Operations on groups of Lie type

Many of the basic operations for Coxeter groups are shortcuts for obtaining information about the underlying root datum (Chapter ROOT DATA FOR LIE THEORY). We list these functions here; see Sections Operators on root data, Properties of root data, Roots, coroots and weights, and Operations on Coxeter groups and elements for more details and examples of their use.

G eq H : GrpLie, GrpLie -> BoolElt

True if G and H are equal.

IsAlgebraicallyIsomorphic( G, H ) : GrpLie, GrpLie -> BoolElt

True if G and H are isomorphic as algebraic groups (ie. they have the same base rings and isomorphic root data).

IsIsogenous( G, H ) : GrpLie, GrpLie -> BoolElt

True if G and H are isogenous.

Generators( G ) : GrpLie ->

Generators for the group of Lie type G as an abstract group. The base ring of G must have a finite set of additive generators (for example, if the base ring is a field, it must have finite degree over its prime field).

AlgebraicGenerators( G ) : GrpLie ->

A set of generators for the group of Lie type G as an algebraic group.

Example GrpLie_Generators (H37E3)

> k<z> := GF(4);
> G := GroupOfLieType( "G2", k );
> Generators(G);
[ x1(1) , x7(1) , x1(z) , x7(z) , x2(1) , x8(1) , x2(z) , x8(z)  ]
> AlgebraicGenerators(G);
[ x1(1) , x2(1) , x7(1) , x8(1)  ]

WeylGroup( G ) : GrpLie -> GrpCox

The Weyl group of the group of Lie type G. This is a crystallographic Coxeter group, see Chapter COXETER GROUPS.

BaseRing( G ) : GrpLie -> Rng

The base ring of the group of Lie type G.

RootDatum( G ) : GrpLie -> RootDtm

The root datum of the group of Lie type G.

Rank( G ) : GrpLie -> RngIntElt

ReductiveRank( G ) : GrpLie -> RngIntElt

The reductive rank of the group of Lie type G, ie. the dimension of the underlying root datum.

SemisimpleRank( G ) : GrpLie -> RngIntElt

The semisimple rank of the group of Lie type G, ie. the rank of the underlying root datum.

CartanMatrix( G ) : GrpLie -> AlgMatElt

The Cartan matrix of the group of Lie type G.

CartanName( G ) : GrpLie -> MonStgElt

The Cartan type of the group of Lie type G.

DynkinDiagram( G ) : GrpLie -> .

Print the Dynkin diagram of the group of Lie type G.

FundamentalGroup( G ) : GrpLie -> RootDtm

The fundamental group of the group of Lie type G.

IsogenyGroup( G ) : GrpLie -> RootDtm

The isogeny group of the group of Lie type G.

CoisogenyGroup( G ) : GrpLie -> RootDtm

The isogeny group of the group of Lie type G.

NumberOfPositiveRoots( G ) : GrpLie -> RngIntElt

NumPosRoots( G ) : GrpLie -> RngIntElt

The number of positive roots of the group of Lie type G.

Roots( G ) : GrpLie -> {@@}

Coroots( G ) : GrpLie -> {@@}

    basis: MonStgElt                    Default: "standard"

An indexed set containing the (co)roots of the group of Lie type G.

PositiveRoots( G ) : GrpLie -> {@@}

PositiveCoroots( G ) : GrpLie -> {@@}

    basis: MonStgElt                    Default: "standard"

An indexed set containing the positive (co)roots of the group of Lie type G.

Root( G, r ) : GrpLie, RngIntElt -> {@@}

Coroot( G, r ) : GrpLie, RngIntElt -> {@@}

    basis: MonStgElt                    Default: "standard"

The rth (co)root of the group of Lie type G.

RootPosition( G, v ) : GrpLie, . -> {@@}

CorootPosition( G, v ) : GrpLie, . -> {@@}

    basis: MonStgElt                    Default: "standard"

If v is a (co)root of the group of Lie type G, this returns its position; otherwise it returns 0.

CoxeterElement( G ) : GrpCox -> GrpPermElt

The Coxeter element of the group of Lie type G, ie. the representative of the Coxeter element in the Weyl group of G.

CoxeterNumber( G ) : GrpCox -> GrpPermElt

The Coxeter number of the simple group of Lie type G.

WeightLattice( G ) : RootDtm -> Lat

CoweightLattice( G ) : RootDtm -> Lat

The (co)weight lattice of the group of Lie type G.

FundamentalWeights( G ) : GrpLie -> SeqEnum

FundamentalCoweights( G ) : GrpLie -> SeqEnum

    basis: MonStgElt                    Default: "standard"

The fundamental (co)weights of the group of Lie type G.