In this section functions for creating other structures from a permutation Coxeter group are briefly listed. See the appropriate chapters of the Handbook for more details.
The finitely presented group isomorphic to the permutation Coxeter group W. See Chapter FINITELY PRESENTED GROUPS.
The reflection group isomorphic to the Coxeter group W. See Chapter REFLECTION GROUPS.
The reductive Lie algebraover the ring R with Weyl group W. If W is noncrystallographic, an error is flagged. See Section Almost Reductive Lie Algebras.
The group of Lie typeover the ring R with Weyl group W. The roots and coroots of W must have integral components. See Chapter GROUPS OF LIE TYPE.