Related Structures

In this section we briefly list functions for creating other structures from Coxeter matrices, Coxeter graphs, Cartan matrices, Dynkin diagrams, and Cartan names. Refer 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 N. If the corresponding Coxeter group is infinite, an error is flagged. See Chapter ROOT SYSTEMS.

RootDatum( C ) : AlgMatElt -> RootDtm

RootDatum( D ) : GrphDir -> RootDtm

RootDatum( N ) : MonStgElt -> RootDtm

The finite root datum of a crystallographic Cartan matrix C, Dynkin digraph D, or Cartan name N. If the corresponding Coxeter group is infinite, an error is flagged. See Chapter ROOT DATA.

CoxeterGroup( GrpFPCox, M ) : Cat, AlgMatElt -> GrpFPCox

CoxeterGroup( GrpFPCox, G ) : Cat, GrphUnd -> GrpFPCox

CoxeterGroup( GrpFPCox, C ) : Cat, AlgMatElt -> GrpFPCox

CoxeterGroup( GrpFPCox, D ) : Cat, GrphDir -> GrpFPCox

CoxeterGroup( GrpFPCox, N ) : Cat, MonStgElt -> GrpFPCox

The Coxeter group of a Coxeter matrix M, Coxeter graph G, Cartan matrix C, Dynkin digraph D, or Cartan name N. See Chapter COXETER GROUPS.

CoxeterGroup( GrpPermCox, M ) : Cat, AlgMatElt -> GrpPermCox

CoxeterGroup( GrpPermCox, G ) : Cat, GrphUnd -> GrpPermCox

CoxeterGroup( GrpPermCox, C ) : Cat, AlgMatElt -> GrpPermCox

CoxeterGroup( GrpPermCox, D ) : Cat, GrphDir -> GrpPermCox

CoxeterGroup( GrpPermCox, N ) : Cat, MonStgElt -> GrpPermCox

The permutation Coxeter group of a Coxeter matrix M, Coxeter graph G, Cartan matrix C, Dynkin digraph D, or Cartan name N. If the corresponding Coxeter group is infinite, an error is flagged. See Chapter COXETER GROUPS AS PERMUTATION GROUPS.

CoxeterGroup( M ) : Cat, AlgMatElt -> GrpPermCox

CoxeterGroup( G ) : Cat, GrphUnd -> GrpPermCox

CoxeterGroup( C ) : Cat, AlgMatElt -> GrpPermCox

CoxeterGroup( D ) : Cat, GrphDir -> GrpPermCox

CoxeterGroup( N ) : Cat, MonStgElt -> GrpPermCox

The Coxeter group of a Coxeter matrix M, Coxeter graph G, Cartan matrix C, Dynkin digraph D, or Cartan name N. If the corresponding 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 N. See Chapter REFLECTION GROUPS.

LieAlgebra( C, k ) : AlgMatElt -> AlgLie

LieAlgebra( D, k ) : GrphDir -> AlgLie

LieAlgebra( N, k ) : MonStgElt -> AlgLie

The Lie algebra over the field k of a crystallographic Cartan matrix C, Dynkin digraph D, or Cartan name N. If the corresponding Coxeter group is infinite, an error is flagged. See Chapter LIE ALGEBRAS.

GroupOfLieType( C, k ) : AlgMatElt -> GrpLie

GroupOfLieType( D, k ) : GrphDir -> GrpLie

GroupOfLieType( N, k ) : MonStgElt -> GrpLie

The group of Lie type over the field k of a crystallographic Cartan matrix C, Dynkin digraph D, or Cartan name N. If the corresponding Coxeter group is infinite, an error is flagged. See Chapter GROUPS OF LIE TYPE.