application: group

The application group provides basic functionality for working with permutation groups. An object 'Group' is defined by a list of generators describing the permutation of some indices. The object 'GroupOfCone' encapsulates symmetry information of a cone. It can act on the indices of rays (DOMAIN=>1), of facets(2), or on coordinates(3).

imports from: common

Objects

Group
A finite permutation group.
Properties of Group

BASE: common::Array<Int>

A base for STRONG_GENERATORS.

DEGREE: common::Int

The degree of the permutation representation.

GENERATORS: common::Array<Array<Int>>

Some generating set for the group. All generators must have the same length DEGREE. FIXME: Concrete description as second row of permutation representation

N_STRONG_GENERATORS: common::Int

The number of strong generators.

ORDER: common::Int

The order of the group.

STRONG_GENERATORS: common::Array<Array<Int>>

Strong generating set with respect to BASE.

TRANSVERSALS: common::Array<Array<Int>>

Transversals along the stabilizer chain.

TRANSVERSAL_SIZES: common::Array<Int>

The number of group elements per transversal.
GroupOfCone
UNDOCUMENTED

derived from: Group
Properties of GroupOfCone

Symmetric Cones

REPRESENTATIVE_FACETS: common::Matrix<Rational, NonSymmetric>

Representatives for the orbits of facets of the cone under this group action. No redundancies are allowed. FIXME: Could cause problems since representatives are not unique! The order should correspond to the order of the orbits of facets!

REPRESENTATIVE_LINEAR_SPAN: common::Matrix<Rational, NonSymmetric>

Representatives for the orbits of the linear span of the cone under this group action. No redundancies are allowed. FIXME: Could cause problems since representatives are not unique! The order should correspond to the order of the orbits of facets!

REPRESENTATIVE_RAYS: common::Matrix<Rational, NonSymmetric>

Representatives for the orbits of rays of the cone under this group action. No redundancies are allowed. FIXME: Could cause problems since representatives are not unique! How to do canonicalization? The order should correspond to the order of the orbits of rays!

Symmetric cones

COORDINATES_IN_ORBITS: common::Array<Set<Int>>

An array that contains a set of indices (of rays) for each orbit of coordinates.

DOMAIN: common::Int

Indicates on which domain the group action is defined, see Enum defined in 'apps/polytope/rules/permlib.rules' . OnRays=1: action on the (indices of the) rays OnFacets=2: action on the facets OnCoords=3: action on the coordinates

FACETS_IN_ORBITS: common::Array<Set<Int>>

An array that contains a set of indices (of facets) for each orbit of facets. The indices of the facets correspond to their indices in the polymake object.

INEQUALITIES_IN_ORBITS: common::Array<Set<Int>>

An array that contains a set of indices (of inequalities) for each orbit of inequalities. The indices of the inequalities correspond to their indices in the polymake object.

INPUT_RAYS_IN_ORBITS: common::Array<Set<Int>>

An array that contains a set of indices (of input rays) for each orbit of rays. The indices of the input rays correspond to their indices in the polymake object.

N_ORBITS_OF_COORDINATES: common::Int

The number of orbits of coordinates.

N_ORBITS_OF_FACETS: common::Int

The number of orbits of facets.

N_ORBITS_OF_INEQUALITIES: common::Int

The number of orbits of inequalities.

N_ORBITS_OF_INPUT_RAYS: common::Int

The number of orbits of input rays.

N_ORBITS_OF_RAYS: common::Int

The number of orbits of rays.

N_REPRESENTATIVE_FACETS: common::Int

The number of facet representatives.

N_REPRESENTATIVE_RAYS: common::Int

The number of ray representatives.

RAYS_IN_ORBITS: common::Array<Set<Int>>

An array that contains a set of indices (of rays) for each orbit of rays. The indices of the rays correspond to their indices in the polymake object.

SYMMETRIC_FACETS: common::Matrix<Rational, NonSymmetric>

The symmetrized facets of the cone. Same order as in the FACETS-section. Explanation: The symmetry group of a cone also act on the facets. If the cone is not full-dimensional, however, the facets may appear unsymmetric. Nevertheless, there always exists some symmetric facet description.

SYMMETRIC_RAYS: common::Matrix<Rational, NonSymmetric>

The symmetrized rays of the cone. Same order as in the RAYS-section. Explanation: The symmetry group of a cone also act on the rays. If the cone has a non-trivial lineality space, however, the rays may appear unsymmetric. Nevertheless, there always exists some symmetric ray description.
GroupOfPolytope
UNDOCUMENTED

derived from: GroupOfCone
Properties of GroupOfPolytope

Symmetric cones

N_ORBITS_OF_POINTS: common::Int

The number of orbits of points. Alias for property N_ORBITS_OF_INPUT_RAYS.

N_ORBITS_OF_VERTICES: common::Int

The number of orbits of vertices. Alias for property N_ORBITS_OF_RAYS.

N_REPRESENTATIVE_VERTICES: common::Int

The number of vertex representatives. Alias for property N_REPRESENTATIVE_RAYS.

POINTS_IN_ORBITS: common::Array<Set<Int>>

An array that contains a set of indices (of points) for each orbit of points. The indices of the points correspond to their indices in the polymake object. Alias for property INPUT_RAYS_IN_ORBITS.

REPRESENTATIVE_AFFINE_HULL: common::Matrix<Rational, NonSymmetric>

Representatives for the orbits of linear span of the polytope under this group action. No redundancies are allowed. Alias for property REPRESENTATIVE_LINEAR_SPAN.

REPRESENTATIVE_VERTICES: common::Matrix<Rational, NonSymmetric>

Representatives for the orbits of vertices of the polytope under this group action. No redundancies are allowed. Alias for property REPRESENTATIVE_RAYS.

SYMMETRIC_VERTICES: common::Matrix<Rational, NonSymmetric>

The symmetrized vertices of the polytope. Same order as in the VERTICES-section. Explanation: The symmetry group of a polytope also act on the vertices. If the polytope has a non-trivial lineality space, however, the vertices may appear unsymmetric. Nevertheless, there always exists some symmetric vertex description.

VERTICES_IN_ORBITS: common::Array<Set<Int>>

An array that contains a set of indices (of vertices) for each orbit of vertices. The indices of the vertices correspond to their indices in the polymake object. Alias for property RAYS_IN_ORBITS.

User Functions

Basics
- group_from_generators (gens, group)
  
  Computes the basic properties 'base', 'strong generators', and 'transversals' and stores the result in a Group object.
  Parameters
  Array gens
  some generators of the group
  Group group
  to fill in the data
Orbits
- are_in_same_orbit (group, vec1, vec2) → bool
  
  Checks whether vector vec1 and vec2 are in the same orbit with respect to the (coordinate) action of group.
  Parameters
  Group group
  the permutation group acting on coordinates
  Vector vec1
  the first vector
  Vector vec2
  the second vector
  Returns
  bool
  true, if vec1 and vec2 are in the same orbit, false otherwise
- compute_complete_coord_act_orbit (group, mat) → ListReturn
  
  Computes the orbit of the set of all vectors of the matrix mat under group, which acts by permuting coordinates. The set of vectors does not have to be complete.
  Parameters
  Group group
  a group of coordinate permutations
  Matrix<Scalar> mat
  some input vectors
  Returns
  ListReturn
  a matrix containing all generated vectors , and an array containing the orbits of generated vectors.
- compute_coord_act_orbits (group, mat) → Array
  
  Computes the orbits of the vectors (homogenized) of a matrix mat by permuting the coordinates of the vectors (skipping the homogenizing coordinate). The group must act on the set of vectors. Choose the function 'compute_complete_coord_act_orbit' if your set is not complete.
  Parameters
  Group group
  a group acting on the cone by permuting the coordinates
  Matrix<Scalar> mat
  a matrix with vectors on which the group acts by coordinate permutation
  Returns
  Array
  an array of the orbits under the action on the coordinates
- compute_domain_orbits (group) → Array
  
  Computes the orbits of the basic set under group.
  Parameters
  Group group
  a group of a cone
  Returns
  Array
- compute_induced_orbits (group, inc) → Array
  
  Computes the orbits of a set on which an action is induced. The incidences between the domain elements and the elements in the set are given by an incidence matrix inc.
  Parameters
  Group group
  a group of a cone
  IncidenceMatrix inc
  the incidences between domain elements and elements on which an action is induced
  Returns
  Array
  an array of the orbits of the induced action
Others
- construct_cyclic_group (degree) → Group
  
  UNDOCUMENTED
  Parameters
  int degree
  of cyclic group
  Returns
  Group
- construct_symmetric_group (degree) → Group
  
  UNDOCUMENTED
  Parameters
  int degree
  of symmetric group
  Returns
  Group
Stabilizers
- stabilizer_of_set (group, set) → Group
  
  Computes the subgroup of group which stabilizes the given set of indices set.
  Parameters
  Group group
  a permutation group
  Set set
  the set to be stabilized
  Returns
  Group
  the stabilizer of set w.r.t. group
- stabilizer_of_vector (group, vec) → Group
  
  Computes the subgroup of group which stabilizes the given vector vec.
  Parameters
  Group group
  a permutation group
  Vector vec
  the vector to be stabilized
  Returns
  Group
  the stabilizer of vec w.r.t. group
Utilities
- group_from_cyclic_notation0 (group) → Group
  
  Constructs a group from a string with generators in cyclic notation. All numbers in the string are 0-based. Example: "(0,2)(1,3)"
  Parameters
  String group
  generators in cyclic notation
  Returns
  Group
  \@see group_from_cyclic_notation1
- group_from_cyclic_notation1 (group) → Group
  
  Constructs a group from a string with generators in cyclic notation. All numbers in the string are 1-based. Example: "(1,3)(2,4)"
  Parameters
  String group
  generators in cyclic notation
  Returns
  Group
  \@see group_from_cyclic_notation0
- group_from_permlib_cyclic_notation (gens, degree) → Group
  
  Constructs a Group from generators given in permlib cyclic notation. (permlib cyc_not: indices seperated by whitespace, generators seperated by commas)
  Parameters
  Array gens
  generators of the permutation group in permlib cyclic notation
  Int degree
  the degree of the permutation group
  Returns
  Group
  the group generated by gens
- group_to_cyclic_notation (the) → String
  
  Returns group generators in 1-based cyclic notation (GAP like, not permlib like notation)
  Parameters
  Group the
  permutation group
  Returns
  String
  group generators, separated by newline and comma

Array	gens	some generators of the group
Group	group	to fill in the data

Group	group	the permutation group acting on coordinates
Vector	vec1	the first vector
Vector	vec2	the second vector

Group	group	a group of coordinate permutations
Matrix<Scalar>	mat	some input vectors

Group	group	a group acting on the cone by permuting the coordinates
Matrix<Scalar>	mat	a matrix with vectors on which the group acts by coordinate permutation

Group	group	a group of a cone
IncidenceMatrix	inc	the incidences between domain elements and elements on which an action is induced

Group	group	a permutation group
Vector	vec	the vector to be stabilized

Array	gens	generators of the permutation group in permlib cyclic notation
Int	degree	the degree of the permutation group

Navigation

application: group

Objects

Properties of Group

Properties of GroupOfCone

Properties of GroupOfPolytope

User Functions

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