.. _SectGrpCoxAction:

.. _action-root-coroot:

Standard Action
===============

Every finite Coxeter group :math:`W` has a standard action. For example, the
standard action group of a Coxeter group of type :math:`A_n` is the symmetric
group of degree :math:`n+1` acting on :math:`\{1,\dots,n\}`.

.. magma:function:: StandardAction(W)
   :input_types: GrpFPCox
   :output_types: Map
   :label: StandardAction_GrpFPCox

.. magma:function:: StandardAction(W)
   :input_types: GrpPermCox
   :output_types: Map
   :label: StandardAction_GrpPermCox

   The standard action of the finite Coxeter group :math:`W`.

.. magma:function:: StandardActionGroup(W)
   :input_types: GrpFPCox
   :output_types: GrpPerm, Map
   :label: StandardActionGroup_GrpFPCox

.. magma:function:: \name{IntrStandardActionGroup}{StandardActionGroup}(W)
   :input_types: GrpPermCox
   :output_types: GrpPerm, Map

   The group :math:`G` of the standard action of the finite Coxeter group
   :math:`W`, together with an isomorphism :math:`W\to G`.

.. magma:example:: Example: Standard Action
   :label: StandardAction

   .. code-block:: magma

      > W := CoxeterGroup("A3");
      > G, h := StandardActionGroup(W);
      > IsSymmetric(G);
      true
      %%a> assert $1;
      > h(W.1); h(W.2); h(W.3);          
      (1, 2)
      (2, 3)
      (3, 4)