86 Algebraic groups and semi-simple elements

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 -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 -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 ] ] );
    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

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 , 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);
    gap> L:=ReflectionSubgroup(G,[1,3]);
    ReflectionSubgroup(RootDatum("sl",4), [ 1, 3 ])
    gap> AlgebraicCentre(L);
      Z0 :=
       SubTorus(ReflectionSubgroup(RootDatum("sl",4), [ 1, 3 ]),[ [ 1, 2, \ 
    1 ] ]),
      AZ := Group( <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;
    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));
    gap> s^G.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);
    gap> s:=SemisimpleElement(G,[0,1/2,0]);
    gap> Centralizer(G,s);

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).


  1. CoxeterGroup (extended form)
  2. RootDatum
  3. Dual for root Data
  4. Torus
  5. FundamentalGroup for algebraic groups
  6. IntermediateGroup
  7. SemisimpleElement
  8. Operations for semisimple elements
  9. Centralizer for semisimple elements
  10. SubTorus
  11. Operations for Subtori
  12. AlgebraicCentre
  13. SemisimpleSubgroup
  14. IsIsolated
  15. IsQuasiIsolated
  16. QuasiIsolatedRepresentatives
  17. SemisimpleCentralizerRepresentatives

86.1 CoxeterGroup (extended form)

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:


the matrix of simple roots


the matrix of simple coroots


the matrices (in row convention --- that is, the matrices operate from the right) of the simple reflections of the Coxeter group.

86.2 RootDatum


This function returns the root datum for the algebraic group described by type and rank. The types understood as of now are: "gl", "sl", "pgl", "sp", "so", "psp", "pso", "halfspin", "spin", "F4" and "G2".

    gap> RootDatum("spin",8);# same as CoxeterGroup("D",4,"sc")

86.3 Dual for root Data


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);
    gap> Dual(W);

86.4 Torus


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);
    gap> ReflectionName(last);

86.5 FundamentalGroup for algebraic groups


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 expained in the context of non-irreducible groups for example in Bon05, \S 3.B.

    gap> W:=CoxeterGroup("A",3);
    gap> FundamentalGroup(W);
    Group( ( 1, 2, 3,12) )
    gap> W:=CoxeterGroup("A",3,"sc");
    gap> FundamentalGroup(W);
    Group( () )

86.6 IntermediateGroup

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
    gap> FundamentalGroup(last);
    Group( ( 1, 2, 3,12) )
    gap> IntermediateGroup(W,[2]);# intermediate
    gap> FundamentalGroup(last);
    Group( ( 1, 3)( 2,12) )

86.7 SemisimpleElement


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:

the given list, taken Mod1 if its elements are rationals.

the parent of the group W.

    gap> G:=CoxeterGroup("A",3);;
    gap> s:=SemisimpleElement(G,[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.

86.8 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);
    gap> s:=SemisimpleElement(G,[0,1/2,0]);
    gap> t:=SemisimpleElement(G,[1/2,1/3,1/7]);
    gap> s*t;
    gap> t^3;
    gap> t^-1;
    gap> t^0;
    gap> String(t);

then a similar example with elements of GF(5):

    gap> s:=SemisimpleElement(G,Z(5)*[1,2,1]);
    gap> t:=SemisimpleElement(G,Z(5)*[2,3,4]);
    gap> s*t;
    gap> t^3;
    gap> t^-1;
    gap> t^0;
    gap> String(t);

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]);
    gap> s^G.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]);
    gap> s^G.roots[4];
    gap> s:=SemisimpleElement(G,Z(5)*[1,1,1]);
    gap> s^G.roots[4];

Frobenius( WF ):

If 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]);
    gap> F:=Frobenius(WF);
    function ( arg ) ... end
    gap> F(s);
    gap> F(s,-1);

86.9 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);
    gap> s:=SemisimpleElement(G,[0,1/2,0]);
    gap> Centralizer(G,s);

86.10 SubTorus


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
    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 ] ])

86.11 Operations for Subtori

The operation in can test if a semisimple element belongs to a subtorus:

    gap> W:=RootDatum("gl",4);;
    gap> r:=AlgebraicCentre(W);
      Z0 := SubTorus(RootDatum("gl",4),[ [ 1, 1, 1, 1 ] ]),
      AZ := Group( <0,0,0,0> ),
      descAZ := [ [ 1 ] ] )
    gap> SemisimpleElement(W,[1/4,1/4,1/4,1/4]) in r.Z0;

The operation Rank gives the rank of the subtorus:

    gap> Rank(r.Z0);

86.12 AlgebraicCentre

AlgebraicCentre( W )

W should be a Weyl group record, or an extended Weyl group record. This function returns a description of the centre Z of the algebraic group defined by W as a record with the following fields:

the neutral component Z0 of Z as a subtorus of T.

representatives of A(Z):=Z/Z0 given as a group of semisimple elements.

if W is not an extended Weyl group, describes the inclusion of A(Z) in the center pi of the corresponding simply connected group. It contains a list elements given as words in the generators of pi which generate A(Z).

    gap> G:=CoxeterGroup("A",3,"sc");;
    gap> L:=ReflectionSubgroup(G,[1,3]);
    ReflectionSubgroup(CoxeterGroup("A",3,"sc"), [ 1, 3 ])
    gap> AlgebraicCentre(L);
      Z0 :=
       SubTorus(ReflectionSubgroup(CoxeterGroup("A",3,"sc"), [ 1, 3 ]),[ [\ 
     1, 2, 1 ] ]),
      AZ := Group( <0,0,1/2> ),
      descAZ := [ [ 1, 2 ] ] )
    gap> G:=CoxeterGroup("A",3);;
    gap> s:=SemisimpleElement(G,[0,1/2,0]);;
    gap> Centralizer(G,s);
    gap> AlgebraicCentre(last);
      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.

86.13 SemisimpleSubgroup

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> ]

86.14 IsIsolated


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> ]

86.15 IsQuasiIsolated


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> ]

86.16 QuasiIsolatedRepresentatives


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> ]

86.17 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);
    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 ] ]

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18 Jun 2018