Let us fix an algebraically closed field K and let G be a connected reductive algebraic group over K. Let T be a maximal torus of G, let X(T) be the character group of T (resp. Y(T) the dual lattice of one-parameter subgroups of T) and Φ (resp Φ∨) the roots (resp. coroots) of G with respect to T.
Then G is determined up to isomorphism by the root datum (X(T),Φ, Y(T),Φ∨). In algebraic terms, this consists in giving a free ℤ-lattice X=X(T) of dimension the rank of T (which is also called the rank of G), and a root system Φ ⊂ X, and giving similarly the dual roots Φ∨⊂ Y=Y(T).
This is obtained by a slight generalization of our setup for a Coxeter
group W. This time we assume the canonical basis of the vector space V
on which W acts is a ℤ-basis of X, and Φ is specified by a
matrix W.simpleRoots
whose lines are the simple roots expressed in this
basis of X. Similarly Φ∨ is described by a matrix
W.simpleCoroots
whose lines are the simple coroots in the basis of Y
dual to the chosen basis of X. The duality pairing between X and Y is
the canonical one, that is the pairing between vectors x∈ X and y∈
Y is given in GAP3 by x*y
. Thus, we must have the relation
W.simpleCoroots*TransposedMat(W.simpleRoots)=CartanMat(W)
.
We get that in CHEVIE by a new form of the function CoxeterGroup
, where
the arguments are the two matrices W.simpleRoots
and W.simpleCoroots
described above. The roots need not generate V, so the matrices need not
be square. For instance, the root datum of the linear group of rank 3 can
be specified as:
gap> W := CoxeterGroup( [ [ -1, 1, 0], [ 0, -1, 1 ] ], > [ [ -1, 1, 0], [ 0, -1, 1 ] ] ); CoxeterGroup([[-1,1,0],[0,-1,1]],[[-1,1,0],[0,-1,1]]) gap> MatXPerm( W, W.1); [ [ 0, 1, 0 ], [ 1, 0, 0 ], [ 0, 0, 1 ] ]
here the symmetric group on 3 letters acts by permutation of the basis of
X. The dimension of X (the length of the vectors in .simpleRoots
) is
the rank and the dimension of the subspace generated by the roots (the
length of .simpleroots
) is called the semi-simple rank. In the example
the rank is 3 and the semisimple rank is 2.
The default form W:=CoxeterGroup("A",2)
defines the adjoint algebraic
group (the group with a trivial center). In that case Φ is a basis of
X, so W.simpleRoots
is the identity matrix and W.simpleCoroots
is the Cartan matrix CartanMat(W)
of the root system. The form
CoxeterGroup("A",2,"sc")
constructs the semisimple simply connected
algebraic group, where W.simpleRoots
is the transposed of CartanMat(W)
and W.simpleCoroots
is the identity matrix.
There is an extreme form of root data which requires another function to
specify: when W is the trivial CoxeterGroup()
and there are thus no
roots (in this case G is a torus), the root datum cannot be determined
by the roots, but is entirely determined by the rank r. The function
Torus(r)
constructs such a root datum.
Finally, there is also a function RootDatum
which understands some
familiar names for the algebraic groups and gives the results that could be
obtained by giving the appropriate matrices W.simpleRoots
and
W.simpleCoroots
:
gap> RootDatum("gl",3); # same as the previous example RootDatum("gl",3)
Semisimple elements
It is also possible to compute with semi-simple elements. The first type
are finite order elements of T, which over an algebraically closed
field K are in bijection with elements of Y⊗ ℚ/ℤ whose
denominator is prime to the characteristic of K. These are represented as
elements of a vector space of rank r over ℚ, taken Mod1
whenever the
need arises, where Mod1
is the function which replaces the numerator of a
fraction with the numerator mod
the denominator; the fraction
p/q represents a primitive q-th root of unity raised to the
p-th power. In this representation, multiplication of roots of unity
becomes addition Mod1
of rationals and raising to the power n becomes
multiplication by n. We call this the ``additive'' representation of
semisimple elements.
Here is an example of computations with semisimple-elements given as list of r elements of ℚ/ℤ.
gap> G:=RootDatum("sl",4); RootDatum("sl",4) gap> L:=ReflectionSubgroup(G,[1,3]); ReflectionSubgroup(RootDatum("sl",4), [ 1, 3 ]) gap> AlgebraicCentre(L); rec( Z0 := SubTorus(ReflectionSubgroup(RootDatum("sl",4), [ 1, 3 ]),[ [ 1, 2, \ 1 ] ]), AZ := Group( <0,0,1/2> ), ZD := Group( <1/2,0,0>, <0,0,1/2> ), descAZ := [ [ 1, 2 ] ] ) gap> SemisimpleSubgroup(last.Z0,3); Group( <1/3,2/3,1/3> ) gap> e:=Elements(last); [ <0,0,0>, <1/3,2/3,1/3>, <2/3,1/3,2/3> ]
First, the group G=SL4 is constructed, then the Levi subgroup L
consisting of block-diagonal matrices of shape 2× 2. The function
AlgebraicCentre
returns a record with : the neutral component Z0 of
the centre Z of L, represented by a basis of Y(Z0), a complement
subtorus S of T to Z0 represented similarly by a basis of Y(S),
and semi-simple elements representing the classes of Z modulo Z0 ,
chosen in S. The classes Z/Z0 also biject to the fundamental group as
given by the field .descAZ
, see AlgebraicCentre for an explanation.
Finally the semi-simple elements of order 3 in Z0 are computed.
gap> e[2]^G.2; <1/3,0,1/3> gap> Orbit(G,e[2]); [ <1/3,2/3,1/3>, <1/3,0,1/3>, <2/3,0,1/3>, <1/3,0,2/3>, <2/3,0,2/3>, <2/3,1/3,2/3> ]
Since over an algebraically closed field K the points of T are in
bijection with Y⊗ K× it is also possible to represent any
point of T over K as a list of r non-zero elements of K. This is
the ``multiplicative'' representation of semisimple elements. here is the
same computation as above performed with semisimple elements whose
coefficients are in the finite field GF(4)
:
gap> s:=SemisimpleElement(G,List([1,2,1],i->Z(4)^i)); <Z(2^2),Z(2^2)^2,Z(2^2)> gap> s^G.2; <Z(2^2),Z(2)^0,Z(2^2)> gap> Orbit(G,s); [ <Z(2^2),Z(2^2)^2,Z(2^2)>, <Z(2^2),Z(2)^0,Z(2^2)>, <Z(2^2)^2,Z(2)^0,Z(2^2)>, <Z(2^2),Z(2)^0,Z(2^2)^2>, <Z(2^2)^2,Z(2)^0,Z(2^2)^2>, <Z(2^2)^2,Z(2^2),Z(2^2)^2> ]
We can compute the centralizer CG(s) of a semisimple element in G:
gap> G:=CoxeterGroup("A",3); CoxeterGroup("A",3) gap> s:=SemisimpleElement(G,[0,1/2,0]); <0,1/2,0> gap> Centralizer(G,s); (A1xA1)<1,3>.(q+1)
The result is an extended reflection group; the reflection group part is the Weyl group of CG0(s) and the extended part are representatives of CG(s) modulo CG0(s) taken as diagram automorphisms of the reflection part. Here is is printed as a coset CG0(s)φ which generates CG(s).
CoxeterGroup( simpleRoots, simpleCoroots )
CoxeterGroup( C[, "sc"] )
CoxeterGroup( type1, n1, ... , typek, nk[, "sc"] )
The above are extended forms of the function CoxeterGroup
allowing to
specify more general root data. In the first form a set of roots is given
explicitly as the lines of the matrix simpleRoots, representing vectors
in a vector space V, as well as a set of coroots as the lines of the
matrix simpleCoroots expressed in the dual basis of V∨. The product
C=simpleCoroots*TransposedMat(simpleRoots)
must be a valid Cartan
matrix. The dimension of V can be greater than Length(C)
. The length
of C is called the semisimple rank of the Coxeter datum, while the
dimension of V is called its rank.
In the second form C is a Cartan matrix, and the call CoxeterGroup(C)
is equivalent to CoxeterGroup(IdentityMat(Length(C)),C)
. When the
optional "sc"
argument is given the situation is reversed: the simple
coroots are given by the identity matrix, and the simple roots by the
transposed of C (this corresponds to the embedding of the root system in
the lattice of characters of a maximal torus in a simply connected
algebraic group).
The argument "sc"
can also be given in the third form with the same
effect.
The following fields in a Coxeter group record complete the description of the corresponding root datum:
simpleRoots
:
simpleCoroots
:
matgens
:
RootDatum(type,n)
This function returns the root datum for the algebraic group described by
type and n. The types understood as of now are: "gl"
, "sl"
,
"pgl"
, "slmod"
, "tgl"
"sp"
, "so"
, "psp"
, "csp"
,
"pso"
, "halfspin"
, "spin"
, "gpin"
, "E6"
, "E6sc "
,
"E7"
, "E8"
, "F4"
, "G2"
. Most of these names should be
familar; let us explain those which may not be:
gpin(2n)
:
slmod(n,q)
:
tgl(n,k)
:
gap> RootDatum("spin",8);# same as CoxeterGroup("D",4,"sc") RootDatum("spin",8)
Dual(W)
This function returns the dual root datum of the root datum W, describing
the Langlands dual algebraic group. The fields .simpleRoots
and
.simpleCoroots
are swapped in the dual compared to W.
gap> W:=CoxeterGroup("B",3); CoxeterGroup("B",3) gap> Dual(W); CoxeterGroup("C",3,sc)
Torus(rank)
This function returns the CHEVIE object corresponding to the notion of a torus of dimension rank, a Coxeter group of semisimple rank 0 and given rank. This corresponds to a split torus; the extension to Coxeter cosets is more useful (see Torus for Coxeter cosets).
gap> Torus(3); Torus(3) gap> ReflectionName(last); "(q-1)^3"
86.5 FundamentalGroup for algebraic groups
FundamentalGroup(W)
This function returns the fundamental group of the algebraic group defined by the Coxeter group record W. This group is returned as a group of diagram automorphisms of the corresponding affine Weyl group, that is as a group of permutations of the set of simple roots enriched by the lowest root of each irreducible component. The definition we take of the fundamental group of a (not necessarily semisimple) reductive group is (P∩ Y(T))/Q where P is the coweight lattice (the dual lattice in Y(T)⊗ℚ of the root lattice) and Q is the coroot latice. The bijection between elements of P/Q and diagram automorphisms is explained in the context of non-irreducible groups for example in Bon05, \S 3.B.
gap> W:=CoxeterGroup("A",3); CoxeterGroup("A",3) gap> FundamentalGroup(W); Group( ( 1,12, 3, 2) ) gap> W:=CoxeterGroup("A",3,"sc"); CoxeterGroup("A",3,"sc") gap> FundamentalGroup(W); Group( () )
IntermediateGroup(W, indices)
This computes a Weyl group record representing a semisimple algebraic group
intermediate between the adjoint group --- obtained by a call like
CoxeterGroup("A",3)
--- and the simply connected semi-simple group ---
obtained by a call like CoxeterGroup("A",3,"sc")
. The group is
specified by specifying a subset of the minuscule weights, which are
weights whose scalar product with every coroot is in -1,0,1 (the weights
are the elements of the weight lattice, the lattice in X(T)⊗ℚ
dual to the coroot lattice). The non-trivial characters of the (algebraic)
center of a semi-simple simply connected algebraic group are in bijection
with the minuscule weights; this set is also in bijection with P/Q where
P is the weight lattice and Q is the root lattice. If W is
irreducible, the minuscule weights are part of the basis of the weight
lattice given by the fundamental weights, which is the dual basis of the
simple coroots. They can thus be specified by an index in the Dynkin
diagram (see PrintDiagram). The constructed group has lattice X(T)
generated by the sum of the root lattice and the weights with the given
indices. If W is not irreducible, a minuscule weight is a sum of
minuscule weights in different components. An element of indices is thus
itself a list, interpreted as representing the sum of the corresponding
weights.
gap> W:=CoxeterGroup("A",3);; gap> IntermediateGroup(W,[]); # adjoint CoxeterGroup("A",3) gap> FundamentalGroup(last); Group( ( 1,12, 3, 2) ) gap> IntermediateGroup(W,[2]);# intermediate CoxeterGroup([[2,0,-1],[0,1,0],[0,0,1]],[[1,-1,0],[-1,2,-1],[1,-1,2]]) gap> FundamentalGroup(last); Group( ( 1, 3)( 2,12) )
WeightInfo(W)
W is a Coxeter group record describing an algebraic group G, or an
irreducible type. The function is independent of the isogeny type of G
(so just depends on ReflectionType(W)
, that is on the root system). It
returns a record with the following fields:
.minusculeWeights
:
.minusculeCoweights
:
.decompositions
:
.moduli
:
.AdjointFundamentalGroup
:
.CenterSimplyConnected
:
.chosenAdaptedBasis
:C*.chosenAdaptedBasis
is almost in Smith normal form (it is
diagonal but the diagonal entries may be permuted compared to the Smith
normal form).
gap> WeightInfo(CoxeterGroup("A",2,"B",2)); rec( minusculeWeights := [ [ 1, 3 ], [ 1 ], [ 2, 3 ], [ 2 ], [ 3 ] ], minusculeCoweights := [ [ 1, 4 ], [ 1 ], [ 2, 4 ], [ 2 ], [ 4 ] ], decompositions := [ [ 2, 1 ], [ 2, 0 ], [ 1, 1 ], [ 1, 0 ], [ 0, 1 ] ], moduli := [ 3, 2 ], chosenAdaptedBasis := [ [ 1, -1, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], CenterSimplyConnected := [ [ 1/3, 2/3, 0, 0 ], [ 0, 0, 1/2, 0 ] ], AdjointFundamentalGroup := [ ( 1,12, 2), ( 4,14) ] )
SemisimpleElement(W,v[,additive])
W should be a root datum, given as a Coxeter group record for a Weyl
group, and v a list of length W.rank
. The result is a semisimple
element record, which has the fields:
.v
:Mod1
if its elements are rationals.
.group
:
gap> G:=CoxeterGroup("A",3);; gap> s:=SemisimpleElement(G,[0,1/2,0]); <0,1/2,0>
If all elements of v are rational numbers, they are converted by Mod1
to fractions between 0 and 1 representing roots of unity, and these
roots of unity are multiplied by adding Mod1
the fractions. In this way
any semisimple element of finite order can be represented.
If the entries are not rational numbers, they are assumed to represent
elements of a field which are multiplied or added normally. To explicitly
control if the entries are to be treated additively or not, a third
argument can be given: if true
the entries are treated additively, or
not if false
. For entries to be treated additively, they must belong to a
domain for which the method Mod1
had been defined.
86.9 Operations for semisimple elements
The arithmetic operations *
, /
and ^
work for semisimple elements.
They also have Print
and String
methods. We first give an element with
elements of ℚ/ℤ representing roots of unity.
gap> G:=CoxeterGroup("A",3); CoxeterGroup("A",3) gap> s:=SemisimpleElement(G,[0,1/2,0]); <0,1/2,0> gap> t:=SemisimpleElement(G,[1/2,1/3,1/7]); <1/2,1/3,1/7> gap> s*t; <1/2,5/6,1/7> gap> t^3; <1/2,0,3/7> gap> t^-1; <1/2,2/3,6/7> gap> t^0; <0,0,0> gap> String(t); "<1/2,1/3,1/7>"
GF(5)
:
gap> s:=SemisimpleElement(G,Z(5)*[1,2,1]); <Z(5),Z(5)^2,Z(5)> gap> t:=SemisimpleElement(G,Z(5)*[2,3,4]); <Z(5)^2,Z(5)^0,Z(5)^3> gap> s*t; <Z(5)^3,Z(5)^2,Z(5)^0> gap> t^3; <Z(5)^2,Z(5)^0,Z(5)> gap> t^-1; <Z(5)^2,Z(5)^0,Z(5)> gap> t^0; <Z(5)^0,Z(5)^0,Z(5)^0> gap> String(t); "<Z(5)^2,Z(5)^0,Z(5)^3>"
The operation ^
also works for applying an element of its defining Weyl
group to a semisimple element, which allows orbit computations:
gap> s:=SemisimpleElement(G,[0,1/2,0]); <0,1/2,0> gap> s^G.2; <1/2,1/2,1/2> gap> Orbit(G,s); [ <0,1/2,0>, <1/2,1/2,1/2>, <1/2,0,1/2> ]
The operation ^
also works for applying a root to a semisimple element:
gap> s:=SemisimpleElement(G,[0,1/2,0]); <0,1/2,0> gap> s^G.roots[4]; 1/2 gap> s:=SemisimpleElement(G,Z(5)*[1,1,1]); <Z(5),Z(5),Z(5)> gap> s^G.roots[4]; Z(5)^2
Frobenius( WF )
:WF
is a Coxeter coset associated to the
Coxeter group W, the function Frobenius
returns the associated
automorphism which can be applied to semisimple elements, see Frobenius.
gap> W:=CoxeterGroup("D",4);;WF:=CoxeterCoset(W,(1,2,4));; gap> s:=SemisimpleElement(W,[1/2,0,0,0]); <1/2,0,0,0> gap> F:=Frobenius(WF); function ( arg ) ... end gap> F(s); <0,0,0,1/2> gap> F(s,-1); <0,1/2,0,0>
86.10 Centralizer for semisimple elements
Centralizer( W, s)
W should be a Weyl group record or and extended reflection group and s a semisimple element for W. This function returns the stabilizer of the semisimple element s in W, which describes also CG(s), if G is the algebraic group described by W. The stabilizer is an extended reflection group, with the reflection group part equal to the Weyl group of CG0(s), and the diagram automorphism part being those induced by CG(s)/CG0(s) on CG0(s).
gap> G:=CoxeterGroup("A",3); CoxeterGroup("A",3) gap> s:=SemisimpleElement(G,[0,1/2,0]); <0,1/2,0> gap> Centralizer(G,s); (A1xA1)<1,3>.(q+1)
SubTorus(W,Y)
The function returns the subtorus S of the maximal torus T of the
reductive group represented by the Weyl group record W such that Y(S)
is the (pure) sublattice of Y(T) generated by the (integral) vectors
Y. A basis of Y(S) adapted to Y(T) is computed and stored in the
field S.generators
of the returned subtorus object. Here, adapted means
that there is a set of integral vectors, stored in S.complement
, such
that M:=Concatenation(S.generators,S.complement)
is a basis of Y(T)
(equivalently M∈GL(ℤrank(W)). An error is raised if
Y does not define a pure sublattice.
gap> W:=CoxeterGroup("A",4);; gap> SubTorus(W,[[1,2,3,4],[2,3,4,1],[3,4,1,2]]); Error, not a pure sublattice in SubTorus( W, [ [ 1, 2, 3, 4 ], [ 2, 3, 4, 1 ], [ 3, 4, 1, 2 ] ] ) called from main loop brk> gap> SubTorus(W,[[1,2,3,4],[2,3,4,1],[3,4,1,1]]); SubTorus(CoxeterGroup("A",4),[ [ 1, 0, 3, -13 ], [ 0, 1, 2, 7 ], [ 0, 0, 4, -3 ] ])
The operation in
can test if a semisimple element belongs to a subtorus:
gap> W:=RootDatum("gl",4);; gap> r:=AlgebraicCentre(W); rec( Z0 := SubTorus(RootDatum("gl",4),[ [ 1, 1, 1, 1 ] ]), AZ := Group( <0,0,0,0> ), ZD := Group( <1/4,1/4,1/4,1/4> ), descAZ := [ [ 1 ] ] ) gap> SemisimpleElement(W,[1/4,1/4,1/4,1/4]) in r.Z0; true
The operation Rank
gives the rank of the subtorus:
gap> Rank(r.Z0); 1
AlgebraicCentre( W )
W should be a Weyl group record, or an extended Weyl group record. This function returns a description of the centre ZG of the algebraic group G defined by W as a record with the following fields:
Z0
:
AZ
:
ZD
:
descAZ
:pi
of the simply connected covering of G.
It contains a list of elements given as words in the generators of pi
which generate the kernel of the quotient map.
gap> G:=CoxeterGroup("A",3,"sc");; gap> L:=ReflectionSubgroup(G,[1,3]); ReflectionSubgroup(CoxeterGroup("A",3,"sc"), [ 1, 3 ]) gap> AlgebraicCentre(L); rec( Z0 := SubTorus(ReflectionSubgroup(CoxeterGroup("A",3,"sc"), [ 1, 3 ]),[ [\ 1, 2, 1 ] ]), AZ := Group( <0,0,1/2> ), ZD := Group( <1/2,0,0>, <0,0,1/2> ), descAZ := [ [ 1, 2 ] ] ) gap> G:=CoxeterGroup("A",3);; gap> s:=SemisimpleElement(G,[0,1/2,0]);; gap> Centralizer(G,s); (A1xA1)<1,3>.(q+1) gap> AlgebraicCentre(last); rec( Z0 := SubTorus(ReflectionSubgroup(CoxeterGroup("A",3), [ 1, 3 ]),), AZ := Group( <0,1/2,0> ) )
Note that in versions of CHEVIE prior to april 2017, the field Z0
was
not a subtorus but what is now Z0.generators
, and there was a field
complement
which is now Z0.complement
.
SemisimpleSubgroup( S, n )
This function returns the subgroup of semi-simple elements of order dividing n in the subtorus S.
gap> G:=CoxeterGroup("A",3,"sc");; gap> L:=ReflectionSubgroup(G,[1,3]);; gap> z:=AlgebraicCentre(L);; gap> z.Z0; SubTorus(ReflectionSubgroup(CoxeterGroup("A",3,"sc"), [ 1, 3 ]),[ [ 1,\ 2, 1 ] ]) gap> SemisimpleSubgroup(z.Z0,3); Group( <1/3,2/3,1/3> ) gap> Elements(last); [ <0,0,0>, <1/3,2/3,1/3>, <2/3,1/3,2/3> ]
IsIsolated(W,s)
s should be a semi-simple element for the algebraic group G specified by the Weyl group record W. A semisimple element s of an algebraic group G is isolated if the connected component CG0(s) does not lie in a proper parabolic subgroup of G. This function tests this condition.
gap> W:=CoxeterGroup("E",6);; gap> QuasiIsolatedRepresentatives(W); [ <0,0,0,0,0,0>, <0,0,0,1/3,0,0>, <0,1/6,1/6,0,1/6,0>, <0,1/2,0,0,0,0>, <1/3,0,0,0,0,1/3> ] gap> Filtered(last,x->IsIsolated(W,x)); [ <0,0,0,0,0,0>, <0,0,0,1/3,0,0>, <0,1/2,0,0,0,0> ]
IsQuasiIsolated(W,s)
s should be a semi-simple element for the algebraic group G specified by the Weyl group record W. A semisimple element s of an algebraic group G is quasi-isolated if CG(s) does not lie in a proper parabolic subgroup of G. This function tests this condition.
gap> W:=CoxeterGroup("E",6);; gap> QuasiIsolatedRepresentatives(W); [ <0,0,0,0,0,0>, <0,0,0,1/3,0,0>, <0,1/6,1/6,0,1/6,0>, <0,1/2,0,0,0,0>, <1/3,0,0,0,0,1/3> ] gap> Filtered(last,x->IsQuasiIsolated(ReflectionSubgroup(W,[1,3,5,6]),x)); [ <0,0,0,0,0,0>, <0,0,0,1/3,0,0>, <0,1/2,0,0,0,0> ]
86.17 QuasiIsolatedRepresentatives
QuasiIsolatedRepresentatives(W[,p])
W should be a Weyl group record corresponding to an algebraic group G. This function returns a list of semisimple elements for G, which are representatives of the G-orbits of quasi-isolated semisimple elements. It follows the algorithm given by C. Bonnafé in Bon05. If a second argument p is given, it gives representatives of those quasi-isolated elements which exist in characteristic p.
gap> W:=CoxeterGroup("E",6);;QuasiIsolatedRepresentatives(W); [ <0,0,0,0,0,0>, <0,0,0,1/3,0,0>, <0,1/6,1/6,0,1/6,0>, <0,1/2,0,0,0,0>, <1/3,0,0,0,0,1/3> ] gap> List(last,x->IsIsolated(W,x)); [ true, true, false, true, false ] gap> W:=CoxeterGroup("E",6,"sc");;QuasiIsolatedRepresentatives(W); [ <0,0,0,0,0,0>, <1/3,0,2/3,0,1/3,2/3>, <1/2,0,0,1/2,0,1/2>, <2/3,0,1/3,0,1/3,2/3>, <2/3,0,1/3,0,2/3,1/3>, <2/3,0,1/3,0,2/3,5/6>, <5/6,0,2/3,0,1/3,2/3> ] gap> List(last,x->IsIsolated(W,x)); [ true, true, true, true, true, true, true ] gap> QuasiIsolatedRepresentatives(W,3); [ <0,0,0,0,0,0>, <1/2,0,0,1/2,0,1/2> ]
86.18 SemisimpleCentralizerRepresentatives
SemisimpleCentralizerRepresentatives(W [,p])
W should be a Weyl group record corresponding to an algebraic group
G. This function returns a list giving representatives H of
G-orbits of reductive subgroups of G which can be the identity
component of the centralizer of a semisimple element. Each group H is
specified by a list h of reflection indices in W such that H
corresponds to ReflectionSubgroup(W,h)
. If a second argument p is
given, only the list of the centralizers which occur in characteristic p
is returned.
gap> W:=CoxeterGroup("G",2); CoxeterGroup("G",2) gap> l:=SemisimpleCentralizerRepresentatives(W); [ [ ], [ 1 ], [ 1, 2 ], [ 1, 5 ], [ 2 ], [ 2, 6 ] ] gap> List(last,h->ReflectionName(ReflectionSubgroup(W,h))); [ "(q-1)^2", "A1.(q-1)", "G2", "A2<1,5>", "~A1<2>.(q-1)", "~A1<2>xA1<6>" ] gap> SemisimpleCentralizerRepresentatives(W,2); [ [ ], [ 1 ], [ 1, 2 ], [ 1, 5 ], [ 2 ] ]
gap3-jm