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 ROOT SYSTEMS.
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 ROOT DATA.
CoxeterGroup(GrpMat, M) : Cat, AlgMatElt -> GrpMat
The Coxeter group (as a matrix group) of a Coxeter or Cartan matrix M. There are variations of this signature. The first argument can be GrpMat, GrpPermCox, GrpPerm, GrpFPCox or GrpFP and 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 GrpPermCox or GrpPerm and the corresponding Coxeter group is infinite, an error is flagged. See Chapter COXETER GROUPS.

CoxeterGroup(M) : Cat, AlgMat -> Grp
The Coxeter group of a Coxeter matrix or a Cartan matrix. If the corresponding Coxeter group is finite, it is returned as a permutation group of type GrpPermCox; otherwise it is returned as a finitely presented group of type GrpFPCox.

There are variations of this signature. The argument can be a Coxeter matrix, a Cartan matrix, a Coxeter graph, a Dynkin digraph, or a Cartan name given by a string.

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
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
CoxeterGroup(M) : AlgMatElt -> GrpPermCox
CoxeterGroup(G) : GrphUnd -> GrpPermCox
CoxeterGroup(C) : AlgMatElt -> GrpPermCox
CoxeterGroup(D) : GrphDir -> GrpPermCox
CoxeterGroup(N) : MonStgElt -> GrpPermCox
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 REFLECTION GROUPS.
LieAlgebra(C, k) : AlgMatElt, Rng -> AlgLie
LieAlgebra(D, k) : GrphDir, Rng -> AlgLie
LieAlgebra(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 LIE ALGEBRAS.
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 LIE ALGEBRAS.
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 GROUPS OF LIE TYPE.
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