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.
The root system corresponding to the root datum R. See Chapter ROOT SYSTEMS.
The Coxeter group (as a matrix group) 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. (See Chapter COXETER GROUPS.) 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.
The permutation Coxeter group with root datum R. See Chapter COXETER GROUPS.
The reflection group of the root datum R. See Chapter REFLECTION GROUPS.
The homomorphism of reductive Lie algebras over the ring k corresponding to the root datum morphism φ. See Chapter LIE ALGEBRAS.
The reductive Lie algebraover the ring k with root datum R. See Chapter LIE ALGEBRAS.
The group of Lie typeover the ring k with root datum R. See Chapter GROUPS OF LIE TYPE.
The algebraic homomorphism of groups of Lie type over the ring k corresponding to the root datum morphism φ. See Chapter GROUPS OF LIE TYPE.
> R := RootDatum("b3");
> SemisimpleType(LieAlgebra(R, Rationals()));
B3
> #CoxeterGroup(R);
48
> GroupOfLieType(R, Rationals());
$: Group of Lie type B3 over Rational Field