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|
# Copyright (c) 1997-2024
# Ewgenij Gawrilow, Michael Joswig, and the polymake team
# Technische Universität Berlin, Germany
# https://polymake.org
#
# This program is free software; you can redistribute it and/or modify it
# under the terms of the GNU General Public License as published by the
# Free Software Foundation; either version 2, or (at your option) any
# later version: http://www.gnu.org/licenses/gpl.txt.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#-------------------------------------------------------------------------------
function induced_orbits<Scalar>(fan::PolyhedralFan<Scalar> $$ { homog_action=>0, return_matrix=>1 }) {
my ($c, $action_name, $generator_name, $options) = @_;
return group::induced_orbits_impl<Scalar>($c, $action_name, $generator_name, (new Scalar()), $options);
}
object PolyhedralFan {
# @category Symmetry
property GROUP : group::Group : multiple {
# @category Symmetry
property MATRIX_ACTION : group::MatrixActionOnVectors<Scalar>;
# @category Symmetry
# One representative from every orbit of the [[RAYS]] under the group action.
property REPRESENTATIVE_RAYS : Matrix<Scalar>;
# @category Symmetry
# One representative from every orbit of the [[MAXIMAL_CONES]] under the group action.
property REPRESENTATIVE_MAXIMAL_CONES : IncidenceMatrix;
# @category Symmetry
# List of all cones of all dimensions of the fan, one from each orbit,
# sorted by dimension.
# Indices in the IncidenceMatrix's refer to [[RAYS]].
property REPRESENTATIVE_CONES : Array<IncidenceMatrix>;
#
# some shortcuts
#
# @category Symmetry
# explicit representatives of equivalence classes of [[fan::INPUT_RAYS]] under a group action
user_method REPRESENTATIVE_INPUT_RAYS = INPUT_RAYS_ACTION.EXPLICIT_ORBIT_REPRESENTATIVE_MATRIX;
# @category Symmetry
# explicit representatives of equivalence classes of [[fan::INPUT_CONES]] under a group action
user_method REPRESENTATIVE_INPUT_CONES = INPUT_CONES_ACTION.EXPLICIT_ORBIT_REPRESENTATIVES;
rule REPRESENTATIVE_RAYS = RAYS_ACTION.EXPLICIT_ORBIT_REPRESENTATIVE_MATRIX;
rule REPRESENTATIVE_MAXIMAL_CONES = MAXIMAL_CONES_ACTION.EXPLICIT_ORBIT_REPRESENTATIVES;
property MAXIMAL_CONES_ACTION {
# @category Symmetry
# dimensions of representatives of maximal cones
property REPRESENTATIVE_COMBINATORIAL_DIMS : Array<Int>;
# @category Symmetry
# counts how many representatives of maximal cones there are in each dimension
property REPRESENTATIVE_F_VECTOR : Array<Int>;
}
} # end property GROUP
# First, some rules that apply to all actions
rule GROUP.ORDER : GROUP.RAYS_ACTION.TRANSVERSAL_SIZES | GROUP.MAXIMAL_CONES_ACTION.TRANSVERSAL_SIZES | GROUP.HOMOGENEOUS_COORDINATE_ACTION.TRANSVERSAL_SIZES {
my $arr = $this->give("GROUP.RAYS_ACTION.TRANSVERSAL_SIZES | GROUP.MAXIMAL_CONES_ACTION.TRANSVERSAL_SIZES | GROUP.HOMOGENEOUS_COORDINATE_ACTION.TRANSVERSAL_SIZES");
my $i = new Integer(1);
$i *= $_ foreach(@{$arr});
$this->GROUP->ORDER = $i;
}
weight 1.10;
rule GROUP.CONJUGACY_CLASS_SIZES : GROUP.RAYS_ACTION.CONJUGACY_CLASSES | GROUP.MAXIMAL_CONES_ACTION.CONJUGACY_CLASSES | \
GROUP.HOMOGENEOUS_COORDINATE_ACTION.CONJUGACY_CLASSES | GROUP.SET_ACTION.CONJUGACY_CLASSES | GROUP.MATRIX_ACTION.CONJUGACY_CLASSES {
my $arr = $this->give("GROUP.RAYS_ACTION.CONJUGACY_CLASSES | GROUP.MAXIMAL_CONES_ACTION.CONJUGACY_CLASSES | GROUP.HOMOGENEOUS_COORDINATE_ACTION.CONJUGACY_CLASSES | GROUP.SET_ACTION.CONJUGACY_CLASSES | GROUP.MATRIX_ACTION.CONJUGACY_CLASSES");
$this->GROUP->CONJUGACY_CLASS_SIZES = [ map {$_->size} @{$arr} ];
}
weight 1.10;
# *_ACTION->IRREDUCIBLE_DECOMPOSITION, in alphabetical order
rule GROUP.HOMOGENEOUS_COORDINATE_ACTION.IRREDUCIBLE_DECOMPOSITION : GROUP.HOMOGENEOUS_COORDINATE_ACTION.CHARACTER, GROUP.CHARACTER_TABLE, GROUP.CONJUGACY_CLASS_SIZES, GROUP.ORDER {
$this->GROUP->HOMOGENEOUS_COORDINATE_ACTION->IRREDUCIBLE_DECOMPOSITION = group::irreducible_decomposition($this->GROUP->HOMOGENEOUS_COORDINATE_ACTION->CHARACTER, $this->GROUP);
}
weight 1.10;
rule GROUP.MATRIX_ACTION.IRREDUCIBLE_DECOMPOSITION : GROUP.MATRIX_ACTION.CHARACTER, GROUP.CHARACTER_TABLE, GROUP.CONJUGACY_CLASS_SIZES, GROUP.ORDER {
$this->GROUP->MATRIX_ACTION->IRREDUCIBLE_DECOMPOSITION = group::irreducible_decomposition($this->GROUP->MATRIX_ACTION->CHARACTER, $this->GROUP);
}
weight 1.10;
rule GROUP.MAXIMAL_CONES_ACTION.IRREDUCIBLE_DECOMPOSITION : GROUP.MAXIMAL_CONES_ACTION.CHARACTER, GROUP.CHARACTER_TABLE, GROUP.CONJUGACY_CLASS_SIZES, GROUP.ORDER {
$this->GROUP->MAXIMAL_CONES_ACTION->IRREDUCIBLE_DECOMPOSITION = group::irreducible_decomposition($this->GROUP->MAXIMAL_CONES_ACTION->CHARACTER, $this->GROUP);
}
weight 1.10;
rule GROUP.RAYS_ACTION.IRREDUCIBLE_DECOMPOSITION : GROUP.RAYS_ACTION.CHARACTER, GROUP.CHARACTER_TABLE, GROUP.CONJUGACY_CLASS_SIZES, GROUP.ORDER {
$this->GROUP->RAYS_ACTION->IRREDUCIBLE_DECOMPOSITION = group::irreducible_decomposition($this->GROUP->RAYS_ACTION->CHARACTER, $this->GROUP);
}
weight 1.10;
rule GROUP.MAXIMAL_CONES_ACTION.REPRESENTATIVE_COMBINATORIAL_DIMS : MAXIMAL_CONES_COMBINATORIAL_DIMS, GROUP.MAXIMAL_CONES_ACTION.ORBIT_REPRESENTATIVES {
my @dims;
foreach (@{$this->GROUP->MAXIMAL_CONES_ACTION->ORBIT_REPRESENTATIVES}) {
push @dims, $this->MAXIMAL_CONES_COMBINATORIAL_DIMS->[$_];
}
$this->GROUP->MAXIMAL_CONES_ACTION->REPRESENTATIVE_COMBINATORIAL_DIMS = \@dims;
}
rule GROUP.MAXIMAL_CONES_ACTION.REPRESENTATIVE_F_VECTOR : COMBINATORIAL_DIM, GROUP.MAXIMAL_CONES_ACTION.REPRESENTATIVE_COMBINATORIAL_DIMS {
my $fvect = new Array<Int>(1 + $this->COMBINATORIAL_DIM);
foreach (@{$this->GROUP->MAXIMAL_CONES_ACTION->REPRESENTATIVE_COMBINATORIAL_DIMS}) {
$fvect->[$_]++;
}
$this->GROUP->MAXIMAL_CONES_ACTION->REPRESENTATIVE_F_VECTOR = $fvect;
}
#
# explicit representatives
#
# rule GROUP.SET_ACTION.EXPLICIT_ORBIT_REPRESENTATIVES : GROUP.SET_ACTION.DOMAIN_NAME, GROUP.SET_ACTION.ORBIT_REPRESENTATIVES, MAX_INTERIOR_SIMPLICES | INTERIOR_RIDGE_SIMPLICES {
# my $domain_name = $this->GROUP->SET_ACTION->DOMAIN_NAME;
# my @reps = map { $this->$domain_name->[$_] } @{$this->GROUP->SET_ACTION->ORBIT_REPRESENTATIVES};
# $this->GROUP->SET_ACTION->EXPLICIT_ORBIT_REPRESENTATIVES = \@reps;
# }
# weight 1.10;
rule GROUP.INPUT_RAYS_ACTION.EXPLICIT_ORBIT_REPRESENTATIVE_MATRIX : GROUP.INPUT_RAYS_ACTION.ORBIT_REPRESENTATIVES, INPUT_RAYS {
$this->GROUP->INPUT_RAYS_ACTION->EXPLICIT_ORBIT_REPRESENTATIVE_MATRIX = $this->INPUT_RAYS->minor($this->GROUP->INPUT_RAYS_ACTION->ORBIT_REPRESENTATIVES, All);
}
weight 1.10;
rule GROUP.RAYS_ACTION.EXPLICIT_ORBIT_REPRESENTATIVE_MATRIX : GROUP.RAYS_ACTION.ORBIT_REPRESENTATIVES, RAYS {
$this->GROUP->RAYS_ACTION->EXPLICIT_ORBIT_REPRESENTATIVE_MATRIX = $this->RAYS->minor($this->GROUP->RAYS_ACTION->ORBIT_REPRESENTATIVES, All);
}
weight 1.10;
#
# induce actions from others
#
#
# induce RAYS_ACTION
#
rule GROUP.INPUT_RAYS_ACTION.GENERATORS : GROUP.HOMOGENEOUS_COORDINATE_ACTION.GENERATORS, INPUT_RAYS {
$this->GROUP->INPUT_RAYS_ACTION = group::induce_permutation_action($this, "HOMOGENEOUS_COORDINATE_ACTION", "INPUT_RAYS", "input_ray_action", "induced from coordinate action", 1);
}
weight 1.10;
rule GROUP.INPUT_RAYS_ACTION.CONJUGACY_CLASS_REPRESENTATIVES: GROUP.HOMOGENEOUS_COORDINATE_ACTION.CONJUGACY_CLASS_REPRESENTATIVES, INPUT_RAYS {
$this->GROUP->INPUT_RAYS_ACTION->CONJUGACY_CLASS_REPRESENTATIVES = group::induced_permutations($this->GROUP->HOMOGENEOUS_COORDINATE_ACTION->CONJUGACY_CLASS_REPRESENTATIVES, $this->INPUT_RAYS, homogeneous_action=>1);
}
weight 1.10;
rule GROUP.RAYS_ACTION.GENERATORS : GROUP.MAXIMAL_CONES_ACTION.GENERATORS, MAXIMAL_CONES_THRU_RAYS {
$this->GROUP->RAYS_ACTION->GENERATORS = group::induce_permutation_action($this, "MAXIMAL_CONES_ACTION", "MAXIMAL_CONES_THRU_RAYS", "ray_action", "induced from maximal cones action", 0)->GENERATORS;
}
weight 1.10;
rule GROUP.RAYS_ACTION.CONJUGACY_CLASS_REPRESENTATIVES: GROUP.MAXIMAL_CONES_ACTION.CONJUGACY_CLASS_REPRESENTATIVES, MAXIMAL_CONES_THRU_RAYS {
$this->GROUP->RAYS_ACTION->CONJUGACY_CLASS_REPRESENTATIVES = group::induced_permutations($this->GROUP->MAXIMAL_CONES_ACTION->CONJUGACY_CLASS_REPRESENTATIVES, $this->MAXIMAL_CONES_THRU_RAYS, homogeneous_action=>0);
}
weight 1.10;
rule GROUP.RAYS_ACTION.GENERATORS : GROUP.HOMOGENEOUS_COORDINATE_ACTION.GENERATORS, RAYS {
$this->GROUP->RAYS_ACTION->GENERATORS = group::induce_permutation_action($this, "HOMOGENEOUS_COORDINATE_ACTION", "RAYS", "ray_action", "induced from homogeneous coordinate action", 1)->GENERATORS;
}
weight 1.10;
rule GROUP.RAYS_ACTION.CONJUGACY_CLASS_REPRESENTATIVES: GROUP.HOMOGENEOUS_COORDINATE_ACTION.CONJUGACY_CLASS_REPRESENTATIVES, RAYS {
$this->GROUP->RAYS_ACTION->CONJUGACY_CLASS_REPRESENTATIVES = group::induced_permutations($this->GROUP->HOMOGENEOUS_COORDINATE_ACTION->CONJUGACY_CLASS_REPRESENTATIVES, $this->RAYS, homogeneous_action=>1);
}
weight 1.10;
#
# induce MAXIMAL_CONES_ACTION
#
rule GROUP.MAXIMAL_CONES_ACTION.GENERATORS : GROUP.RAYS_ACTION.GENERATORS, MAXIMAL_CONES {
$this->GROUP->MAXIMAL_CONES_ACTION->GENERATORS = group::induce_permutation_action($this, "RAYS_ACTION", "MAXIMAL_CONES", "maximal_cones_action", "induced from ray action", 0)->GENERATORS;
}
weight 1.10;
rule GROUP.MAXIMAL_CONES_ACTION.CONJUGACY_CLASS_REPRESENTATIVES: GROUP.RAYS_ACTION.CONJUGACY_CLASS_REPRESENTATIVES, MAXIMAL_CONES {
$this->GROUP->MAXIMAL_CONES_ACTION->CONJUGACY_CLASS_REPRESENTATIVES = group::induced_permutations($this->GROUP->RAYS_ACTION->CONJUGACY_CLASS_REPRESENTATIVES, $this->MAXIMAL_CONES, homogeneous_action=>0);
}
weight 1.10;
#
# induce MATRIX_ACTION
#
rule GROUP.MATRIX_ACTION.GENERATORS : GROUP.INPUT_RAYS_ACTION.GENERATORS, INPUT_RAYS {
group::induce_matrix_action_generators($this, "MATRIX_ACTION", "INPUT_RAYS_ACTION", "INPUT_RAYS", new Matrix<Scalar>());
}
rule GROUP.MATRIX_ACTION.CONJUGACY_CLASS_REPRESENTATIVES : GROUP.INPUT_RAYS_ACTION.CONJUGACY_CLASS_REPRESENTATIVES, INPUT_RAYS {
group::induce_matrix_action_conjugacy_class_representatives($this, "MATRIX_ACTION", "INPUT_RAYS_ACTION", "INPUT_RAYS", new Matrix<Scalar>());
}
rule GROUP.MATRIX_ACTION.GENERATORS : GROUP.RAYS_ACTION.GENERATORS, RAYS {
group::induce_matrix_action_generators($this, "MATRIX_ACTION", "RAYS_ACTION", "RAYS", new Matrix<Scalar>());
}
rule GROUP.MATRIX_ACTION.CONJUGACY_CLASS_REPRESENTATIVES : GROUP.RAYS_ACTION.CONJUGACY_CLASS_REPRESENTATIVES, RAYS {
group::induce_matrix_action_conjugacy_class_representatives($this, "MATRIX_ACTION", "RAYS_ACTION", "RAYS", new Matrix<Scalar>());
}
#
# Others
#
rule GROUP.FACET_NORMALS_ACTION.GENERATORS : GROUP.HOMOGENEOUS_COORDINATE_ACTION.GENERATORS, FACET_NORMALS {
$this->GROUP->FACET_NORMALS_ACTION->GENERATORS = group::induce_permutation_action($this, "HOMOGENEOUS_COORDINATE_ACTION", "FACET_NORMALS", "facet_normals_action", "induced from homogeneous_coordinate_action", 1, \&canonicalize_facets)->GENERATORS;
}
weight 1.10;
rule GROUP.FACET_NORMALS_ACTION.CONJUGACY_CLASS_REPRESENTATIVES: GROUP.HOMOGENEOUS_COORDINATE_ACTION.CONJUGACY_CLASS_REPRESENTATIVES, FACET_NORMALS {
$this->GROUP->FACET_NORMALS_ACTION->CONJUGACY_CLASS_REPRESENTATIVES = group::induced_permutations($this->GROUP->HOMOGENEOUS_COORDINATE_ACTION->CONJUGACY_CLASS_REPRESENTATIVES, $this->FACET_NORMALS, homogeneous_action=>1);
}
weight 1.10;
#
# Orbits induced by coordinate actions
#
rule INPUT_RAYS, GROUP.INPUT_RAYS_ACTION.EXPLICIT_ORBIT_REPRESENTATIVE_MATRIX : GROUP.HOMOGENEOUS_COORDINATE_ACTION.GENERATORS, GROUP.HOMOGENEOUS_COORDINATE_ACTION.INPUT_RAYS_GENERATORS {
my ($pts, $a) = induced_orbits($this, "HOMOGENEOUS_COORDINATE_ACTION", "INPUT_RAYS_GENERATORS", homog_action => 1);
$this->INPUT_RAYS = $pts;
$this->GROUP->INPUT_RAYS_ACTION = $a;
}
weight 2.10;
# incurs PointsPerm;
method rays_from_action($) {
my ($this, $action_type) = @_;
my $is_homogeneous = ($action_type eq "HOMOGENEOUS_COORDINATE_ACTION");
my ($pts, $a) = induced_orbits($this, $action_type, "RAYS_GENERATORS", homog_action => $is_homogeneous, return_matrix => 0);
# make the given RAYS_GENERATORS come first in the list of RAYS
my $rep_set = new HashSet<Vector<Scalar>>;
my @pts_in_order;
foreach (@{$this->GROUP->$action_type->RAYS_GENERATORS}) {
$rep_set += $_;
push @pts_in_order, $_;
}
foreach (@{$pts}) {
if ( ! $rep_set->contains($_) ) {
push @pts_in_order, $_;
}
}
my $all_rays = new Matrix<Scalar>(\@pts_in_order);
# we find the induced permutation action on the rows of this matrix
# before assigning it to RAYS
# because that assignment homogenizes the first column and sabotages
# the attempt to find the induced permutation
my $ia = group::induce_permutation_action($this, $action_type, $all_rays, "induced RAYS_ACTION", "induced from $action_type", $is_homogeneous);
$this->RAYS = $all_rays;
$a->GENERATORS = $ia->GENERATORS;
if (defined $ia->lookup("CONJUGACY_CLASS_REPRESENTATIVES")) {
$a->CONJUGACY_CLASS_REPRESENTATIVES = $ia->CONJUGACY_CLASS_REPRESENTATIVES;
}
$this->GROUP->RAYS_ACTION = $a;
}
rule RAYS, GROUP.RAYS_ACTION.GENERATORS, GROUP.RAYS_ACTION.EXPLICIT_ORBIT_REPRESENTATIVE_MATRIX : GROUP.HOMOGENEOUS_COORDINATE_ACTION.GENERATORS, GROUP.HOMOGENEOUS_COORDINATE_ACTION.RAYS_GENERATORS {
$this->rays_from_action("HOMOGENEOUS_COORDINATE_ACTION");
}
weight 2.10;
incurs RaysPerm;
rule RAYS, GROUP.RAYS_ACTION.GENERATORS, GROUP.RAYS_ACTION.EXPLICIT_ORBIT_REPRESENTATIVE_MATRIX : GROUP.MATRIX_ACTION.GENERATORS, GROUP.MATRIX_ACTION.RAYS_GENERATORS {
$this->rays_from_action("MATRIX_ACTION");
}
weight 2.10;
incurs RaysPerm;
rule MAXIMAL_CONES : RAYS, GROUP.RAYS_ACTION.GENERATORS, GROUP.MAXIMAL_CONES_ACTION.MAXIMAL_CONES_GENERATORS {
my $gens = $this->GROUP->RAYS_ACTION->GENERATORS;
my $all_cones = new HashSet<Set<Int>>;
foreach (@{$this->GROUP->MAXIMAL_CONES_ACTION->MAXIMAL_CONES_GENERATORS}) {
foreach(@{group::orbit($gens, new Set($_))}) {
$all_cones += $_;
}
}
my $max_cones = new IncidenceMatrix($all_cones->size(), $gens->[0]->size());
my $i=0;
foreach (@{$all_cones}) {
$max_cones->row($i++) += $_;
}
$this->MAXIMAL_CONES = $max_cones;
}
weight 2.10;
incurs ConesPerm;
} # end PolyhedralFan
object PolyhedralComplex {
property GROUP {
# @category Symmetry
property VERTICES_ACTION = override RAYS_ACTION;
# @category Symmetry
property COORDINATE_ACTION = override HOMOGENEOUS_COORDINATE_ACTION {
# @category Symmetry
property POINTS_GENERATORS = override INPUT_RAYS_GENERATORS;
# @category Symmetry
property N_POINTS_GENERATORS = override N_INPUT_RAYS_GENERATORS;
# @category Symmetry
property VERTICES_GENERATORS = override RAYS_GENERATORS;
# @category Symmetry
property N_VERTICES_GENERATORS = override N_RAYS_GENERATORS;
}
# @category Symmetry
property POINTS_ACTION = override INPUT_RAYS_ACTION;
# @category Symmetry
property MAXIMAL_POLYTOPES_ACTION = override MAXIMAL_CONES_ACTION {
# @category Symmetry
property POINTS_GENERATORS = override INPUT_RAYS_GENERATORS;
# @category Symmetry
property N_POINTS_GENERATORS = override N_INPUT_RAYS_GENERATORS;
# @category Symmetry
property VERTICES_GENERATORS = override RAYS_GENERATORS;
# @category Symmetry
property N_VERTICES_GENERATORS = override N_RAYS_GENERATORS;
# @category Symmetry
property MAXIMAL_POLYTOPES_GENERATORS = override MAXIMAL_CONES_GENERATORS;
}
# @category Symmetry
property MATRIX_ACTION_ON_COMPLEX = override MATRIX_ACTION {
# overriding this name is silly, but if the name of the action is not overridden, assigning to GENERATORS breaks.
# to test this, delete the "override" and execute the example for
# orbit_complex<Scalar>(fan::PolyhedralComplex<Scalar>, group::MatrixActionOnVectors<Scalar>)
# @category Symmetry
property VERTICES_GENERATORS = override RAYS_GENERATORS;
}
# @category Symmetry
property REPRESENTATIVE_VERTICES = override REPRESENTATIVE_RAYS;
# @category Symmetry
property REPRESENTATIVE_MAXIMAL_POLYTOPES = override REPRESENTATIVE_MAXIMAL_CONES;
}
# induced actions
rule GROUP.POINTS_ACTION.GENERATORS : GROUP.COORDINATE_ACTION.GENERATORS , POINTS {
$this->GROUP->POINTS_ACTION->GENERATORS = group::induce_permutation_action($this, "COORDINATE_ACTION", "POINTS", "points_action", "induced from coordinate_action")->GENERATORS;
}
weight 1.10;
rule GROUP.POINTS_ACTION.CONJUGACY_CLASS_REPRESENTATIVES: GROUP.COORDINATE_ACTION.CONJUGACY_CLASS_REPRESENTATIVES, POINTS {
$this->GROUP->POINTS_ACTION->CONJUGACY_CLASS_REPRESENTATIVES = group::induced_permutations($this->GROUP->COORDINATE_ACTION->CONJUGACY_CLASS_REPRESENTATIVES, $this->POINTS, homogeneous_action=>0);
}
weight 1.10;
rule GROUP.POINTS_ACTION.GENERATORS, GROUP.POINTS_ACTION.DEGREE : GROUP.PERMUTATION_ACTION.GENERATORS, GROUP.PERMUTATION_ACTION.DEGREE {
$this->GROUP->POINTS_ACTION->GENERATORS = $this->GROUP->PERMUTATION_ACTION->GENERATORS;
$this->GROUP->POINTS_ACTION->DEGREE = $this->GROUP->PERMUTATION_ACTION->DEGREE;
}
weight 0.10;
precondition : GROUP.PERMUTATION_ACTION.DEGREE, N_POINTS { $this->GROUP->PERMUTATION_ACTION->DEGREE == $this->N_POINTS }
rule GROUP.VERTICES_ACTION.GENERATORS : GROUP.COORDINATE_ACTION.GENERATORS, VERTICES {
$this->GROUP->VERTICES_ACTION->GENERATORS = group::induce_permutation_action($this, "COORDINATE_ACTION", "VERTICES", "vertices_action", "induced from coordinate_action")->GENERATORS;
}
weight 1.10;
rule GROUP.VERTICES_ACTION.CONJUGACY_CLASS_REPRESENTATIVES: GROUP.COORDINATE_ACTION.CONJUGACY_CLASS_REPRESENTATIVES, VERTICES {
$this->GROUP->VERTICES_ACTION->CONJUGACY_CLASS_REPRESENTATIVES = group::induced_permutations($this->GROUP->COORDINATE_ACTION->CONJUGACY_CLASS_REPRESENTATIVES, $this->VERTICES, homogeneous_action=>0);
}
weight 1.10;
rule GROUP.VERTICES_ACTION.GENERATORS, GROUP.VERTICES_ACTION.DEGREE : GROUP.PERMUTATION_ACTION.GENERATORS, GROUP.PERMUTATION_ACTION.DEGREE {
$this->GROUP->VERTICES_ACTION->GENERATORS = $this->GROUP->PERMUTATION_ACTION->GENERATORS;
$this->GROUP->VERTICES_ACTION->DEGREE = $this->GROUP->PERMUTATION_ACTION->DEGREE;
}
weight 0.10;
precondition : GROUP.PERMUTATION_ACTION.DEGREE, N_VERTICES { $this->GROUP->PERMUTATION_ACTION->DEGREE == $this->N_VERTICES }
# induced orbits
rule POINTS, GROUP.POINTS_ACTION.EXPLICIT_ORBIT_REPRESENTATIVE_MATRIX : GROUP.COORDINATE_ACTION.GENERATORS, GROUP.COORDINATE_ACTION.POINTS_GENERATORS {
my ($pts, $a) = induced_orbits($this, "COORDINATE_ACTION", "POINTS_GENERATORS", homog_action => 0);
$this->POINTS = $pts;
$this->GROUP->POINTS_ACTION = $a;
}
weight 2.10;
# incurs PointsPerm;
rule VERTICES, GROUP.VERTICES_ACTION.GENERATORS, GROUP.VERTICES_ACTION.EXPLICIT_ORBIT_REPRESENTATIVE_MATRIX : GROUP.COORDINATE_ACTION.GENERATORS, GROUP.COORDINATE_ACTION.VERTICES_GENERATORS {
$this->rays_from_action("COORDINATE_ACTION");
}
weight 2.10;
incurs RaysPerm;
rule GROUP.POINTS_ACTION.EXPLICIT_ORBIT_REPRESENTATIVE_MATRIX : GROUP.POINTS_ACTION.ORBIT_REPRESENTATIVES, POINTS {
$this->GROUP->POINTS_ACTION->EXPLICIT_ORBIT_REPRESENTATIVE_MATRIX = $this->POINTS->minor($this->GROUP->POINTS_ACTION->ORBIT_REPRESENTATIVES, All);
}
weight 1.10;
# Generates an array with a specified number of different colors.
# @param Int number the number of colors
# @return array the colors as strings
sub generateColors {
my $number=$_[0];
# not really implemented yet!
my @FIFTEEN_COLORS=("chocolate1","salmon1","plum1","LightGreen","azure","LightSlateGrey","MidnightBlue","DarkOliveGreen","IndianRed","LavenderBlush","orange","green","red","blue","yellow");
my @colors=();
for(my $i=0;$i<$number;$i++) {
if($i<15) {
$colors[$i]=$FIFTEEN_COLORS[14-$i];
} else {
$colors[$i]="black";
}
}
return @colors;
}
# @category Visualization
# Visualizes the graph of a symmetric cone:
# All nodes belonging to one orbit get the same color.
# @return Visual::PolytopeGraph
user_method VISUAL_ORBIT_COLORED_GRAPH(%Visual::Graph::decorations, { seed => undef }) {
my ($this, $decor, $seed)=@_;
my @colors=generateColors($this->GROUP->RAYS_ACTION->N_ORBITS);
my @nodeColors;
for(my $i=0; $i<$this->GROUP->RAYS_ACTION->N_ORBITS; $i++){
foreach my $ray_index(@{$this->GROUP->RAYS_ACTION->ORBITS->[$i]}){
$nodeColors[$ray_index]=$colors[$i]; #each ray gets the color(=number) of its orbit
}
}
my $VG=$this->GRAPH->VISUAL( Name => $this->name,
(exists $decor->{NodeBorderColor} || exists $decor->{VertexColor} ? () : (NodeColor => \@nodeColors)),
$decor, $seed );
visualize( new Visual::PolytopeGraph( Name => "ORBIT_COLORED_GRAPH of " . $this->name,
Polytope => $this,
$VG ));
}
} # end PolyhedralComplex
# @category Symmetry
# Constructs the orbit fan of a given fan //input_fan//
# with respect to a given set of generators //gens//.
# @param fan::PolyhedralFan input_fan the generating fan of the orbit fan
# @param Array<Array<Int>> gens the generators of a permutation group that acts on the coordinates of the ambient space
# @return fan::PolyhedralFan the orbit fan of //input_fan// w.r.t. the coordinate action generated by //gens//
# @example [prefer ppl] To calculate an orbit fan, follow these steps:
# First specify a seed fan:
# > $f=new PolyhedralFan(RAYS=>[[1,1],[1,0],[-1,-1]], MAXIMAL_CONES=>[[0,1],[1,2]]);
# Then define the orbit fan by specifying coordinate permutations:
# > $of = orbit_fan($f,[[1,0]]);
# The only properties of $of defined so far reside in GROUP:
# > print $of->GROUP->properties();
# | name: unnamed#0
# | type: Group as PolyhedralFan<Rational>::GROUP
# |
# | HOMOGENEOUS_COORDINATE_ACTION
# | type: PermutationAction<Int, Rational>
# |
# | MAXIMAL_CONES_ACTION
# | type: PermutationAction<Int, Rational> as PolyhedralFan<Rational>::GROUP::MAXIMAL_CONES_ACTION
# |
# Now you can calculate the [[RAYS]] and [[MAXIMAL_CONES]] of the orbit fan.
# > print $of->RAYS;
# | 1 1
# | 1 0
# | -1 -1
# | 0 1
# > print $of->N_MAXIMAL_CONES;
# | 4
user_function orbit_fan(fan::PolyhedralFan, can_convert_to<Array<Array<Int>>>) {
my ($input_fan, $gens) = @_;
my $ra = new group::PermutationAction(GENERATORS=>$gens, RAYS_GENERATORS=>$input_fan->give("RAYS | INPUT_RAYS"));
my $ca = new group::PermutationAction(MAXIMAL_CONES_GENERATORS=>$input_fan->MAXIMAL_CONES);
my $g = new group::Group;
my $p = new fan::PolyhedralFan;
$p->add("GROUP", $g, HOMOGENEOUS_COORDINATE_ACTION=>$ra, MAXIMAL_CONES_ACTION=>$ca);
return $p;
}
# @category Symmetry
# Constructs the orbit fan of a given fan //input_fan//
# with respect to a given set of matrix group generators //gens//.
# @param fan::PolyhedralFan input_fan the generating fan of the orbit fan
# @param Array<Matrix<Scalar>> gens the generators of a matrix group that acts on the ambient space
# @tparam Scalar underlying number type
# @return fan::PolyhedralFan the orbit fan of //input_fan// w.r.t. the matrix action generated by //gens//
# @example To calculate an orbit fan, follow these steps:
# First specify a seed fan:
# > $f=new PolyhedralFan(RAYS=>[[1,1,1],[1,1,0],[1,1/2,1/4]],MAXIMAL_CONES=>[[0,2],[1,2]]);
# Then define the orbit fan by specifying a matrix group action:
# > $of = orbit_fan($f,polytope::cube(2,group=>1)->GROUP->MATRIX_ACTION);
# The only properties of $of defined so far reside in GROUP:
# > print $of->GROUP->properties();
# | name: unnamed#0
# | type: Group as PolyhedralFan<Rational>::GROUP
# |
# | MATRIX_ACTION
# | type: MatrixActionOnVectors<Rational>
# |
# | MAXIMAL_CONES_ACTION
# | type: PermutationAction<Int, Rational> as PolyhedralFan<Rational>::GROUP::MAXIMAL_CONES_ACTION
# |
# Now you can calculate the [[RAYS]] and [[MAXIMAL_CONES]] of the orbit fan.
# > print $of->RAYS;
# | 1 1 1
# | 1 1 0
# | 1 1/2 1/4
# | 1 -1 -1
# | 1 -1 1
# | 1 1 -1
# | 1 -1 0
# | 1 0 -1
# | 1 0 1
# | 1 -1/2 -1/4
# | 1 -1/2 1/4
# | 1 -1/4 -1/2
# | 1 -1/4 1/2
# | 1 1/4 -1/2
# | 1 1/4 1/2
# | 1 1/2 -1/4
# > print $of->N_MAXIMAL_CONES;
# | 16
user_function orbit_fan<Scalar>(fan::PolyhedralFan<Scalar>, group::MatrixActionOnVectors<Scalar>) {
my ($input_fan, $a) = @_;
my $ra = new group::MatrixActionOnVectors<Scalar>(GENERATORS=>$a->GENERATORS, RAYS_GENERATORS=>$input_fan->give("RAYS | INPUT_RAYS"));
my $ca = new group::MatrixActionOnVectors<Scalar>(MAXIMAL_CONES_GENERATORS=>$input_fan->MAXIMAL_CONES);
my $g = new group::Group;
my $p = new fan::PolyhedralFan;
$p->add("GROUP", $g, MATRIX_ACTION=>$ra, MAXIMAL_CONES_ACTION=>$ca);
return $p;
}
# @category Symmetry
# Constructs the orbit complex of a given polyhedral complex //input_complex//
# with respect to a given set of generators //gens//.
# @param fan::PolyhedralComplex input_complex the generating complex of the orbit complex
# @param Array<Array<Int>> gens the generators of a permutation group that acts on the coordinates of the ambient space
# @return fan::PolyhedralComplex the orbit complex of //input_complex// w.r.t. the coordinate action generated by //gens//
# @example [prefer ppl] To calculate an orbit complex with respect to a group of coordinate permutations, follow these steps:
# First specify a seed complex:
# > $f=new PolyhedralComplex(VERTICES=>[[1,1,1],[1,1,0],[1,-1,-1]], MAXIMAL_POLYTOPES=>[[0,1],[1,2]]);
# Then define the orbit complex by specifying a permutation action on coordinates:
# > $oc = orbit_complex($f, [[1,0]]);
# The only properties of $oc defined so far reside in GROUP:
# > print $oc->GROUP->properties();
# | type: Group as PolyhedralComplex<Rational>::GROUP
# |
# | COORDINATE_ACTION
# | type: PermutationAction<Int, Rational> as PolyhedralComplex<Rational>::GROUP::COORDINATE_ACTION
# |
# | MAXIMAL_POLYTOPES_ACTION
# | type: PermutationAction<Int, Rational> as PolyhedralComplex<Rational>::GROUP::MAXIMAL_POLYTOPES_ACTION
#
# Now you can calculate the [[VERTICES]] and [[MAXIMAL_POLYTOPES]] of the orbit fan.
# > print $oc->VERTICES;
# | 1 1 1
# | 1 1 0
# | 1 -1 -1
# | 1 0 1
# > print $oc->N_MAXIMAL_POLYTOPES;
# | 4
user_function orbit_complex(fan::PolyhedralComplex, can_convert_to<Array<Array<Int>>>) {
my ($input, $gens) = @_;
return new fan::PolyhedralComplex(GROUP => new group::Group,
"GROUP.COORDINATE_ACTION" => new group::PermutationAction(),
"GROUP.COORDINATE_ACTION.GENERATORS" => $gens,
"GROUP.COORDINATE_ACTION.VERTICES_GENERATORS" => $input->give("VERTICES | POINTS"),
"GROUP.MAXIMAL_POLYTOPES_ACTION" => new group::PermutationAction(),
"GROUP.MAXIMAL_POLYTOPES_ACTION.MAXIMAL_POLYTOPES_GENERATORS" => $input->MAXIMAL_POLYTOPES);
}
# @category Symmetry
# Constructs the orbit complex of a given polyhedral complex //input_complex//
# with respect to a given group action //a//.
# @param fan::PolyhedralComplex input_complex the generating complex of the orbit complex
# @param group::PermutationAction a the action of a permutation group on the coordinates of the ambient space
# @return fan::PolyhedralComplex the orbit complex of //input_complex// w.r.t. the action //a//
# @example To calculate an orbit complex with respect to a group of coordinate permutations, follow these steps:
# First specify a seed complex:
# > $f=new PolyhedralComplex(VERTICES=>[[1,1,1],[1,1,0],[1,1/2,1/4]], MAXIMAL_POLYTOPES=>[[0,2],[1,2]]);
# Then define the orbit complex by specifying a matrix group action on the coordinates:
# > $oc = orbit_complex($f, polytope::cube(2,group=>1)->GROUP->MATRIX_ACTION);
# The only properties of $oc defined so far reside in GROUP:
# | type: Group as PolyhedralComplex<Rational>::GROUP
# |
# | MATRIX_ACTION_ON_COMPLEX
# | type: MatrixActionOnVectors<Rational> as PolyhedralComplex<Rational>::GROUP::MATRIX_ACTION_ON_COMPLEX
# |
# | MAXIMAL_POLYTOPES_ACTION
# | type: PermutationAction<Int, Rational> as PolyhedralComplex<Rational>::GROUP::MAXIMAL_POLYTOPES_ACTION
#
# Now you can calculate the [[VERTICES]] and [[MAXIMAL_POLYTOPES]] of the orbit fan.
# > print $oc->VERTICES;
# | 1 1 1
# | 1 1 0
# | 1 1/2 1/4
# | 1 -1 -1
# | 1 -1 1
# | 1 1 -1
# | 1 -1 0
# | 1 0 -1
# | 1 0 1
# | 1 -1/2 -1/4
# | 1 -1/2 1/4
# | 1 -1/4 -1/2
# | 1 -1/4 1/2
# | 1 1/4 -1/2
# | 1 1/4 1/2
# | 1 1/2 -1/4
# > print $oc->N_MAXIMAL_POLYTOPES;
# | 16
user_function orbit_complex<Scalar>(fan::PolyhedralComplex<Scalar>, group::MatrixActionOnVectors<Scalar>) {
my ($input, $a) = @_;
my $ma = new group::MatrixActionOnVectors<Scalar>();
return new fan::PolyhedralComplex(GROUP => new group::Group,
"GROUP.MATRIX_ACTION_ON_COMPLEX" => $ma,
"GROUP.MATRIX_ACTION_ON_COMPLEX.GENERATORS" => $a->GENERATORS,
"GROUP.MATRIX_ACTION_ON_COMPLEX.VERTICES_GENERATORS" => $input->give("VERTICES | POINTS"),
"GROUP.MAXIMAL_POLYTOPES_ACTION" => new group::PermutationAction(),
"GROUP.MAXIMAL_POLYTOPES_ACTION.MAXIMAL_POLYTOPES_GENERATORS" => $input->MAXIMAL_POLYTOPES);
}
# @category Symmetry
# Returns the permutation action induced by the symmetry group of the fan //f// on the set of //k//-dimensional cones.
# This action is not stored as a property of //f//, because polymake doesn't support dynamic names of properties.
# Be aware that the set of //k//-dimensional cones itself is ''%% $f->CONES->[$k-1] %%''.
# @param fan::PolyhedralFan f the input fan
# @param Int k the dimension of the cones to induce the action on
# @return group::PermutationAction a the action induced by //Aut//(//f//) on the set of //k//-dimensional cones
# @example Consider a 3-cube //c//. To calculate the induced action of //Aut//(//c//) on the set of 2-dimensional cones of the normal fan, type
# > $f = fan::normal_fan(polytope::cube(3, group=>1));
# > print fan::cones_action($f,2)->properties();
# | name: CONES_ACTION(2)
# | type: PermutationAction<Int, Rational>
# | description: action induced on 2-dimensional cones
# |
# | GENERATORS
# | 0 3 4 1 2 5 7 6 8 10 9 11
# | 1 0 2 5 6 3 4 7 9 8 11 10
# | 0 2 1 4 3 8 9 10 5 6 7 11
# > print $f->CONES->[1];
# | {2 4}
# | {0 4}
# | {0 2}
# | {1 4}
# | {1 2}
# | {3 4}
# | {0 3}
# | {1 3}
# | {2 5}
# | {0 5}
# | {1 5}
# | {3 5}
user_function cones_action(fan::PolyhedralFan, $) {
my ($this, $k) = @_;
my $a = new group::PermutationAction("CONES_ACTION($k)");
my $cones = $this->HASSE_DIAGRAM->cones_of_dim($k);
$a->GENERATORS = group::induced_permutations($this->GROUP->RAYS_ACTION->GENERATORS, $cones);
if (defined(my $cc = $this->lookup("GROUP.RAYS_ACTION.CONJUGACY_CLASS_REPRESENTATIVES"))) {
$a->CONJUGACY_CLASS_REPRESENTATIVES = group::induced_permutations($cc, $cones);
}
$a->description = "action induced on $k-dimensional cones";
return $a;
}
# @category Symmetry
# Compute the combinatorial symmetries (i.e., automorphisms of the face lattice) of
# a given fan //f//. They are stored in terms of a GROUP.RAYS_ACTION and a GROUP.MAXIMAL_CONES_ACTION
# property in //f//, and the GROUP.MAXIMAL_CONES_ACTION is also returned.
# @param PolyhedralFan f
# @return group::PermutationAction the action of the combinatorial symmetry group on the rays
# @example To get the ray symmetry group of the square and print its generators, type the following:
# > print combinatorial_symmetries(normal_fan(polytope::cube(2)))->GENERATORS;
# | 2 3 0 1
# | 1 0 3 2
# | 0 2 1 3
# > $f = normal_fan(polytope::cube(2)); combinatorial_symmetries($f);
# > print $f->GROUP->RAYS_ACTION->GENERATORS;
# | 0 1 3 2
# | 1 0 2 3
# | 2 3 0 1
# > print $f->GROUP->MAXIMAL_CONES_ACTION->GENERATORS;
# | 2 3 0 1
# | 1 0 3 2
# | 0 2 1 3
user_function combinatorial_symmetries(fan::PolyhedralFan) {
my ($f) = @_;
return group::combinatorial_symmetries_impl($f, $f->MAXIMAL_CONES, "MAXIMAL_CONES_ACTION", "RAYS_ACTION");
}
# Local Variables:
# mode: perl
# cperl-indent-level: 3
# indent-tabs-mode:nil
# End:
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