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.
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.
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.
The Coxeter group in the category given by the first argument, which can be GrpMat (a reflection group), GrpFPCox, GrpFP (a finitely presented group), GrpPermCox 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 GrpPermCox or GrpPerm and the corresponding Coxeter group is infinite, an error is flagged. See Chapter COXETER GROUPS.
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.
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.
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.
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.
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.