application: group

The application group provides basic functionality for working with permutation groups. An object of type 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 (DOMAIN=>2), or on coordinates (DOMAIN=>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::Integer

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
A permutation group which acts on a Cone by permuting (1) its RAYS (DOMAIN=>1), (2) its FACETS (DOMAIN=>2), or (3) the coordinates (DOMAIN=>3).

derived from: Group
Properties of GroupOfCone

Combinatorics

These properties capture combinatorial information of the object. Combinatorial properties only depend on combinatorial data of the object like, e.g., the face lattice.

Contained in extension bundled:group.

REPRESENTATIVE_INTERIOR_RIDGE_SIMPLICES: common::Array<boost_dynamic_bitset>

The (d-1)-dimensional simplices in the interior.

REPRESENTATIVE_MAX_BOUNDARY_SIMPLICES: common::Array<boost_dynamic_bitset>

The boundary (d-1)-dimensional simplices of a cone of combinatorial dimension d.

REPRESENTATIVE_MAX_INTERIOR_SIMPLICES: common::Array<boost_dynamic_bitset>

The interior d-dimensional simplices of a cone of combinatorial dimension d.

SIMPLEXITY_LOWER_BOUND: common::Int

The symmetrized version of SIMPLEXITY_LOWER_BOUND.

Symmetry

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

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.

N_REPRESENTATIVE_SIMPLICES: common::Array<Int>

An array of size COMBINATORIAL_DIM whose k-th entry stores the number of symmetry classes of k-dimensional simplices in the cone.

Contained in extension bundled:group.

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.

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!

REPRESENTATIVE_SIMPLICES: common::Array<Array<boost_dynamic_bitset>>

An array of size COMBINATORIAL_DIM whose k-th entry stores one representative for each symmetry class of k-dimensional simplices in the cone.

Contained in extension bundled:group.

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.

SYMMETRIZED_COCIRCUIT_EQUATIONS: common::SparseMatrix<Rational, NonSymmetric>

A SparseMatrix whose rows are the sum of all cocircuit equations corresponding to a fixed symmetry class of interior ridge.

Contained in extension bundled:group.
GroupOfPolytope
A permutation group which acts on a Polytope by permuting (1) its VERTICES (DOMAIN=>1), (2) its FACETS (DOMAIN=>2), or (3) the coordinates (DOMAIN=>3).

derived from: GroupOfCone
Properties of GroupOfPolytope

Symmetry

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

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

Orbits
These functions are concerned with orbits in groups.

Contained in extension bundled:group.
- 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
  Vector vec2
  Returns
  Bool
- orbit (G, S) → Set
  
  The orbit of a set S under a group G.
  Parameters
  Group G
  Set S
  Returns
  Set
- orbit (G, S) → Set
  
  The orbit of a set S of sets under a group G.
  Parameters
  Group G
  Set<Set> S
  Returns
  Set
- orbits_coord_action_complete (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.
- orbits_induced_action (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
- orbits_of_domain (group) → Array
  
  Computes the orbits of the basic set under group.
  Parameters
  Group group
  a group of a cone
  Returns
  Array
- orbit_coord_action (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 'orbits_coord_action_complete' 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
Producing a group
With these clients you can produce objects of type Group -- groups from certain parameterized families, as stabilizers of sets in other groups or from different kinds of cycle notations.
- alternating_group (degree) → Group
  
  Constructs an alternating group of given degree.
  Parameters
  Int degree
  of the alternating group
  Returns
  Group
- cyclic_group (degree) → Group
  
  Constructs a cyclic group of given degree.
  Parameters
  Int degree
  of the cyclic group
  Returns
  Group
- 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)"
  Contained in extension bundled:group.
  Parameters
  String group
  generators in cyclic notation
  Returns
  Group
- 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)"
  Contained in extension bundled:group.
  Parameters
  String group
  generators in cyclic notation
  Returns
  Group
- group_from_permlib_cyclic_notation (gens, degree) → Group
  
  Constructs a Group from generators given in permlib cyclic notation, i.e., indices separated by whitespace, generators separated by commas.
  Contained in extension bundled:group.
  Parameters
  Array<String> 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
- stabilizer_of_set (group, set) → Group
  
  Computes the subgroup of group which stabilizes the given set of indices set.
  Contained in extension bundled:group.
  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.
  Contained in extension bundled:group.
  Parameters
  Group group
  a permutation group
  Vector vec
  the vector to be stabilized
  Returns
  Group
  the stabilizer of vec w.r.t. group
- symmetric_group (degree) → Group
  
  Constructs a symmetric group of given degree.
  Parameters
  Int degree
  of the symmetric group
  Returns
  Group
Symmetry
These functions capture information of the object that is concerned with the action of permutation groups.
- col_to_row_action (M, of) → Array
  
  If the action of some permutations on the entries of the rows maps each row of a matrix to another row we obtain an induced action on the set of rows of the matrix. Considering the rows as points this corresponds to the action on the points induced by the action of some permutations on the coordinates.
  Parameters
  Matrix M
  Array of
  permutations
  Returns
  Array
  of permutations
Utilities
Miscellaneous functions.

Contained in extension bundled:group.
- all_group_elements (group) → Array
  
  Compute all elements of a given group.
  Parameters
  Group group
  the permutation group
  Returns
  Array
  <Array<Int> contains all group elements
- group_from_generators (gens, group)
  
  Computes the basic properties BASE, STRONG_GENERATORS, and TRANSVERSALS and stores the result in the given Group object group.
  Parameters
  Array<Array<Int>> gens
  some generators of the group
  Group group
  to fill in the data
- group_to_cyclic_notation (g) → String
  
  Returns group generators in 1-based cyclic notation (GAP like, not permlib like notation)
  Parameters
  Group g
  the permutation group
  Returns
  String
  group generators, separated by newline and comma

Group	group	the permutation group acting on coordinates
Vector	vec1
Vector	vec2

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

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

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

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

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

Matrix	M
Array	of	permutations

Navigation

application: group

Objects

Properties of Group

Properties of GroupOfCone

Properties of GroupOfPolytope

User Functions

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