Let W be a finite reflection group on the vector space V over a subfield k of the complex numbers. An efficient representation for computational purposes that we use in CHEVIE for such a group, is a permutation representation on a 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 that all reflections in W are the reflections with root r and coroot r^{∨} for some pair in R (see Reflection). This generalizes the usual construction for Coxeter groups (the special case k=ℝ) where to each reflection of W is associated two roots, a positive and a negative one. For complex reflection groups, more roots may lie on the same line --- we need at least as many as the order of the center of W. 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. In CHEVIE, in almost all cases, all the matrices for elements of W have coefficients in k. Further, this set of matrices is globally invariant under the Galois group of k/ℚ, thus the Galois action induces automorphisms of W. The exceptions are G_{22}, G_{27} 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, where the matrices given (the usual model over k) are nor globally invariant. A degree 2 extension would be needed for an invariant model.
• The Cartan matrix (see CartanMat) for the generating roots has entries in the ring ℤ_{k} of integers of k, and the roots (resp. coroots) are linear combination with coefficients in ℤ_{k} of a linearly independent subset of them.
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 G_{4} to G_{37}. They cover the
exceptional Coxeter groups, e.g., CoxeterGroup("E",8)
is the same as
G_{37}.
CHEVIE provides functions to build any finite reflection group,
described in terms of the classification, or given by generating
reflections described in term of roots; the output is a permutation group
of a root system (see ComplexReflectionGroup
and PermRootGroup
). Of
course, 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).
The finite reflection groups are 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:
roots
:
simpleCoroots
:.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 f_{1},...,f_{}dim V. The polynomials f_{i} are not uniquely determined but their degrees are. The f_{i} 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.
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.
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.
EltWord
:.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.
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 brk> 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 A_{3}.
gap> W:=PermRootGroup([[1,0,-1],[-1,1,0],[1,0,1]]); PermRootGroup([[1,0,-1],[-1,1,0],[1,0,1]]) gap> ReflectionDegrees(W); [ 2, 3, 4 ]
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 G_{n}), 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).
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 ) ); "A2xA1" gap> ReflectionName( ReflectionType( CartanMat( "I", 2, 7 ) ) ); "I2(7)" gap> ReflectionName(ReflectionSubgroup(CoxeterGroup("E",8),[2,3,6,7])); "A1<2>xA1<3>xA2<6,7>.(q-1)^4"
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.
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])); "A2+2A1"
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 ); ComplexReflectionGroup(4) gap> ReflectionDegrees( G ); [ 4, 6 ] gap> Size( G ); 24 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); CoxeterGroup("B",6)
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 C_{W}(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.
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 ] ]
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; true
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 ] ]
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); ComplexReflectionGroup(4) gap> F:=InvariantForm(W); [ [ 1, 0 ], [ 0, 2 ] ] gap> List(W.matgens,m->m*F*ComplexConjugate(TransposedMat(m))=F); [ true, true ]
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); 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 ] ]
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); CoxeterGroup("A",4) gap> ReflectionLength(W,LongestCoxeterElement(W)); 2 gap> ReflectionLength(W,EltWord(W,[1,2,3,4])); 4
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); CoxeterGroup("A",4) gap> ReflectionWord(W,LongestCoxeterElement(W)); [ 6, 10 ] gap> ReflectionWord(W,EltWord(W,[1,2,3,4])); [ 1, 2, 3, 4 ]
HyperplaneOrbits( W )
returns a list of records, one for each hyperplane orbit of W, containing the following fields for each orbit:
.s
:
.e_s
:
.classno
:w=W.generators[.s]
returns
List([1..e_s-1],i->PositionClass(W,w^i)
.N_s
:
.det_s
:[1..e_s-1]
, position in CharTable of (det_s)^i
gap> W:=CoxeterGroup("B",2); 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 ] ) ]
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); 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.
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
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 ); 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 ]
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 ); CoxeterGroup("A",3) gap> ReflectionCharacter(W); [ 3, 1, -1, 0, -1 ]
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 d_{1},...,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); CoxeterGroup("H",4) gap> ReflectionDegrees( W ); [ 2, 12, 20, 30 ] gap> Size( W ); 14400
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 ]
GenericOrder(W,q)
returns the "compact" generic order of W
as a polynomial in q
. This
is q^{Nh}∏_{i}(q^{di}-1) where d_{i} are the reflection degrees and
N_{h} 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
TorusOrder(W,i,q)
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)) ]
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 ] ]
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 e_{1},...,e_{n} is a basis of V and f
is the
GAP3 function, then the value of the polynomial function on
a_{1}e_{1}+...+a_{n} e_{n} is obtained by calling f(a_{1},...,a_{n}). 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(a_{1},...,a_{n})=
f(v_{1},...,v_{n}) where [v_{1},...,v_{n}]=[a_{1},...,a_{n}]× m. This
action is implemented on Mvp
s by the function OnPolynomials
(see
OnPolynomials).
gap> W:=CoxeterGroup("A",2); CoxeterGroup("A",2) gap> i:=Invariants(W); [ function ( arg ) ... end, function ( arg ) ... end ] gap> x:=X(Rationals);;x.name:="x";; gap> y:=X(RationalsPolynomials);;y.name:="y";; gap> i[1](x,y); (-2*x^0)*y^2 + (2*x)*y + (-2*x^2) gap> i[2](x,y); (-6*x)*y^2 + (6*x^2)*y
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); -42x1^2x2x3-12x1^2x2^2+21/2x1^2x3^2-9/2x2^2x3^2-6x2^3x3+14x1^4+18/7x2^\ 4-21/8x3^4 gap> OnPolynomials(W.matgens[1],last)-last; 0
Catalan( n )
returns the n-th Catalan number.
gap> Catalan(8); 1430
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)); 1430
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 s_{1},...,s_{i} of simples in the dual monoid where s_{j} divides s_{j+1}. For these groups, it is also equal to ∏_{j}(ih+d_{j})/d_{j} 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); 16
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); 1+2x^12+3x^24+4x^36+3x^48+2x^60+x^72Previous Up Next
gap3-jm