectionSubgroup(CoxeterGroup("E",8),[2,3,6,7]));

84.6 ComplexReflectionGroup

ComplexReflectionGroup( STnumber )

ComplexReflectionGroup( p, q, r )

The first form of ComplexReflectionGroup returns the complex reflection group which has Shephard-Todd number STnumber, see ST54. The second form returns the imprimitive complex reflection group G(p,q,r).

    gap> G := ComplexReflectionGroup( 4 );
    gap> ReflectionDegrees( G );
    [ 4, 6 ]
    gap> Size( G );
    gap> q := X( Cyclotomics );; q.name := "q";;
    gap> FakeDegrees( G, q );
    [ q^0, q^4, q^8, q^7 + q^5, q^5 + q^3, q^3 + q, q^6 + q^4 + q^2 ]
    gap> ComplexReflectionGroup(2,1,6);

84.7 Reflections

Reflections( W )

returns the list of distinguished reflections of W, as elements of W. We recall that a reflection is distinguished (see "Reflections, and reflection groups") if it has eigenvalue E(e) where e is the cardinality of the cyclic subgroup CW(H), where H is the hyperplane of fixed points of the reflection (all reflections are distinguished if W is generated by reflections of order 2). The generating reflections of W are Reflections(W){W.generatingReflections}.

    gap> W := CoxeterGroup( "B", 2 );;
    gap> Reflections( W );
    [ (1,5)(2,4)(6,8), (1,3)(2,6)(5,7), (2,8)(3,7)(4,6), (1,7)(3,5)(4,8) ]

the code needed to obtain in general all reflections of W (not only the distinguished ones) is:

    gap> l:=Reflections(W);;
    gap> l:=Concatenation(List(l,s->List([1..Order(W,s)-1],i->s^i)));;

for finite Coxeter groups, Reflections(W) are in the same order as the positive roots. For general complex reflection groups the relationship with roots is only guaranteed for the generating reflections, that is Reflections(W){W.generatingReflections} are the reflections with respect to W.roots{W.generatingReflections}. The other reflections are not in the same order as the roots.

84.8 MatXPerm

MatXPerm( W, w )

Let w be a permutation of the roots of the finite reflection group W with reflection representation V. The function MatXPerm returns the matrix of w acting on V. This is the linear transformation of V which acts trivially on the orthogonal of the coroots and has same effect as w on the simple roots. The function makes sense more generally for an element of the normalizer of W in the whole permutation group of the roots.

    gap> W := CoxeterGroup(
    > [ [ 2, 0,-1, 0, 0, 0, 1 ], [ 0, 2, 0,-1, 0, 0, 0 ],
    >   [-1, 0, 2,-1, 0, 0,-1 ], [ 0,-1,-1, 2,-1, 0, 0 ],
    >   [ 0, 0, 0,-1, 2,-1, 0 ], [ 0, 0, 0, 0,-1, 2, 0 ] ],
    > [ [ 1, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0 ],
    >   [ 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0 ],
    >   [ 0, 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 0, 1, 0 ] ] );;
    gap> w0 := LongestCoxeterElement( W );;
    gap> mx := MatXPerm( W, w0 );
    [ [ 0, 0, 0, 0, 0, -1, 1 ], [ 0, -1, 0, 0, 0, 0, 2 ],
      [ 0, 0, 0, 0, -1, 0, 3 ], [ 0, 0, 0, -1, 0, 0, 4 ],
      [ 0, 0, -1, 0, 0, 0, 3 ], [ -1, 0, 0, 0, 0, 0, 1 ],
      [ 0, 0, 0, 0, 0, 0, 1 ] ]

84.9 PermMatX

PermMatX( W, M )

Let M be a linear transformation of reflection representation of W which preserves the set of roots, and thus normalizes W (remember that matrices act on the right in GAP3). PermMatX returns the corresponding permutation of the roots; it returns false if M does not normalize the set of roots.

We continue the example from MatXPerm and obtain:

    gap> PermMatX( W, mx ) = w0;

84.10 MatYPerm

MatYPerm( W, w )

Let w be a permutation of the roots of the finite reflection group W with reflection representation V. The function MatYPerm returns the matrix of w acting on the dual vector space V. This is the linear transformation of V which acts trivially on the orthogonal of the roots and has same effect as w on the simple coroots. The function makes sense more generally for an element of the normalizer of W in the whole permutation group of the roots.

    gap>  W:=ReflectionSubgroup(CoxeterGroup("E",7),[1..6]);
    ReflectionSubgroup(CoxeterGroup("E",7), [ 1, 2, 3, 4, 5, 6 ])
    gap>  w0:=LongestCoxeterElement(W);;
    gap> my:=MatYPerm(W,w0);
    [ [ 0, 0, 0, 0, 0, -1, 2 ], [ 0, -1, 0, 0, 0, 0, 2 ],
      [ 0, 0, 0, 0, -1, 0, 3 ], [ 0, 0, 0, -1, 0, 0, 4 ],
      [ 0, 0, -1, 0, 0, 0, 3 ], [ -1, 0, 0, 0, 0, 0, 2 ],
      [ 0, 0, 0, 0, 0, 0, 1 ] ]

84.11 InvariantForm for finite reflection groups

InvariantForm( W )

This function returns the matrix F of an Hermitian form invariant under the action of the reflection group W. That is, if M is the matrix of an element of W, then M*F*Transpose(ComplexConjugate(M))=F.

    gap> W:=ComplexReflectionGroup(4);
    gap> F:=InvariantForm(W);
    [ [ 1, 0 ], [ 0, 2 ] ]
    gap> List(W.matgens,m->m*F*ComplexConjugate(TransposedMat(m))=F);
    [ true, true ]

84.12 ReflectionEigenvalues

ReflectionEigenvalues( W [, c])

Let W be a reflection group on the vector space V. ReflectionEigenvalues( W) returns the list for each conjugacy classes of the eigenvalues of an element of that class acting on V. This is returned as a list of fractions i/n, where such a fraction represents the eigenvalue E(n)^i (the reason for returning such a representation instead of E(n)^i is that in GAP3 it is trivial to compute E(n)^i given i/n, but the converse is more expensive). If a second argument c is given, returns only the list of eigenvalues of an element of the cth conjugacy class.

    gap> W:=CoxeterGroup("A",2);
    gap> ReflectionEigenvalues(W,3);
    [ 1/3, 2/3 ]
    gap> ReflectionEigenvalues(CoxeterGroup("B",2));
    [ [ 0, 0 ], [ 1/2, 0 ], [ 1/2, 1/2 ], [ 1/2, 0 ], [ 1/4, 3/4 ] ]

84.13 ReflectionLength

ReflectionLength( W, w )

This function returns the number of eigenvalues of w in the reflection representation which are not equal to 1. For a finite Coxeter group, this is equal to the minimum number of reflections of which w is a product. This also holds in general in a well-generated complex reflection group if w divides a Coxeter element for the reflection length.

    gap> W:=CoxeterGroup("A",4);
    gap> ReflectionLength(W,LongestCoxeterElement(W));
    gap> ReflectionLength(W,EltWord(W,[1,2,3,4]));

84.14 ReflectionWord

ReflectionWord( W, w [, refs])

This function return a list of minimal length of reflections of which w is the product. The reflections are represented as their index in the list of reflections (which is the index of the corresponding positive root in the list of roots). If a third argument is given, it must be a list of reflections and only these reflections are tried, and the index is with respect to this list of reflections. This function works for all elements of a Coxeter group when no third argument is given, or for w a simple of the dual braid monoid if W is a well-generated complex reflection group and refs is the list of atoms of this monoid.

    gap> W:=CoxeterGroup("A",4);
    gap> ReflectionWord(W,LongestCoxeterElement(W));
    [ 6, 10 ]
    gap> ReflectionWord(W,EltWord(W,[1,2,3,4]));
    [ 1, 2, 3, 4 ]

84.15 HyperplaneOrbits

HyperplaneOrbits( W )

returns a list of records, one for each hyperplane orbit of W, containing the following fields for each orbit:


index of first generator in orbit


order of s


if w=W.generators[.s] returns List([1..e_s-1],i->PositionClass(W,w^i)


Size of orbit


for i in [1..e_s-1], position in CharTable of (det_s)^i

    gap> W:=CoxeterGroup("B",2);
    gap> HyperplaneOrbits(W);
    [ rec(
          s := 1,
          e_s := 2,
          classno := [ 2 ],
          N_s := 2,
          det_s := [ 5 ] ), rec(
          s := 2,
          e_s := 2,
          classno := [ 4 ],
          N_s := 2,
          det_s := [ 1 ] ) ]

84.16 BraidRelations

BraidRelations( W )

this function returns the relations which present the braid group of W. These are homogeneous (both sides of the same length) relations between generators in bijection with the generating reflections of W. A presentation of W is obtained by adding relations specifying the order of the generators.

    gap> W:=ComplexReflectionGroup(29);
    gap> BraidRelations(W);
    [ [ [ 1, 2, 1 ], [ 2, 1, 2 ] ], [ [ 2, 4, 2 ], [ 4, 2, 4 ] ],
      [ [ 3, 4, 3 ], [ 4, 3, 4 ] ], [ [ 2, 3, 2, 3 ], [ 3, 2, 3, 2 ] ],
      [ [ 1, 3 ], [ 3, 1 ] ], [ [ 1, 4 ], [ 4, 1 ] ],
      [ [ 4, 3, 2, 4, 3, 2 ], [ 3, 2, 4, 3, 2, 4 ] ] ]

each relation is represented as a pair of lists, specifying that the product of the generators according to the indices on the left side is equal to the product according to the indices on the right side. See also PrintDiagram.

84.17 PrintDiagram

PrintDiagram( W ) PrintDiagram( type )

This is a purely descriptive routine, which, by printing a diagram as in BMR98 for W or the given reflection type (a Dynkin diagram for Weyl groups) shows how the generators of W are labeled in the CHEVIE presentation.

    gap> PrintDiagram(ComplexReflectionGroup(31));
    G31 4 - 2 - 5
         \ /3\ /
          1 - 3     i.e. A_5 on 14253 plus 123=231=312

84.18 ReflectionCharValue

ReflectionCharValue( W, w )

Returns the trace of the element w of the reflection group W as an endomorphism of the vector space V on which W acts. This could also be obtained (less efficiently) by TraceMat(MatXPerm(W,w)).

    gap> W := CoxeterGroup( "A", 3 );
    gap> List( Elements( W ), x -> ReflectionCharValue( W, x ) );
    [ 3, 1, 1, 1, 0, 0, 0, -1, 0, -1, -1, 1, 1, -1, -1, -1, 0, 0, 0, 0,
      -1, -1, 1, -1 ]

84.19 ReflectionCharacter

ReflectionCharacter( W )

Returns the reflection character of the reflection group W. This could also be obtained (less efficiently) by List(ConjugacyClasses(W),c->ReflectionCharValue(W,c)). When W is irreducible, it can also be written CharTable(W).irreducibles[ChevieCharInfo(W).extRefl[2]]

    gap> W := CoxeterGroup( "A", 3 );
    gap> ReflectionCharacter(W);
    [ 3, 1, -1, 0, -1 ]

84.20 ReflectionDegrees

ReflectionDegrees( W )

returns a list holding the degrees of W as a reflection group on the vector space V on which it acts. These are the degrees d1,...,d dim V of the basic invariants of W in SV. They reflect various properties of W; in particular, their product is the size of W.

    gap> W := ComplexReflectionGroup(30);
    gap> ReflectionDegrees( W );
    [ 2, 12, 20, 30 ]
    gap> Size( W );

84.21 ReflectionCoDegrees

ReflectionCoDegrees( W )

returns a list holding the codegrees of W as a reflection group on the vector space V on which it acts. These are one less than the degrees d*1,...,d*dim V of the basic derivations of W on SV⊗ V.

    gap> W := ComplexReflectionGroup(4);;
    gap> ReflectionCoDegrees( W );
    [ 0, 2 ]

84.22 GenericOrder


returns the "compact" generic order of W as a polynomial in q. This is qNhi(qdi-1) where di are the reflection degrees and Nh the number of reflecting hyperplanes. For a Weyl group, it is the order of the associated semisimple finite reductive group over the field with q elements.

    gap> q:=X(Rationals);;q.name:="q";;
    gap> GenericOrder(ComplexReflectionGroup(4),q);
    q^14 - q^10 - q^8 + q^4

84.23 TorusOrder


returns as a polynomial in q the toric order of the i-th conjugacy class of W. This is the characteristic polynomial of an element of that class on the reflection representation of W. It is the same as the generic order of the reflection subcoset of W determined by the trivial subgroup and a representative of the i-th conjugacy class.

    gap> W:=ComplexReflectionGroup(4);;
    gap> q:=X(Cyclotomics);;q.name:="q";;
    gap> List([1..NrConjugacyClasses(W)],i->TorusOrder(W,i,q));
    [ q^2 - 2*q + 1, q^2 + 2*q + 1, q^2 + 1, q^2 + (-E(3))*q + (E(3)^2),
      q^2 + (E(3))*q + (E(3)^2), q^2 + (E(3)^2)*q + (E(3)),
      q^2 + (-E(3)^2)*q + (E(3)) ]

84.24 ParabolicRepresentatives for reflection groups

ParabolicRepresentatives(W [, r])

Returns a list of subsets of W.reflectionsLabels describing representatives of orbits of parabolic subgroups under conjugation by W. If a second argument r is given, returns only representatives of the parabolic subgroups of semisimple rank r. Contrary to the case of Coxeter groups, it may happen that for some orbits no representative can be chosen all of whose elements are standard generators.

    gap> ParabolicRepresentatives(ComplexReflectionGroup(3,3,3));
    [ [  ], [ 1 ], [ 1, 2 ], [ 1, 3 ], [ 1, 20 ], [ 2, 3 ], [ 1, 2, 3 ] ]
    gap> ParabolicRepresentatives(ComplexReflectionGroup(3,3,3),2);
    [ [ 1, 2 ], [ 1, 3 ], [ 1, 20 ], [ 2, 3 ] ]

84.25 IsParabolic


whether the reflection group W is a parabolic subgroup of Parent(W).

    gap> W:=ComplexReflectionGroup(7);
    gap> IsParabolic(ReflectionSubgroup(W,[1,2]));
    gap> IsParabolic(ReflectionSubgroup(W,[1]));

84.26 ParabolicClosure


I should be a list of indices of reflections of W. Returns J such that ReflectionSubgroup(W,J) is the smallest parabolic subgroup of W containing ReflectionSubgroup(W,I).

    gap> W:=ComplexReflectionGroup(7);
    gap> ParabolicClosure(W,[1]);
    [ 1 ]
    gap> ParabolicClosure(W,[1,2]);
    [ 1, 2, 3 ]

84.27 Invariants

Invariants( W )

returns the fundamental invariants of W in its reflection representation V. That is, returns a set of algebraically independent elements of the symmetric algebra of the dual of V which generate the W-invariant polynomial functions on V. Each such invariant function is returned as a GAP3 function: if e1,...,en is a basis of V and f is the GAP3 function, then the value of the polynomial function on a1e1+...+an en is obtained by calling f(a1,...,an). This function depends on the classification, and is dependent on the exact reflection representation of W. So for the moment it is only implemented when the reflection representation for the irreducible components has the same Cartan matrix as the one provided by CHEVIE for the corresponding irreducible group. The polynomials are invariant for the natural action of the group elements as matrices; that is, if m is MatXPerm(W,w) for some w in W, then an invariant f satisfies f(a1,...,an)= f(v1,...,vn) where [v1,...,vn]=[a1,...,an]× m. This action is implemented on Mvps by the function OnPolynomials (see OnPolynomials).

    gap> W:=CoxeterGroup("A",2);
    gap> i:=Invariants(W);
    [ function ( arg ) ... end, function ( arg ) ... end ]
    gap> a:=X(Rationals);;a.name:="a";;
    gap> b:=X(RationalsPolynomials);;b.name:="b";;
    gap> i[1](a,b);
    (-2*a^0)*b^2 + (2*a)*b + (-2*a^2)
    gap> i[2](a,b);
    (-6*a)*b^2 + (6*a^2)*b

Another example using Mvp from the package VKCURVE.

    gap> W:=ComplexReflectionGroup(24);;
    gap> i:=Invariants(W);;
    gap> v:=List([1..3],i->Mvp(SPrint("x",i)));
    [ x1, x2, x3 ]
    gap> ApplyFunc(i[1],v);
    gap> OnPolynomials(W.matgens[1],last)-last;

84.28 Discriminant

Discriminant( W )

returns the discriminant of the complex reflection group W, as a polynomial in the fundamental invariants. The discriminant is the invariant obtained by taking the product of the linear forms describing the reflecting hyperplanes of W, each raised to the order of the corresponding reflection. The discriminant is returned as a GAP3 function f such that the discriminant in the variables a1,...,an is obtained by calling f(a1,...,an). For the moment, this function is implemented only for the exceptional complex reflection groups G4 to G33.

    gap> W:=ComplexReflectionGroup(4);
    gap> Discriminant(W)(x,y);

84.29 Catalan

Catalan( n )

returns the n-th Catalan number.

    gap> Catalan(8);

Catalan( W )

returns the Catalan Number of the irreducible complex reflection group W. For well-generated groups, this number is equal to the number of simples in the dual Braid monoid. For other groups it was defined by Gordon and Griffeth (gg12). For Weyl groups, it also counts the number of antichains of roots.

    gap> Catalan(CoxeterGroup("A",7));

Catalan( W, i)

returns the i-th Fuss-Catalan Number of the irreducible complex reflection group W. For well-generated groups, this number is equal to the number of chains s1,...,si of simples in the dual monoid where sj divides sj+1. For these groups, it is also equal to j(ih+dj)/dj where the product runs over the reflection degrees of W, and where h is the Coxeter number of W. For non-well generated groups, the definition is in gg12.

    gap> Catalan(ComplexReflectionGroup(7),2);

Catalan( W, q), resp. Catalan( W, i, q)

where q is a variable (an indeterminate or an Mvp) returns the q-Catalan number (resp. the i-th q-Fuss Catalan number) of W. Again the definitions in general are in gg12.

    gap> Catalan(ComplexReflectionGroup(7),2,x);
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02 Dec 2021