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Reference documentation for older polymake versions: release 3.4, release 3.3, release 3.2

# BigObject Action

from application 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

### Symmetry

These properties capture information of the object that is concerned with the action of permutation groups.

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:
Array<GeneratorType>

CHARACTER

The character of the action. The ordering corresponds to the columns of the CHARACTER_TABLE

Type:
Vector<QuadraticExtension<Rational>>

CONJUGACY_CLASSES

The conjugacy classes themselves

Type:
Array<Set<GeneratorType>>

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 CHARACTER_TABLE.

Type:
Array<GeneratorType>

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:
Int

DOMAIN_NAME

the name of the property that we act on, for example MAX_INTERIOR_SIMPLICES or INTERIOR_RIDGES

Type:
String

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:
Matrix<OrbitGeneratorScalarType,NonSymmetric>

EXPLICIT_ORBIT_REPRESENTATIVES

the representatives of orbits explicitly, not via their indices

Type:
Array<DomainType>

EXPLICIT_ORBIT_REPRESENTATIVE_MATRIX

the representatives of orbits explicitly, not via their indices

Type:
Matrix<OrbitGeneratorScalarType,NonSymmetric>

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:
Matrix<OrbitGeneratorScalarType,NonSymmetric>

GENERATORS

The generators of the group action

Type:
Array<GeneratorType>

IMAGES

The images of all domain elements under each group element: [ [ g(x) for x in D ] for g in G ]

Type:
Array<Array<DomainType>>

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:
HashMap<DomainType,Int>

INDUCED_FROM

the name of a different action that this action is induced from

Type:
String

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:
Matrix<OrbitGeneratorScalarType,NonSymmetric>

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:
Matrix<OrbitGeneratorScalarType,NonSymmetric>

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:
Matrix<OrbitGeneratorScalarType,NonSymmetric>

IRREDUCIBLE_DECOMPOSITION

The multiplicities of each irreducible representation in this action. The ordering corresponds to the rows of the CHARACTER_TABLE

Type:
Vector<Int>

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:
Matrix<OrbitGeneratorScalarType,NonSymmetric>

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:
IncidenceMatrix<NonSymmetric>

N_EQUATIONS_GENERATORS

The number of generators for orbits.

Type:
Int

N_FACETS_GENERATORS

The number of generators for orbits.

Type:
Int

N_INEQUALITIES_GENERATORS

The number of generators for orbits.

Type:
Int

N_INPUT_LINEALITY_GENERATORS

The number of generators for orbits.

Type:
Int

N_INPUT_RAYS_GENERATORS

The number of generators for orbits.

Type:
Int

N_LINEALITY_SPACE_GENERATORS

The number of generators for orbits.

Type:
Int

N_MAXIMAL_CONES_GENERATORS

The number of generators for orbits.

Type:
Int

N_ORBITS

The number of orbits in the domain under the group action

Type:
Int

N_ORBIT_REPRESENTATIVES

the number of representatives of orbits

Type:
Int

N_RAYS_GENERATORS

The number of generators for orbits.

Type:
Int

ORBITS

The orbits of the domain, represented via their indices

Type:
Array<Set<Int>>

ORBIT_REPRESENTATIVES

A collection of representatives for each orbit, represented via their indices

Type:
Array<Int>

ORBIT_REPRESENTATIVE_LABELS

Labels for the orbit representatives

Type:
Array<String>

ORBIT_SIZES

The cardinality of each orbit

Type:
Array<Int>

PERMUTATION_TO_ORBIT_ORDER

A permutation that orders the domain elements by orbits

Type:
Array<Int>

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:
Matrix<OrbitGeneratorScalarType,NonSymmetric>

• documentation/latest/group/action.txt