.. _SectGrpCoxRelated:

.. _related:

Related Structures
==================

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.

.. magma:function:: CoxeterGroup(GrpFP, W)
   :input_types: Cat, GrpPermCox
   :output_types: GrpFPCox
   :label: CoxeterGroup_Cat_GrpPermCox

.. magma:function:: Presentation(W)
   :input_types: GrpPermCox
   :output_types: GrpFPCox
   :label: Presentation_GrpPermCox

   The finitely presented group isomorphic to the permutation Coxeter group
   :math:`W`. See Chapter :ref:`ChapGrpFP`.

.. magma:function:: ReflectionGroup(W)
   :input_types: GrpPermCox
   :output_types: GrpMat
   :label: ReflectionGroup_GrpPermCox

.. magma:function:: CoxeterGroup(GrpMat, W)
   :input_types: Cat, GrpPermCox
   :output_types: GrpFPCox
   :label: CoxeterGroup_Cat_GrpPermCox_2

   The reflection group isomorphic to the Coxeter group :math:`W`. See
   Chapter :ref:`ChapGrpRfl`.

.. magma:function:: LieAlgebra(W, R)
   :input_types: GrpPermCox, Rng
   :output_types: AlgLie
   :label: LieAlgebra_GrpPermCox_Rng

   The reductive Lie algebra over the ring :math:`R` with Weyl group :math:`W`. If
   :math:`W` is noncrystallographic, an error is flagged. See
   Section :ref:`SubsectAlgLieConstrRed`.

.. magma:function:: GroupOfLieType(W, R)
   :input_types: GrpPermCox, Rng
   :output_types: GrpLie
   :label: GroupOfLieType_GrpPermCox_Rng

   The group of Lie type over the ring :math:`R` with Weyl group :math:`W`. The
   roots and coroots of :math:`W` must have integral components. See
   Chapter :ref:`ChapGrpLie`.