Related Structures

In this section functions for creating other structures from Coxeter matrices, Coxeter graphs, Cartan matrices, Dynkin diagrams, and Cartan names are listed. The reader is referred to the appropriate sections of the Handbook for more details.

RootSystem(M): AlgMatElt → RootSys

RootSystem(G): GrphUnd → RootSys

RootSystem(C): AlgMatElt → RootSys

RootSystem(D): GrphDir → RootSys

RootSystem(N): MonStgElt → RootSys

The finite root system of a Coxeter matrix \(M\), Coxeter graph \(G\), Cartan matrix \(C\), Dynkin digraph \(D\), or Cartan name given by the string \(N\). If the corresponding Coxeter group is infinite, an error is flagged. See Chapter ChapRootSys.

RootDatum(C): AlgMatElt → RootDtm

RootDatum(M): AlgMatElt → RootDtm

RootDatum(G): GrphUnd → RootDtm

RootDatum(D): GrphDir → RootDtm

RootDatum(N): MonStgElt → RootDtm

The finite root datum of a crystallographic Cartan matrix \(C\), Coxeter matrix \(M\), Coxeter graph \(G\), Dynkin digraph \(D\), or Cartan name given by the string \(N\). If the corresponding Coxeter group is infinite, an error is flagged. See Chapter ChapRootDtm.

CoxeterGroup(grpcat, M): Cat, AlgMatElt → grpcat

CoxeterGroup(grpcat, G): Cat, GrphUnd → grpcat

CoxeterGroup(grpcat, D): Cat, GrphDir → grpcat

CoxeterGroup(grpcat, N): Cat, MonStgElt → grpcat

The Coxeter group in the category given by the first argument, which can be (a reflection group), , GrpFP (a finitely presented group), or GrpPerm (a permutation group).

The second argument can be a Coxeter matrix, a Cartan matrix, a Coxeter graph, a Dynkin digraph, or a Cartan name given by a string.

If the first argument is or GrpPerm and the corresponding Coxeter group is infinite, an error is flagged. See Chapter ChapGrpCox.

CoxeterGroup(M): AlgMatElt → Grp

CoxeterGroup(C): AlgMatElt → Grp

CoxeterGroup(G): GrphUnd → Grp

CoxeterGroup(D): GrphDir → Grp

CoxeterGroup(N): MonStgElt → Grp

The Coxeter group of a Coxeter matrix, a Cartan matrix, Coxeter graph, a Dynkin digraph or the Cartan name given by a string. If the resulting Coxeter group is finite, it is returned as a permutation group; otherwise it is returned as a finitely presented group.

ReflectionGroup(M): AlgMatElt → GrpMat

ReflectionGroup(G): GrphUnd → GrpMat

ReflectionGroup(C): AlgMatElt → GrpMat

ReflectionGroup(D): GrphDir → GrpMat

ReflectionGroup(N): MonStgElt → GrpMat

The reflection group of a Coxeter matrix \(M\), Coxeter graph \(G\), Cartan matrix \(C\), Dynkin digraph \(D\), or Cartan name given by the string \(N\). See Chapter ChapGrpRfl.

LieAlgebra(C, k): AlgMatElt, Rng → AlgLie

LieAlgebra(D, k): GrphDir, Rng → AlgLie

LieAlgebra(N, k): MonStgElt, Rng → AlgLie

Isogeny : Any                          Default: ``Ad"

The Lie algebra over the ring \(k\) of a crystallographic Cartan matrix \(C\), Dynkin digraph \(D\), or Cartan name given by the string \(N\). The Lie algebra is constructed from a root datum with the given Isogeny (see Subsection Isogeny of Split Reduced Root Data). If the corresponding Coxeter group is infinite, an error is flagged. See Chapter ChapAlgLie.

MatrixLieAlgebra(C, k): AlgMatElt, Rng → AlgLie

MatrixLieAlgebra(D, k): GrphDir, Rng → AlgLie

MatrixLieAlgebra(N, k): MonStgElt, Rng → AlgLie

The Lie algebra over the ring \(k\) of a crystallographic Cartan matrix \(C\), Dynkin digraph \(D\), or Cartan name given by the string \(N\). If the corresponding Coxeter group is infinite, an error is flagged. See Chapter ChapAlgLie.

GroupOfLieType(C, k): AlgMatElt, Rng → GrpLie

GroupOfLieType(D, k): GrphDir, Rng → GrpLie

GroupOfLieType(N, k): MonStgElt, Rng → GrpLie

The group of Lie type over the ring \(k\) of a crystallographic Cartan matrix \(C\), Dynkin digraph \(D\), or Cartan name given by the string \(N\). If the corresponding Coxeter group is infinite, an error is flagged. See Chapter ChapGrpLie.