Related Structures

In this section we briefly list functions for creating other structures from a root system. Refer to the appropriate chapters of the Handbook for more details.

RootDatum( R ) : RootSys -> RootDtm

The (split) root datum corresponding to the root system R. The coefficients of the simple roots and coroots must be integral; otherwise an error is signalled.

CoxeterGroup( GrpFPCox, R ) : Cat, RootSys -> RngIntElt

The Coxeter group with root system R. See Chapter COXETER GROUPS. The braid group and pure braid group can be computed from the Coxeter group using the commands in the Section Braid Groups.

CoxeterGroup( R ) : RootSys -> RngIntElt

CoxeterGroup( GrpPermCox, R ) : Cat, RootSys -> RngIntElt

The permutation Coxeter group with root system R. See Chapter COXETER GROUPS AS PERMUTATION GROUPS.

ReflectionGroup( R ) : RootSys -> GrpMat

CoxeterGroup( GrpMat, W ) : Cat, RootSys -> GrpPermCox

The reflection group of the root system R. See Chapter REFLECTION GROUPS.

LieAlgebra( R, k ) : RootSys -> GrpMat

The Lie algebra of the root system R over the base ring k. See Chapter LIE ALGEBRAS.

Example RootSys_Related (H79E17)

> R := RootSystem( "b3" );
> SemiSimpleType( LieAlgebra( R, Rationals() ) );
B3
> #CoxeterGroup( R );
48