====== BigObject Action ======
//from application [[..:group|group]]//\\
\\
 parametrized object used for encoding group actions (representations).
   * //GeneratorType// is 
     * ''Array<Int>'' for permutation groups, or 
     * ''Matrix<Scalar>'' for matrix groups.
   * //DomainType// is 
     * ''Int'' for permutation groups,
     * ''Vector<Scalar> ''for matrix groups acting on facets or vertices,
     * ''Set<Int>'' or ''Bitset'' for groups acting on simplices,
     * ''IncidenceMatrix'' for groups acting on (maximal) cones of a fan
   * //OrbitGeneratorScalarType// is the type of the entries of vectors that generate a geometric orbit; default ''Rational''

===== Properties =====

==== Symmetry ====
 These properties capture information of the object that is concerned with the action of permutation groups.
----
{{anchor:all_group_elements:}}
  ?  **''ALL_GROUP_ELEMENTS''**
  :: All elements of the group, as expressed in the present action Notice that this is a temporary property; it will not be stored in any file.
    ? Type:
    :''[[..:common#Array |Array]]<GeneratorType>''


----
{{anchor:character:}}
  ?  **''CHARACTER''**
  :: The character of the action. The ordering corresponds to the columns of the CHARACTER_TABLE
    ? Type:
    :''[[..:common#Vector |Vector]]<[[..:common#QuadraticExtension |QuadraticExtension]]<[[..:common#Rational |Rational]]%%>>%%''


----
{{anchor:conjugacy_classes:}}
  ?  **''CONJUGACY_CLASSES''**
  :: The conjugacy classes themselves
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Set |Set]]<GeneratorType%%>>%%''


----
{{anchor:conjugacy_class_representatives:}}
  ?  **''CONJUGACY_CLASS_REPRESENTATIVES''**
  :: A set of representatives for each conjugacy class. The order of these representatives must agree with the implicit order of the columns of the [[..:group:Group#CHARACTER_TABLE |CHARACTER_TABLE]].
    ? Type:
    :''[[..:common#Array |Array]]<GeneratorType>''


----
{{anchor:degree:}}
  ?  **''DEGREE''**
  :: The degree of the representation. For permutation groups, this is the number of permuted indices, for matrix groups it is the dimension of the vector space the group acts on
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:domain_name:}}
  ?  **''DOMAIN_NAME''**
  :: the name of the property that we act on, for example MAX_INTERIOR_SIMPLICES or INTERIOR_RIDGES
    ? Type:
    :''[[..:common#String |String]]''


----
{{anchor:equations_generators:}}
  ?  **''EQUATIONS_GENERATORS''**
  :: A set of generators for equations, stored as the rows of a matrix. The list of generators may be redundant and non-canonical.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<OrbitGeneratorScalarType,[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:explicit_orbit_representatives:}}
  ?  **''EXPLICIT_ORBIT_REPRESENTATIVES''**
  :: the representatives of orbits explicitly, not via their indices
    ? Type:
    :''[[..:common#Array |Array]]<DomainType>''


----
{{anchor:explicit_orbit_representative_matrix:}}
  ?  **''EXPLICIT_ORBIT_REPRESENTATIVE_MATRIX''**
  :: the representatives of orbits explicitly, not via their indices
    ? Type:
    :''[[..:common#Matrix |Matrix]]<OrbitGeneratorScalarType,[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:facets_generators:}}
  ?  **''FACETS_GENERATORS''**
  :: A set of generators for facets, stored as the rows of a matrix. The list of generators may be redundant and non-canonical.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<OrbitGeneratorScalarType,[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:generators:}}
  ?  **''GENERATORS''**
  :: The generators of the group action
    ? Type:
    :''[[..:common#Array |Array]]<GeneratorType>''


----
{{anchor:images:}}
  ?  **''IMAGES''**
  :: The images of all domain elements under each group element: [ [ g(x) for x in D ] for g in G ]
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Array |Array]]<DomainType%%>>%%''


----
{{anchor:index_of:}}
  ?  **''INDEX_OF''**
  :: The map giving the index of each group element This is a temporary property; it will not be stored in any file.
    ? Type:
    :''[[..:common#HashMap |HashMap]]<DomainType,[[..:common#Int |Int]]>''


----
{{anchor:induced_from:}}
  ?  **''INDUCED_FROM''**
  :: the name of a different action that this action is induced from
    ? Type:
    :''[[..:common#String |String]]''


----
{{anchor:inequalities_generators:}}
  ?  **''INEQUALITIES_GENERATORS''**
  :: A set of generators for inequalities, stored as the rows of a matrix. The list of generators may be redundant and non-canonical.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<OrbitGeneratorScalarType,[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:input_lineality_generators:}}
  ?  **''INPUT_LINEALITY_GENERATORS''**
  :: A set of generators for input lineality, stored as the rows of a matrix. The list of generators may be redundant and non-canonical.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<OrbitGeneratorScalarType,[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:input_rays_generators:}}
  ?  **''INPUT_RAYS_GENERATORS''**
  :: A set of generators for input rays, stored as the rows of a matrix. The list of generators may be redundant and non-canonical.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<OrbitGeneratorScalarType,[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:irreducible_decomposition:}}
  ?  **''IRREDUCIBLE_DECOMPOSITION''**
  :: The multiplicities of each irreducible representation in this action. The ordering corresponds to the rows of the CHARACTER_TABLE
    ? Type:
    :''[[..:common#Vector |Vector]]<[[..:common#Int |Int]]>''


----
{{anchor:lineality_space_generators:}}
  ?  **''LINEALITY_SPACE_GENERATORS''**
  :: A set of generators for input lineality, stored as the rows of a matrix. The list of generators may be redundant and non-canonical.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<OrbitGeneratorScalarType,[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:maximal_cones_generators:}}
  ?  **''MAXIMAL_CONES_GENERATORS''**
  :: A set of generators for the maximal cones of a fan, stored in terms of indices of vertex generators. The list of generators may be redundant and non-canonical.
    ? Type:
    :''[[..:common#IncidenceMatrix |IncidenceMatrix]]<[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:n_equations_generators:}}
  ?  **''N_EQUATIONS_GENERATORS''**
  :: The number of generators for orbits.
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:n_facets_generators:}}
  ?  **''N_FACETS_GENERATORS''**
  :: The number of generators for orbits.
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:n_inequalities_generators:}}
  ?  **''N_INEQUALITIES_GENERATORS''**
  :: The number of generators for orbits.
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:n_input_lineality_generators:}}
  ?  **''N_INPUT_LINEALITY_GENERATORS''**
  :: The number of generators for orbits.
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:n_input_rays_generators:}}
  ?  **''N_INPUT_RAYS_GENERATORS''**
  :: The number of generators for orbits.
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:n_lineality_space_generators:}}
  ?  **''N_LINEALITY_SPACE_GENERATORS''**
  :: The number of generators for orbits.
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:n_maximal_cones_generators:}}
  ?  **''N_MAXIMAL_CONES_GENERATORS''**
  :: The number of generators for orbits.
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:n_orbits:}}
  ?  **''N_ORBITS''**
  :: The number of orbits in the domain under the group action
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:n_orbit_representatives:}}
  ?  **''N_ORBIT_REPRESENTATIVES''**
  :: the number of representatives of orbits
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:n_rays_generators:}}
  ?  **''N_RAYS_GENERATORS''**
  :: The number of generators for orbits.
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:orbits:}}
  ?  **''ORBITS''**
  :: The orbits of the domain, represented via their indices
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%%''


----
{{anchor:orbit_representatives:}}
  ?  **''ORBIT_REPRESENTATIVES''**
  :: A collection of representatives for each orbit, represented via their indices
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Int |Int]]>''


----
{{anchor:orbit_representative_labels:}}
  ?  **''ORBIT_REPRESENTATIVE_LABELS''**
  :: Labels for the orbit representatives
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#String |String]]>''


----
{{anchor:orbit_sizes:}}
  ?  **''ORBIT_SIZES''**
  :: The cardinality of each orbit
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Int |Int]]>''


----
{{anchor:permutation_to_orbit_order:}}
  ?  **''PERMUTATION_TO_ORBIT_ORDER''**
  :: A permutation that orders the domain elements by orbits
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Int |Int]]>''


----
{{anchor:rays_generators:}}
  ?  **''RAYS_GENERATORS''**
  :: A set of generators for rays, stored as the rows of a matrix. The list of generators may be redundant and non-canonical.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<OrbitGeneratorScalarType,[[..:common#NonSymmetric |NonSymmetric]]>''


----