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/* Copyright (c) 1997-2015
Ewgenij Gawrilow, Michael Joswig (Technische Universitaet Berlin, Germany)
http://www.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.
--------------------------------------------------------------------------------
*/
#ifndef __GROUP_REPRESENTATIONS_H
#define __GROUP_REPRESENTATIONS_H
#include "polymake/Map.h"
#include "polymake/Matrix.h"
#include "polymake/SparseMatrix.h"
#include "polymake/ListMatrix.h"
#include "polymake/SparseVector.h"
#include "polymake/linalg.h"
#include "polymake/IncidenceMatrix.h"
#include "polymake/Vector.h"
#include "polymake/Rational.h"
namespace polymake { namespace group {
namespace {
template<typename SetType>
class InducedAction {
protected:
int degree;
const Array<SetType>& domain;
const Map<SetType, int>& index_of;
Array<int> inverse_permutation(const Array<int>& perm) const {
Array<int> inv_perm(perm.size());
for (int i=0; i<perm.size(); ++i)
inv_perm[perm[i]] = i;
return inv_perm;
}
public:
InducedAction(int degree,
const Array<SetType>& domain,
const Map<SetType, int>& index_of)
: degree(degree)
, domain(domain)
, index_of(index_of)
{}
int index_of_image(const Array<int>& perm,
const SetType& elt) const {
SetType image;
image.resize(perm.size());
for (typename Entire<SetType>::const_iterator sit = entire(elt); !sit.at_end(); ++sit)
image += perm[*sit];
return index_of[image];
}
int index_of_inverse_image(const Array<int>& perm,
const SetType& elt) const {
Array<int> inv_perm(inverse_permutation(perm));
SetType inv_image;
inv_image.resize(inv_perm.size());
for (typename Entire<SetType>::const_iterator sit = entire(elt); !sit.at_end(); ++sit)
inv_image += inv_perm[*sit];
return index_of[inv_image];
}
int index_of_inverse_image(const Array<int>& perm,
const int elt_index) const {
return index_of_inverse_image(perm, domain[elt_index]);
}
SparseMatrix<Rational> rep(const Array<int>& perm) const {
SparseMatrix<Rational> rep(degree, degree);
int col_index(0);
for (typename Entire<Array<SetType> >::const_iterator dit = entire(domain); !dit.at_end(); ++dit, ++col_index) {
rep(index_of_image(perm, *dit), col_index) = 1;
}
return rep;
}
};
template<typename InducedAction, typename RowType>
SparseMatrix<Rational> isotypic_projector_impl(const RowType& character,
const InducedAction& induced_action,
int degree,
const Array<Set<Array<int> > >& conjugacy_classes,
int order)
{
SparseMatrix<Rational> isotypic_projector(degree, degree);
for (int i=0; i<conjugacy_classes.size(); ++i) {
for (Entire<Set<Array<int> > >::const_iterator cit = entire(conjugacy_classes[i]); !cit.at_end(); ++cit) {
isotypic_projector +=
character[i] // FIXME: conjugate here, once complex character tables are implemented
* induced_action.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 InducedAction, typename RowType>
ListMatrix<SparseVector<Rational> >
isotypic_basis_impl(const RowType& character,
const InducedAction& induced_action,
int degree,
const Array<Set<Array<int> > >& conjugacy_classes,
int order)
{
ListMatrix<SparseVector<Rational> >
isotypic_basis(0, degree),
kernel_so_far(unit_matrix<Rational>(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<Rational> new_row(degree);
for (int i=0; i<conjugacy_classes.size(); ++i) {
for (Entire<Set<Array<int> > >::const_iterator cit = entire(conjugacy_classes[i]); !cit.at_end(); ++cit) {
for (int j=0; j<degree; ++j)
if (induced_action.index_of_inverse_image(*cit, 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 SparseMatrixType, typename InducedAction>
IncidenceMatrix<> isotypic_supports_impl(const SparseMatrixType& S,
const Matrix<Rational>& character_table,
const InducedAction& IA,
const Array<Set<Array<int> > >& conjugacy_classes,
int order,
int degree)
{
const int n_irreps = character_table.rows();
IncidenceMatrix<> supp(S.rows(), n_irreps);
for (int i=0; i<n_irreps; ++i) {
const SparseMatrix<Rational> image = isotypic_projector_impl(character_table[i], IA, degree, conjugacy_classes, order) * T(S);
int j(0);
for (Entire<Cols<SparseMatrix<Rational> > >::const_iterator cit = entire(cols(image)); !cit.at_end(); ++cit, ++j)
if (*cit != zero_vector<Rational>(degree))
supp(j,i) = 1;
}
return supp;
}
} // end anonymous namespace
} }
#endif // __GROUP_REPRESENTATIONS_H
// Local Variables:
// mode:C++
// c-basic-offset:3
// indent-tabs-mode:nil
// End:
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