The application group provides basic functionality for working with permutation groups. An object of type Group records the abstract properties of the groups that do not depend on any particular representation, such as ORDER, CHARACTER_TABLE and CONJUGACY_CLASS_SIZES. Moreover, it can contain several subobjects of type PermutationAction, PermutationActionOnSets or ImplicitActionOnSets that encode representations of the group. (We use the terms action and representation interchangeably). If the group object is not contained inside a Cone or Polytope, the representation in question is encoded as a PERMUTATION_ACTION, SET_ACTION or IMPLICIT_SET_ACTION, otherwise it may be encoded more specifically as a RAY_ACTION, FACET_ACTION, COORDINATE_ACTION, etc. Different representations of the same group are convertible among each other either via specific rules, or by using the function induced_action, for example.
imports from:
Action:
parametrized object used for encoding group actions (representations).
Array<Int> for permutation groups, or
Matrix<Scalar> for matrix groups.
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
Rational
Group:
Object encoding a finite group.
ImplicitActionOnSets:
derived object encoding an action of a permutation group on a collection of sets, but where only one representative of each orbit is stored explicitly
LayeredPermutationAction:
object for encoding the separate actions of a group on a collection of sets, induced by the same generators
MatrixActionOnVectors:
derived object for matrix groups
PermutationAction:
derived objects for permutation representations
PermutationActionOnSets:
derived object for permuting collections of sets
These functions are concerned with orbits in groups.
are_in_same_orbit(PermutationAction a, Vector v1, Vector v2)Checks whether vectors v1 and v2 are in the same orbit with respect to the (coordinate) action of group.
PermutationAction a: the permutation group acting on coordinates
Vector v1
Vector v2
orbit<action_type>(Array G, Any O)The orbit of an object O under a group generated by G. The action_type defines how the elements of the group work on O.
action_type: controls the on_container the default. The group acts on the elements of a given container, which also could be container. on_elements for this the group will act on the deepest elements inside a given container with respect to the type of the group elements on_rows For this O should be a matrix so that the group can act on the rows of O. on_cols like on_rows this require a matrix so that the group can act on its columns on_nonhomog_cols also require that O is a matrix on whose columns the group generated by G acts. The 0-th column stay fixed.
Array G: Group generators
Any O: (mostly Array, Vector, Matrix or Set)
Set<Any>
> $permutationgroup = new Array<Array<Int>>([[1,2,3,0]]); > $set = new Set<Set<Int>>([[0,1],[2,3]]); > print orbit<on_elements>($permutationgroup, $set); {{{0 1} {2 3}} {{0 3} {1 2}}}
orbit<action_type>(Group G, Any C)The orbit of a container C under a group G.
action_type: by default on_container can also be on_elements, on_rows, on_cols or on_nonhomog_cols. Call help “<action_type>”; for more information about this.
Group G
Any C
Set<Any>
orbit<Scalar>(Array<Matrix<Scalar>> G, Vector<Scalar> V)The orbit of a vector V under a group generated by G.
Scalar: S the number type
Vector<Scalar> V
orbit_permlib(PermutationAction G, Set S)The orbit of a set S under a group defined by G.
orbit_permlib(PermutationAction G, Set<Set> S)The orbit of a set S of sets under a group given by G.
orbit_representatives(Array<GeneratorType> G)The indices of one representative for each orbit under the group generated by G.
Array<GeneratorType> G: Group generators
orbits_in_orbit_order(PermutationAction a, Matrix M)Computes the orbit of the rows of the matrix mat under the permutation action on coordinates action.
PermutationAction a: an action of a group of coordinate permutations
Matrix M: some input vectors
orbits_of_action(PermutationAction a)Computes the orbits of the basic set under a.
PermutationAction a: a permutation action of a group
orbits_of_coordinate_action(PermutationAction a, Matrix<Scalar> M)Computes the orbits of the vectors (homogenized) of the rows of a matrix M by permuting the coordinates of the vectors (skipping the homogenizing coordinate). The group must act on the set of vectors, and the rows of the matrix must contain the entire orbit.
PermutationAction a: an action of a group acting by permuting the coordinates
Matrix<Scalar> M: a matrix on whose columns the group acts by coordinate permutation
orbits_of_induced_action(PermutationAction a, IncidenceMatrix I)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.
PermutationAction a: an action of a group
IncidenceMatrix I: the incidences between domain elements and elements on which an action is induced
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(Int d)Constructs an alternating group of given degree d.
Int d: degree of the alternating group
cube_group(Int d)Constructs the symmetry group of a d-cube, acting on vertices, for 1 ⇐ d ⇐ 6 (for the moment), along with the corresponding character table and conjugacy class representatives.
cyclic_group(Int d)Constructs a cyclic group of given degree d.
Int d: degree of the cyclic group
dihedral_group(Int o)Constructs a dihedral group of a given order o. If the order is 2, 4, 6, 8, 10, 12, 16, 20 or 24, the character table is exact, otherwise some entries are mutilated rational approximations of algebraic numbers.
Int o: order of the dihedral group that acts on a regular (o/2)-gon
group_from_cyclic_notation0(String generators)Constructs a group from a string with generators in cyclic notation. All numbers in the string are 0-based. Example: “(0,2)(1,3)”
group_from_cyclic_notation1(String generators)Constructs a group from a string with generators in cyclic notation. All numbers in the string are 1-based.
group_from_permlib_cyclic_notation(Array<String> gens, Int degree)Constructs a Group from generators given in permlib cyclic notation, i.e., indices separated by whitespace, generators separated by commas.
Int degree: the degree of the permutation group
stabilizer_of_set(PermutationAction a, Set S)Computes the subgroup of group which stabilizes the given set of indices set.
PermutationAction a: the action of a permutation group
Set S: the set to be stabilized
stabilizer_of_vector(PermutationAction a, Vector v)Computes the subgroup of G which stabilizes the given vector v.
PermutationAction a: the action of a permutation group
Vector v: the vector to be stabilized
symmetric_group(Int d)Constructs a symmetric group of given degree d.
Int d: degree of the symmetric group
These functions are dealing with symmetries.
all_group_elements<Scalar>(MatrixActionOnVectors<Scalar> action)Compute all elements of a given group, expressed as a matrix group action.
Scalar: S the underlying number type
MatrixActionOnVectors<Scalar> action: the action of a permutation group
To generate all elements of the regular representation of S_3, type
> print all_group_elements(symmetric_group(3)->REGULAR_REPRESENTATION); <0 0 1 0 1 0 1 0 0 > <0 0 1 1 0 0 0 1 0 > <0 1 0 0 0 1 1 0 0 > <0 1 0 1 0 0 0 0 1 > <1 0 0 0 0 1 0 1 0 > <1 0 0 0 1 0 0 0 1 >
automorphism_group(GraphAdjacency graph)Find the automorphism group of the graph.
GraphAdjacency graph
automorphism_group(IncidenceMatrix<NonSymmetric> m, Bool on_rows)Find the automorphism group of the incidence matrix.
Bool on_rows: true (default) for row action
col_to_row_action(Matrix M, Array<Array> p)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.
Matrix M
conjugacy_class<Element>(Action action, Element e)Compute the conjugacy class of a group element under a given action
conjugacy_class<Scalar>(MatrixActionOnVectors<Scalar> action, Matrix<Scalar> e)Compute the conjugacy class of a group element under a given action
Scalar: E the underlying number type
MatrixActionOnVectors<Scalar> action: the action of the group
Matrix<Scalar> e: the element to be acted upon
group_left_multiplication_table(Group G)Calculate the left multiplication table of a group action, in which GMT[g][h] = hg
group_right_multiplication_table(Group G)Calculate the right multiplication table of a group action, in which GMT[g][h] = gh
implicit_character(ImplicitActionOnSets A)Calculate character of an implicit action
ImplicitActionOnSets A: the given action
induce_implicit_action(PermutationAction original_action, String property)Construct an implicit action of the action induced on a collection of sets. Only a set of orbit representatives is stored, not the full induced action.
PermutationAction original_action: the action of the group on indices
String property: the name of a property that describes an ordered list of sets on which the group should act
To construct the implicit action of the symmetry group of a cube on its maximal simplices, type:
> $c=cube(3, group=>1, character_table=>0); > group::induce_implicit_action($c, $c->GROUP->VERTICES_ACTION, $c->GROUP->REPRESENTATIVE_MAX_INTERIOR_SIMPLICES, "MAX_INTERIOR_SIMPLICES")->properties(); name: induced_implicit_action_of_ray_action_on_MAX_INTERIOR_SIMPLICES type: ImplicitActionOnSets description: induced from ray_action on MAX_INTERIOR_SIMPLICES DOMAIN_NAME MAX_INTERIOR_SIMPLICES EXPLICIT_ORBIT_REPRESENTATIVES {0 1 2 4} {0 1 2 5} {0 1 2 7} {0 3 5 6} GENERATORS 1 0 3 2 5 4 7 6 0 2 1 3 4 6 5 7 0 1 4 5 2 3 6 7
induce_set_action(Cone c, PermutationAction a, String domain)Construct the induced action of a permutation action on a property that is an ordered collection of sets, such as MAX_INTERIOR_SIMPLICES
Cone c: the cone or polytope
PermutationAction a: a permutation action on, for example, the vertex indices
String domain: the property the induced action should act upon
> $c=cube(3, group=>1, character_table=>0); > group::induce_set_action($c, $c->GROUP->VERTICES_ACTION, "MAX_INTERIOR_SIMPLICES")->properties(); name: induced_set_action_of_ray_action_on_MAX_INTERIOR_SIMPLICES type: PermutationActionOnSets description: induced from ray_action on MAX_INTERIOR_SIMPLICES DOMAIN_NAME MAX_INTERIOR_SIMPLICES GENERATORS 5 4 7 6 1 0 3 2 11 10 9 8 30 29 32 31 38 40 39 41 33 36 35 34 37 43 42 45 44 13 12 15 14 20 23 22 21 24 16 18 17 19 26 25 28 27 49 48 47 46 55 54 57 56 51 50 53 52 0 2 1 3 12 14 13 15 16 17 18 19 4 6 5 7 8 9 10 11 21 20 22 24 23 25 27 26 28 29 31 30 32 34 33 35 37 36 46 47 48 49 50 52 51 53 38 39 40 41 42 44 43 45 54 56 55 57 0 4 8 9 1 5 10 11 2 3 6 7 16 20 25 26 12 17 21 27 13 18 22 23 28 14 15 19 24 33 38 42 43 29 34 35 39 44 30 36 40 45 31 32 37 41 50 51 54 55 46 47 52 56 48 49 53 57
induced_action(PermutationAction a, Array<Set<Int>> domain)Given a permutation action a on some indices and an ordered list domain of sets containing these indices, we ask for the permutation action induced by a on this list of sets.
PermutationAction a: a permutation action that acts on some indices
Consider the symmetry group of the 3-cube acting on vertices, and induce from it the action on the facets:
> $a = cube_group(3)->PERMUTATION_ACTION; > $f = new Array<Set>([[0,2,4,6],[1,3,5,7],[0,1,4,5],[2,3,6,7],[0,1,2,3],[4,5,6,7]]); > print $a->GENERATORS; 1 0 3 2 5 4 7 6 0 2 1 3 4 6 5 7 0 1 4 5 2 3 6 7
> print induced_action($a,$f)->GENERATORS; 1 0 2 3 4 5 2 3 0 1 4 5 0 1 4 5 2 3
To see what the permutation [0,2,1] induces on the array [{0,1},{0,2}], do the following:
> $a = new Array<Set<Int>>(2); > $a->[0] = new Set<Int>(0,1); > $a->[1] = new Set<Int>(0,2); > $p = new PermutationAction(GENERATORS=>[[0,2,1]]); > print induced_action($p,$a)->properties; type: PermutationActionOnSets GENERATORS 1 0
induced_permutations(Array<Array<Int>> gens, Matrix M)gives the permutations that are induced on the rows of a matrix M by the action of gens on the columns of M
induced_permutations(Array<Matrix<Scalar>> gens, Matrix M)gives the permutations that are induced on the rows of a matrix M by the action of gens on the columns of M. The matrices in gens define the permutations by multiplicate the rows from the right side on it.
Matrix M: the matrix acted upon
Bool homogeneous_action: should the generators also act on the homogeneous column? Default False
> $gens = new Array<Matrix>([new Matrix([[0,1,0],[0,0,1],[1,0,0]]), new Matrix([[0,0,1],[1,0,0],[0,1,0]])]); > $M = new Matrix([[1,2,3],[3,1,2],[2,3,1]]); > print induced_permutations($gens, $M, homogeneous_action=>1); 2 0 1 1 2 0
induced_permutations(Array<Array<Int>> gens, Array<DomainType> S)gives the permutations that are induced on an ordered collection S by the action of gens on the elements of S
Array<DomainType> S: the collection acted upon
induced_permutations(Array<Array<Int>> a, IncidenceMatrix M)gives the permutations that are induced on the rows of an incidence matrix M by the action of gens on the columns of M
IncidenceMatrix M: the matrix acted upon
induced_permutations_set_set(Array<Array<Int>> gens, Array<Set<Set>> domain)gives the permutations that are induced on an Array<Set<Set» by permuting the elements of the inner set
induced_rep(Cone C, PermutationActionOnSets A, Array<Int> g)Calculate the representation of a group element
Cone C: the cone or polytope containing the sets acted upon
PermutationActionOnSets A: the action in question
invariant_polynomials(MatrixActionOnVectors a, Int d)Given a matrix action a of a group G on some n-dimensional vector space and a total degree d, calculate the G-invariant polynomials of total degree 0 < deg ≤ d in n variables. This is done by calculating the a-orbit of all monomials of total degree at most d. By a theorem of Noether, for d = the order of G the output is guaranteed to generate the entire ring of G-invariant polynomials. No effort is made to calculate a basis of the ideal generated by these invariants.
MatrixActionOnVectors a: the matrix action
Int d: the maximal degree of the sought invariants
Bool action_is_affine: is the action a affine, ie., ignore the first row and column of the generating matrices? Default yes
To calculate the invariants of degree at most six of the matrix action of the symmetry group of the 3-cube, type
> $c = cube(3, group=>1); > print join "\n", @{group::invariant_polynomials($c->GROUP->MATRIX_ACTION, 6)}; x_0^2 + x_1^2 + x_2^2 x_0^2*x_1^2 + x_0^2*x_2^2 + x_1^2*x_2^2 x_0^2*x_1^2*x_2^2 x_0^4 + x_1^4 + x_2^4 x_0^4*x_1^2 + x_0^4*x_2^2 + x_0^2*x_1^4 + x_0^2*x_2^4 + x_1^4*x_2^2 + x_1^2*x_2^4 x_0^6 + x_1^6 + x_2^6
irreducible_decomposition<Scalar>(Vector<Scalar> character, Group G)Calculate the decomposition into irreducible components of a given representation
Scalar: the number type of the character
Vector<Scalar> character: the character of the given representation
Group G: the given group; it needs to know its CHARACTER_TABLE and CONJUGACY_CLASS_SIZES.
Remember that in polymake, we use the terms action and representation interchangeably. To calculate the irreducible decomposition of the vertex action of the symmetry group of the 3-cube, type
> $g = cube_group(3); $a = $g->PERMUTATION_ACTION; > print irreducible_decomposition($a->CHARACTER, $g); 1 0 0 1 0 0 0 0 1 1
Thus, the action of the symmetry group on the vertices decomposes into one copy of each of the irreducible representations corresponding to the rows 0,3,8,9 of the character table:
> print $g->CHARACTER_TABLE->minor([0,3,8,9],All); 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 3 1 0 -1 1 -1 0 -1 -1 3 3 1 0 1 -1 1 0 -1 -1 -3
The first entries of these rows say that a decomposes into two 1-dimensional irreps and two 3-dimensional ones. This correctly brings the dimension of the representation a to 8, the number of vertices of the 3-cube.
isotypic_basis(Group G, PermutationActionOnSets A, Int i)Calculate a basis of the isotypic component given by the i-th irrep
Group G: the acting group, which needs to know its CHARACTER_TABLE
PermutationActionOnSets A: the representation in question, which needs to know its corresponding CONJUGACY_CLASSES
Int i: the index of the sought irrep
Bool permute_to_orbit_order: Should the rows and columns be ordered by orbits? Default 1
Consider the action of the symmetry group of the 3-cube on the set of facets:
> $g = cube_group(3); > $f = new Array<Set>([[0,2,4,6],[1,3,5,7],[0,1,4,5],[2,3,6,7],[0,1,2,3],[4,5,6,7]]); > $a = induced_action($g->PERMUTATION_ACTION, $f); > print irreducible_decomposition($a->CHARACTER, $g) 1 0 0 0 0 1 0 0 0 1
Now we can calculate a basis of the 5th irrep:
> print isotypic_basis($g, $a, 5); -1/6 -1/6 1/3 1/3 -1/6 -1/6 1/3 1/3 -1/6 -1/6 -1/6 -1/6
isotypic_basis(Group G, PermutationAction A, Int i)Calculate a basis of the isotypic component given by the i-th irrep
Group G: the acting group, which needs to know its CHARACTER_TABLE
PermutationAction A: the action in question, which needs to know its corresponding CONJUGACY_CLASSES
Int i: the index of the sought irrep
Bool permute_to_orbit_order: Should the rows and columns be ordered by orbits? Default 1
Consider the action of the symmetry group of the 3-cube on its vertices. We first calculate its decomposition into irreducible representations via
> $g = cube_group(3); $a = $g->PERMUTATION_ACTION; > print irreducible_decomposition($a->CHARACTER, $g); 1 0 0 1 0 0 0 0 1 1
We now calculate a basis of the 3rd irrep:
> print isotypic_basis($g,$a,3); 1/8 -1/8 -1/8 1/8 -1/8 1/8 1/8 -1/8
isotypic_basis<Scalar>(Group G, MatrixActionOnVectors<Scalar> A, Int i)Calculate a basis of the isotypic component given by the i-th irrep
Scalar: S the underlying number type
Group G: the acting group, which needs to know its CHARACTER_TABLE
MatrixActionOnVectors<Scalar> A: the matrix action in question, which needs to know its corresponding CONJUGACY_CLASSES
Int i: the index of the sought irrep
isotypic_projector(Group G, PermutationAction A, Int i)Calculate the projection map into the isotypic component given by the i-th irrep. The map is given by a block matrix, the rows and columns of which are indexed by the domain elements; by default, these are ordered by orbits.
Group G: the acting group
PermutationAction A: the action in question
Int i: the index of the sought irrep
Bool permute_to_orbit_order: Should the rows and columns be ordered by orbits? Default 1
Consider the action of the symmetry group of the 3-cube on its vertices. We first calculate its decomposition into irreducible representations via
> $g = cube_group(3); $a = $g->PERMUTATION_ACTION; > print irreducible_decomposition($a->CHARACTER, $g); 1 0 0 1 0 0 0 0 1 1
We now calculate the projection matrices to the irreps number 3 and 8:
> $p3 = isotypic_projector($g,$a,3); print $p3, "\n", rank($p3); 1/8 -1/8 -1/8 1/8 -1/8 1/8 1/8 -1/8 -1/8 1/8 1/8 -1/8 1/8 -1/8 -1/8 1/8 -1/8 1/8 1/8 -1/8 1/8 -1/8 -1/8 1/8 1/8 -1/8 -1/8 1/8 -1/8 1/8 1/8 -1/8 -1/8 1/8 1/8 -1/8 1/8 -1/8 -1/8 1/8 1/8 -1/8 -1/8 1/8 -1/8 1/8 1/8 -1/8 1/8 -1/8 -1/8 1/8 -1/8 1/8 1/8 -1/8 -1/8 1/8 1/8 -1/8 1/8 -1/8 -1/8 1/8 1
> $p8 = isotypic_projector($g,$a,8); print $p8, "\n", rank($p8); 3/8 -1/8 -1/8 -1/8 -1/8 -1/8 -1/8 3/8 -1/8 3/8 -1/8 -1/8 -1/8 -1/8 3/8 -1/8 -1/8 -1/8 3/8 -1/8 -1/8 3/8 -1/8 -1/8 -1/8 -1/8 -1/8 3/8 3/8 -1/8 -1/8 -1/8 -1/8 -1/8 -1/8 3/8 3/8 -1/8 -1/8 -1/8 -1/8 -1/8 3/8 -1/8 -1/8 3/8 -1/8 -1/8 -1/8 3/8 -1/8 -1/8 -1/8 -1/8 3/8 -1/8 3/8 -1/8 -1/8 -1/8 -1/8 -1/8 -1/8 3/8 3
From this we deduce that the irrep indexed 3 has dimension 1, and the one indexed 8 has dimension 3. This is consistent with the information collected in the character table:
> print $g->CHARACTER_TABLE->minor([3,8],All); 1 1 1 -1 -1 -1 -1 1 1 -1 3 1 0 -1 1 -1 0 -1 -1 3
In effect, the first entries in these rows are 1 and 3, respectively.
isotypic_projector<Scalar>(Group G, MatrixActionOnVectors<Scalar> A, Int i)Calculate the projection map into the isotypic component given by the i-th irrep. Note that the option permute_to_orbit_order makes no sense for matrix actions, so it is always set to 0.
Scalar: S the underlying number type
Group G: the acting group
MatrixActionOnVectors<Scalar> A: the action in question
Int i: the index of the sought irrep
We first construct a matrix action:
> $s = symmetric_group(3); $a = $s->REGULAR_REPRESENTATION; > print irreducible_decomposition($a->CHARACTER, $s); 0 1 1
Since we now know that the irreps indexed 1 and 2 appear in the regular representation, we project to one of them:
> print isotypic_projector($s, $a, 1); 2/3 -1/3 -1/3 -1/3 2/3 -1/3 -1/3 -1/3 2/3
isotypic_supports(PermutationActionOnSets a, Array<Set> A)For each isotypic component of a representation a, which of a given array A of sets are supported on it?
PermutationActionOnSets a: the representation in question
Bool permute_to_orbit_order: Should the columns be ordered by orbits? Default 1
isotypic_supports(PermutationActionOnSets a, SparseMatrix M)For each row of a given SparseMatrix M, to which isotypic components of a representation a does it have a non-zero projection? The columns of the SparseMatrix correspond, in order, to the sets of the representation.
PermutationActionOnSets a: the representation in question
SparseMatrix M: the given matrix
Bool permute_to_orbit_order: Should the columns be ordered by orbits? Default 1
lex_min_representative(Group G, Set S)Computes the lexicographically smallest representative of a given set with respect to a group
orbit_reps_and_sizes<Container>(Array<Array<Int>> generators, Container S)Computes the lexicographically smallest representatives of a given array of sets with respect to a group, along with the corresponding orbit sizes
Container: a container of sets, for example Array<Set> or IncidenceMatrix
Container S: a container of sets, for example Array<Set> or IncidenceMatrix
To calculate the orbits and orbit sizes of an icosidodecahedron, type
> $q=polytope::icosidodecahedron(); > print orbit_reps_and_sizes($q->GROUP->VERTICES_ACTION->GENERATORS,$q->VERTICES_IN_FACETS); <{0 1 2 4 6} {0 1 3} > 12 20
partition_representatives(Array<Array<Int>> gens, Set<Int> S)Partition a group into translates of a set stabilizer
regular_representation(PermutationAction a)Calculate the regular representation of a permutation action
PermutationAction a: a permutation action
To calculate the regular representation of the symmetric group S_3, type
> $s = symmetric_group(3); $s->REGULAR_REPRESENTATION; > print $s->REGULAR_REPRESENTATION->properties(); type: MatrixActionOnVectors<Rational> CONJUGACY_CLASS_REPRESENTATIVES <1 0 0 0 1 0 0 0 1 > <0 1 0 1 0 0 0 0 1 > <0 0 1 1 0 0 0 1 0 > GENERATORS <0 1 0 1 0 0 0 0 1 > <1 0 0 0 0 1 0 1 0 >
row_support_sizes(SparseMatrix M)How many non-zero entries are there in each row of a SparseMatrix?
SparseMatrix M: the given matrix
span_same_subspace(Array<HashMap<SetType,Rational>> S1, Array<HashMap<SetType,Rational>> S2)Do two collections S1, S2 of sparse vectors span the same subspace?
spans_invariant_subspace(ImplicitActionOnSets a, Array<HashMap<Bitset,Rational>> S)Does a set S of sparse vectors span an invariant subspace under an implicit group action a?
ImplicitActionOnSets a: the given action
Bool verbose: give a certificate if the answer is False
sparse_isotypic_basis(PermutationActionOnSets rep, Int i)Calculate a sparse representation of a basis for an isotypic component. For this, the sets in the representation are listed in order by orbit. In this basis, the projection matrix to the isotypic component decomposes into blocks, one for each orbit.
PermutationActionOnSets rep: the given representation
Int i: the index of the irrep that defines the isotypic component
Bool use_double: use inexact arithmetic for reducing the basis; default 0
String filename: if defined, the basis will be written to a file with this name, but not returned. Use this option if you expect very large output.
sparse_isotypic_spanning_set(PermutationActionOnSets rep, Int i)Calculate a sparse representation of a spanning set for an isotypic component. For this, the sets in the representation are listed in order by orbit. In this basis, the projection matrix to the isotypic component decomposes into blocks, one for each orbit.
PermutationActionOnSets rep: the given representation
Int i: the index of the irrep that defines the isotypic component
String filename: if defined, the basis will be written to a file with this name, but not returned. Use this option if you expect very large output.
sparse_isotypic_support(PermutationActionOnSets rep, Int i)Calculate the support of a sparse representation of a spanning set for an isotypic component.
PermutationActionOnSets rep: the given representation
Int i: the index of the irrep that defines the isotypic component
String filename: if defined, the basis will be written to a file with this name, but not returned. Use this option if you expect very large output.
Miscellaneous functions.
action<action_type>(Any g, Any O)The image of an object O under a group element g. The action_type defines how g acts on O. For more informations see the help function on the action types.
action_type: must be compatible with O by default on_container can also be on_elements, on_rows, on_cols, on_nonhomog_cols
Any g: Group element
Any O: Container type like Array, Set, Matrix or Vector
Any
> $g = new Array<Int>([1,2,0]); > $O = new Matrix<Int>([[0,0],[1,1],[2,2]]); > print $O; 0 0 1 1 2 2
> print action<on_rows>($g, $O); 1 1 2 2 0 0
action_from_generators(Array<Array<Int>> gens, Group action)Computes groups with a permutation action with the basic properties BASE, STRONG_GENERATORS, and TRANSVERSALS.
Group action: the generated action
action_inv<action_type>(Array<Int> p, Any O)The image of an object O under the inverse of a permutation p. The way it acts on O is given by the action_type.
action_type: must be compatible with O, by default on_container can also be on_elements, on_rows, on_cols or on_nonhomog_cols
Any O: Container type like Array, Set, Matrix or Vector
Any
> $p = new Array<Int>([1,2,0]); > $O = new Array<Int>([2,3,1]); > print action_inv($p,$O); 1 2 3
action_to_cyclic_notation(PermutationAction a)Returns group generators in 1-based cyclic notation (GAP like, not permlib like notation)
PermutationAction a: the action of the permutation group
all_group_elements(PermutationAction a)Compute all elements of a given group, expressed as a permutation action.
PermutationAction a: the action of a permutation group
SwitchTableUNDOCUMENTED
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