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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.
--------------------------------------------------------------------------------
*/
#pragma once
#include "polymake/Map.h"
#include "polymake/Matrix.h"
#include "polymake/IncidenceMatrix.h"
#include "polymake/ListMatrix.h"
#include "polymake/SparseVector.h"
#include "polymake/linalg.h"
#include "polymake/group/orbit.h"
#include "polymake/group/group_tools.h"
namespace polymake { namespace group {
template <typename Perm>
SparseMatrix<Rational> permutation_matrix(const Perm& perm,
const Array<Int>& coordinate_permutation)
{
SparseMatrix<Rational> permutation_matrix(degree(perm), degree(perm));
Int i = 0;
for (auto pit = entire(perm); !pit.at_end(); ++pit, ++i)
permutation_matrix(coordinate_permutation[*pit],
coordinate_permutation[i]) = 1;
return permutation_matrix;
}
template <typename Perm>
Int inverse_perm_at(const Perm& perm, Int k)
{
assert(k < perm.size());
Int i = 0;
for (auto pit = entire(perm); !pit.at_end(); ++pit, ++i)
if (*pit == k)
return i;
std::ostringstream msg;
wrap(msg) << "The array " << perm << " is not a permutation.";
throw std::runtime_error(msg.str());
}
/*
template <typename InducedAction, typename RowType>
SparseMatrix<Rational>
isotypic_projector_impl(const RowType& character,
const InducedAction& induced_action,
Int degree,
const ConjugacyClasses& conjugacy_classes,
Int order)
{
SparseMatrix<Rational> isotypic_projector(degree, degree);
for (Int i = 0; i<conjugacy_classes.size(); ++i) {
for (auto cit = entire(conjugacy_classes[i]); !cit.at_end(); ++cit) {
isotypic_projector +=
character[i] // FIXME: conjugate here, once complex character tables are implemented
* induced_action.induced_rep(*cit);
}
}
// chi(G.identity())/G.order() * sum([chi(g).conjugate() * rep.rho(g) for g in G])
return isotypic_projector * character[0] / order;
}
*/
template <typename E>
auto matrix_rep(const Array<E>& g, const Array<Int>& permutation_to_orbit_order)
{
return permutation_matrix(g, permutation_to_orbit_order);
}
template <typename Scalar>
auto matrix_rep(const Matrix<Scalar>& g, const Array<Int>&)
{
return g;
}
/*
Implement the projector to the isotypic component given by the character //chi// using the formula
pi_chi = chi(G.identity())/G.order() * sum([chi(g).conjugate() * rep.rho(g) for g in G])
*/
template <typename RowType, typename Element, typename Scalar>
SparseMatrix<CharacterNumberType>
isotypic_projector_impl(const RowType& chi,
const ConjugacyClasses<Element>& conjugacy_classes,
const Array<Int>& permutation_to_orbit_order,
Int group_order,
const Scalar& )
{
const Int deg(degree(conjugacy_classes[0][0]));
SparseMatrix<CharacterNumberType> isotypic_projector(deg, deg);
for (Int i = 0; i < conjugacy_classes.size(); ++i) {
if (is_zero(chi[i])) continue;
for (const auto& cc: conjugacy_classes[i]) {
isotypic_projector +=
chi[i] // FIXME: conjugate here, once complex character tables are implemented
* matrix_rep(cc, permutation_to_orbit_order);
}
}
isotypic_projector *= chi[0] / group_order;
return isotypic_projector;
}
template <typename RowType, typename Element>
SparseMatrix<double>
isotypic_projector_impl(const RowType& chi,
const ConjugacyClasses<Element>& conjugacy_classes,
const Array<Int>& permutation_to_orbit_order,
Int group_order,
const double& )
{
const Int deg = degree(conjugacy_classes[0][0]);
SparseMatrix<double> isotypic_projector(deg, deg);
for (Int i = 0; i < conjugacy_classes.size(); ++i) {
if (is_zero(chi[i])) continue;
for (const auto& cc: conjugacy_classes[i]) {
isotypic_projector +=
chi[i] // FIXME: conjugate here, once complex character tables are implemented
* matrix_rep(cc, permutation_to_orbit_order);
}
}
isotypic_projector *= chi[0] / double(group_order);
return isotypic_projector;
}
template <typename InducedAction, typename RowType, typename Element>
ListMatrix<SparseVector<CharacterNumberType> >
isotypic_basis_impl(const RowType& character,
const InducedAction& induced_action,
Int degree,
const ConjugacyClasses<Element>& conjugacy_classes,
Int order)
{
ListMatrix<SparseVector<CharacterNumberType>>
isotypic_basis(0, degree),
kernel_so_far(unit_matrix<CharacterNumberType>(degree));
// we fill the matrix row-wise. The entire matrix is
// chi(G.identity())/G.order() * sum([chi(g).conjugate() * rep.rho(g) for g in G])
// and rep.rho(g) (k, g^{-1}(k)) = 1.
for (Int k = 0; k < degree; ++k) {
SparseVector<CharacterNumberType> new_row(degree);
for (Int i = 0; i < conjugacy_classes.size(); ++i) {
for (const auto& cc: conjugacy_classes[i]) {
for (Int j = 0; j < degree; ++j)
if (induced_action.index_of_inverse_image(cc, j) == k) {
new_row[j] +=
character[i]; // FIXME: conjugate here, once complex character tables are implemented
}
}
}
add_row_if_rowspace_increases(isotypic_basis, new_row, kernel_so_far);
}
return isotypic_basis * character[0] / order;
}
template <typename RowType, typename Element>
ListMatrix<SparseVector<CharacterNumberType> >
isotypic_basis_impl(const RowType& character,
const ConjugacyClasses<Element>& conjugacy_classes,
const Array<Int>& permutation_to_orbit_order,
Int order)
{
const Int deg = degree(conjugacy_classes[0][0]);
ListMatrix<SparseVector<CharacterNumberType>>
isotypic_basis(0, deg),
kernel_so_far(unit_matrix<CharacterNumberType>(deg));
// we fill the matrix row-wise. The entire matrix is
// chi(G.identity())/G.order() * sum([chi(g).conjugate() * rep.rho(g) for g in G])
// and rep.rho(g) (k, g^{-1}(k)) = 1.
for (Int k = 0; k < deg; ++k) {
SparseVector<CharacterNumberType> new_row(deg);
for (Int i = 0; i < conjugacy_classes.size(); ++i) {
if (is_zero(character[i])) continue;
for (auto cit = entire(conjugacy_classes[i]); !cit.at_end(); ++cit) {
new_row[permutation_to_orbit_order[inverse_perm_at(*cit,k)]] +=
character[i]; // FIXME: conjugate here, once complex character tables are implemented
}
}
add_row_if_rowspace_increases(isotypic_basis, new_row, kernel_so_far);
}
return isotypic_basis * (character[0] / order);
}
template <typename SparseMatrixType, typename Element>
IncidenceMatrix<>
isotypic_supports_impl(const SparseMatrixType& S,
const Matrix<CharacterNumberType>& character_table,
const ConjugacyClasses<Element>& conjugacy_classes,
const Array<Int>& permutation_to_orbit_order,
Int order)
{
const Int n_irreps = character_table.rows();
IncidenceMatrix<> supp(S.rows(), n_irreps);
for (Int i = 0; i < n_irreps; ++i) {
const SparseMatrix<CharacterNumberType> image = isotypic_projector_impl(character_table[i], conjugacy_classes, permutation_to_orbit_order, order, CharacterNumberType()) * T(S);
Int j = 0;
for (auto cit = entire(cols(image)); !cit.at_end(); ++cit, ++j)
if (!is_zero(*cit))
supp(j,i) = 1;
}
return supp;
}
} }
// Local Variables:
// mode:C++
// c-basic-offset:3
// indent-tabs-mode:nil
// End:
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