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). Such functions are listed here; see Sections Operations on Root Data, Properties of Root Data, Roots, Coroots, and Weights, and Operations on Coxeter Groups for more details and examples of their use.

G eq H: GrpLie, GrpLie -> BoolElt: Returns true iff the groups of Lie type \(G\) and \(H\) are equal.

G subset H: GrpLie, GrpLie -> BoolElt: Returns true iff the group of Lie type \(G\) is a subset of \(H\).

IsAlgebraicallyIsomorphic(G, H): GrpLie, GrpLie -> BoolElt, Map: Returns true if the semisimple groups \(G\) and \(H\) are isomorphic as algebraic groups (i.e. they have the same base rings and isomorphic root data). If true, then the second value returned is an isomorphism.

IsIsogenous(G, H): GrpLie, GrpLie -> BoolElt: Returns true if \(G\) and \(H\) are isogenous. The groups must be semisimple and defined over the same field. If true, the subsequent values returned are: the corresponding adjoint group \(G_{ad}\), the homomorphisms \(G_{ad}\to G\) and \(G_{ad}\to H\), the corresponding simply connected root datum \(G_{sc}\), and the homomorphisms \(G\to G_{sc}\) and \(H\to G_{sc}\).

IsCartanEquivalent(G, H): GrpLie, GrpLie -> BoolElt: Returns true if, and only if, the groups of Lie type \(G\) and \(H\) are Cartan equivalent, i.e. they have isomorphic Dynkin diagrams and defined over the same ring.

BaseRing(G): GrpLie -> Rng
CoefficientRing(G): GrpLie -> Rng: The base ring \(k\) of the group of Lie type \(G\).

BaseExtend(G, K): GrpLie, Rng -> GrpLie, Map: Given a group of Lie type \(G\) with base ring \(k\) and a larger ring \(K\), return the group \(G(K)\) gotten by extending the base ring and the injection \(G\to G(K)\).

ChangeRing(G, K): GrpLie, Rng -> GrpLie: Given a group of Lie type \(G\) and a ring \(K\), return the group with the same root datum, but defined over a different ring.

Generators(G): GrpLie -> {}: Generators for the group of Lie type \(G\) as an abstract group. This is currently only implemented when the base ring is a finite field.

NumberOfGenerators(G): GrpLie -> RngIntElt
Ngens(G): GrpLie -> RngIntElt: The number of generators for the group of Lie type \(G\) as an abstract group. This is currently only implemented when the base ring is a finite field.

AlgebraicGenerators(G): GrpLie -> {}: A set of generators for the group of Lie type \(G\) as an algebraic group.

NumberOfAlgebraicGenerators(G): GrpLie -> RngIntElt
Nalggens(G): GrpLie -> RngIntElt: The number of generators for the group of Lie type \(G\) as an algebraic group.

Example: Generators of a Group of Lie Type (ex-a929c8)

> k<z> := GF(4);
> G := GroupOfLieType("A2", k : Normalising:=false);
> Generators(G);
[ x1(1) , x4(1) , x1(z) , x4(z) , x2(1) , x5(1) , x2(z) , x5(z) , ( z   1) ,
(1   z)  ]
> AlgebraicGenerators(G);
[ x1(1) , x2(1) , x4(1) , x5(1) , ( z   1) , ( 1   z)  ]

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Order(G): GrpLie -> RngIntElt
# G: GrpLie -> RngIntElt: The order of the group of Lie type \(G\).

FactoredOrder(G): GrpLie -> RngIntElt: The factored order of the group of Lie type \(G\).

Dimension(G): GrpLie -> RngIntElt: The dimension of the group of Lie type \(G\), considered as an algebraic variety.

Example: Orders of Groups of Lie Type (ex-1128fd)

> G := GroupOfLieType("G2", 3);
> Order(G);
4245696
> FactoredOrder(G);
[ <2, 8>, <13, 1>, <3, 6>, <7, 1> ]
> G := GroupOfLieType("G2", Rationals());
> Order(G);
Infinity
> Dimension(G);
14

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CartanName(G): GrpLie -> Mtrx: The Cartan name of the group of Lie type \(G\).

RootDatum(G): GrpLie -> RootDtm: The root datum of the group of Lie type \(G\).

DynkinDiagram(G): GrpLie: Print the Dynkin diagram of the group of Lie type \(G\).

CoxeterDiagram(G): GrpLie: Print the Coxeter diagram of the group of Lie type \(G\).

CoxeterMatrix(G): GrpLie -> AlgMatElt: The Coxeter matrix of the group of Lie type \(G\).

CoxeterGraph(G): GrpLie -> GrphUnd: The Coxeter graph of the group of Lie type \(G\).

CartanMatrix(G): GrpLie -> GrphUnd: The Cartan matrix of the group of Lie type \(G\).

DynkinDigraph(G): GrpLie -> GrphUnd: The Dynkin digraph of the group of Lie type \(G\).

Rank(G): GrpLie -> RngIntElt
ReductiveRank(G): GrpLie -> RngIntElt: The reductive rank of the group of Lie type \(G\), i.e. the dimension of the underlying root datum.

SemisimpleRank(G): GrpLie -> RngIntElt: The semisimple rank of the group of Lie type \(G\), i.e. the rank of the underlying root datum.

CoxeterNumber(G): GrpLie -> RngIntElt: The Coxeter number of the group of Lie type \(G\), i.e. the order of the Coxeter element in the Weyl group of \(G\).

WeylGroup(G): GrpLie -> GrpPermCox
WeylGroup(GrpPermCox, G): Cat, GrpLie -> GrpPermCox: The Weyl group of the group of Lie type \(G\) as a permutation Coxeter group. This is a crystallographic Coxeter group, see Chapter Coxeter Groups.

WeylGroup(GrpFPCox, G): Cat, GrpLie -> GrpFPCox: The Weyl group of the group of Lie type \(G\) as a finitely presented Coxeter group. This is a crystallographic Coxeter group, see Chapter Coxeter Groups.

WeylGroup(GrpMat, G): Cat, GrpLie -> GrpMat: The Weyl group of the group of Lie type \(G\) as a reflection group. This is a crystallographic Coxeter group, see Chapter Reflection Groups.

FundamentalGroup(G): GrpLie -> GrpAb, Map: The fundamental group of the group of Lie type \(G\), together with the projection of the weight lattice onto the fundamental group.

IsogenyGroup(G): GrpLie -> GrpAb, Map: The isogeny group of the group of Lie type \(G\), together with its injection into the fundamental group.

CoisogenyGroup(G): GrpLie -> GrpAb, Map: The coisogeny group of the group of Lie type \(G\), together with its projection onto the fundamental group.