Related Structures

In this section functions for creating other structures from a root datum are briefly listed. See the appropriate chapters of the Handbook for more details.

RootSystem(R): RootDtm → RootSys

The root system corresponding to the root datum \(R\). See Chapter ChapRootSys.

CoxeterGroup(grpcat, R): Cat, RootDtm → grpcat

The Coxeter group (of type grpcat) of a root datum \(R\). There are variations of this signature. The first argument can be GrpMat, GrpPermCox, GrpPerm, GrpFPCox or GrpFP and the second argument can be a root system or root datum. (See Chapter ChapGrpCox.) If the first argument is GrpFPCox the braid group and pure braid group can be computed from the Coxeter group using the commands in Section Braid Groups.

CoxeterGroup(R): RootDtm → GrpPermCox

WeylGroup(R): RootDtm → GrpPermCox

The permutation Coxeter group with root datum \(R\). See Chapter ChapGrpPermCox.

CoxeterGroup(GrpPermCox, R): Cat, RootDtm → GrpPermCox

ReflectionGroup(R): RootDtm → GrpMat

The reflection group of the root datum \(R\). See Chapter ChapGrpRfl.

LieAlgebraHomorphism(phi, k): Map, Rng → AlgLie

The homomorphism of reductive Lie algebras over the ring \(k\) corresponding to the root datum morphism \(\phi\). See Chapter ChapAlgLie.

LieAlgebra(R, k): RootDtm, Rng → AlgLie

The reductive Lie algebra over the ring \(k\) with root datum \(R\). See Chapter ChapAlgLie.

GroupOfLieType(R, k): RootDtm, Rng → GrpLie

The group of Lie type over the ring \(k\) with root datum \(R\). See Chapter ChapGrpLie.

GroupOfLieTypeHomomorphism(phi, k): Map, Rng → GrpLie

The algebraic homomorphism of groups of Lie type over the ring \(k\) corresponding to the root datum morphism \(\phi\). See Chapter ChapGrpLie.

Example: Related

> R := RootDatum("b3");
> SemisimpleType(LieAlgebra(R, Rationals()));
B3
> #CoxeterGroup(R);
48
%%a> assert $1 eq 48;
> GroupOfLieType(R, Rationals());
\$: Group of Lie type B3 over Rational Field