Related Structures

In this section functions for creating other structures from a root system are briefly listed. The reader is referred 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. See Chapter ChapRootDtm

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

The Coxeter group (of type grpcat) of a root system \(R\). 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): RootSys → GrpPermCox

WeylGroup(R): RootSys → GrpPermCox

The permutation Coxeter group with root system \(R\). See Chapter ChapGrpCox.

CoxeterGroup(GrpPermCox, R): Cat, RootSys → RngIntElt

ReflectionGroup(R): RootSys → GrpMat

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

CoxeterGroup(GrpPermCox, W): Cat, RootSys → GrpPermCox

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

The Lie algebra of the root system \(R\) over the base ring \(k\). See Chapter ChapAlgLie.

MatrixLieAlgebra(R, k): RootSys → GrpMat

The matrix Lie algebra of the root system \(R\) over the base ring \(k\). See Chapter ChapAlgLie.

Example: Related

> R := RootSystem("b3");
> SemisimpleType(LieAlgebra(R, Rationals()));
B3
> #CoxeterGroup(R);
48
%%a> assert $1 eq 48;