Word ThRightNormalForm::getShortWord( ) const
{
Word result;
Permutation omega = Permutation::getHalfTwistPermutation( getRank( ) );
int power = getPower( );
if( power<0 ) {
const list< Permutation >& decomp = getDecomposition( );
list<Permutation>:: const_iterator it = decomp.begin( );
for( int j=0 ; it!=decomp.end( ) ; ++it, ++j ) {
int n = j - decomp.size( ) - power;
if( n<0 ) {
vector< int > gd = (*it).geodesic();
for( size_t t=0 ; t<gd.size() ; ++t )
result.push_back( gd[t]+1 );
} else {
Permutation p = ( n%2 == 1 ? (*it).inverse() * omega : omega * (*it).inverse() );
vector<int> gd = p.geodesic();
for( int t=gd.size( )-1 ; t>=0 ; --t )
result.push_back( -gd[t]-1 );
}
}
Word omega_w = Word(omega.geodesicWord( ));
omega_w = -omega_w;
for( int j=decomp.size( ) ; j<-power ; ++j )
result = omega_w*result;
} else
result = getWord( );
return result;
}
Word ThLeftNormalForm::getReducedWord() const {
if (theOmegaPower >= 0 || theDecomposition.size() == 0)
return getWord();
const auto p = -theOmegaPower;
const auto d = theDecomposition.size();
const auto a = p < d ? p : d;
Word result;
// 1. Process omega
const Permutation omega = Permutation::getHalfTwistPermutation(theRank);
Word omegaWord = Word(omega.geodesicWord());
omegaWord = -omegaWord;
result = omegaWord.power(p - a);
// 2. Cancel omega^-1 with positive permutations
auto it = theDecomposition.begin();
for (int i = 0; i < a; ++i, ++it) {
auto perm = (-(*it)) * omega;
if ((a - i - 1) % 2 != 0)
perm = perm.flip();
result *= -Word(perm.geodesicWord());
}
// 3. process the rest of positive permutations
for (; it != theDecomposition.end(); ++it) {
result *= Word((*it).geodesicWord());
}
return result;
}
Word ThLeftNormalForm::getWord() const {
Word result;
// A. Take care of Deltas
if( theOmegaPower!=0 ) {
const Permutation omega = Permutation::getHalfTwistPermutation( theRank );
Word omegaWord = Word(omega.geodesicWord());
if( theOmegaPower<0 )
omegaWord = -omegaWord;
result = omegaWord.power(abs(theOmegaPower));
}
// B. Take care of permutation braids
for (const auto &d : theDecomposition) {
result *= Word(d.geodesicWord());
}
return result;
}
Word ThLeftNormalForm::getReducedWord2() const {
if (theOmegaPower >= 0 || theDecomposition.size() == 0)
return getWord();
const auto p = -theOmegaPower;
const auto d = theDecomposition.size();
auto a = p < d ? p : d;
Word result;
// 1. Process omega
const Permutation omega = Permutation::getHalfTwistPermutation(theRank);
Word omegaWord = Word(omega.geodesicWord());
omegaWord = -omegaWord;
result = omegaWord.power(p - a);
// 2. Sort permutations and find the cut_value
vector<int> sizes;
for (const auto &perm : getDecomposition())
sizes.push_back(perm.geodesic().size());
sort(sizes.begin(), sizes.end(), [](const int &lhs, const int &rhs) { return lhs > rhs; });
const int cut_value = sizes[a - 1];
// 3. Cancel omega^-1 with positive permutations
for (auto it = theDecomposition.begin(); it != theDecomposition.end(); ++it) {
auto perm = *it;
if (a > 0 && perm.geodesic().size() >= cut_value) {
perm = (-perm) * omega;
if ((--a) % 2 != 0)
perm = perm.flip();
result *= -Word(perm.geodesicWord());
} else {
if (a % 2 != 0)
perm = perm.flip();
result *= Word(perm.geodesicWord());
}
}
return result;
}
