.. _SectGrpFPCoxRel:

.. _related:

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

In this section we briefly list functions for creating other structures from a
reflection group. See the appropriate chapters of the Handbook for more details.

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

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

.. magma:function:: CoxeterGroup(GrpPermCox, W)
   :input_types: Cat, GrpMat
   :output_types: GrpPermCox
   :label: CoxeterGroup_Cat_GrpMat_2

   The permutation Coxeter group isomorphic to the finite real reflection group
   :math:`W`. See Chapter :ref:`ChapGrpPermCox`.

.. magma:function:: LieAlgebra(W, R)
   :input_types: GrpMat, Rng
   :output_types: AlgLie
   :label: LieAlgebra_GrpMat_Rng

   The reductive Lie algebra over the ring :math:`R` with Weyl group :math:`W`.
   Unless :math:`W` is finite, real, and crystallographic, an error is flagged. See
   Section :ref:`SubsectAlgLieConstrRed`.

.. magma:function:: GroupOfLieType(W, k)
   :input_types: GrpMat, Rng
   :output_types: GrpLie
   :label: GroupOfLieType_GrpMat_Rng

   The group of Lie type over the field :math:`k` with Weyl group :math:`W`. Unless
   :math:`W` is finite, real, and crystallographic, an error is flagged. The roots
   and coroots of :math:`W` must also have integral components. See
   Chapter :ref:`ChapGrpLie`.