atlas::rootdata Namespace Reference

Classes
struct	atlas::rootdata::RootBasisTag
class	atlas::rootdata::RootDatum
	Based root datum for a complex reductive group. More...
class	atlas::rootdata::RootIterator< I >
	Iterator for traversing a set of roots. More...
Typedefs
typedef unsigned long	RootNbr
	Integer representing the location of a root on the list maintained by a RootDatum.
typedef latticetypes::LatticeElt	Root
	Type for a root, coroot, character, or cocharacter: an element of the lattice Z^n.
typedef std::vector< RootNbr >	RootList
	List of _numbers_ of roots, referring to the list of roots in RootDatum.
typedef RootIterator< RootList::const_iterator >	WRootIterator
typedef bitmap::BitMap	RootSet
	BitMap representing a set of roots (such as the set of positive roots, or noncompact roots).
typedef std::vector< RootSet >	RootSetList
Functions
void	cartanMatrix (latticetypes::LatticeMatrix &c, const RootDatum &rd)
	Puts in c the Cartan matrix of the root datum.
void	cartanMatrix (latticetypes::LatticeMatrix &c, const RootList &rb, const RootDatum &rd)
	Puts in c the Cartan matrix of the root basis rb.
void	dualBasedInvolution (LT::LatticeMatrix &di, const LT::LatticeMatrix &i, const RootDatum &rd)
	Puts i^vee in di.
LT::LatticeMatrix	dualBasedInvolution (const LT::LatticeMatrix &i, const RootDatum &rd)
void	lieType (lietype::LieType &lt, const RootList &rb, const RootDatum &rd)
	puts in lt the Lie type of the root system spanned by rb.
void	longest (LT::LatticeMatrix &q, const RootDatum &rd)
	Puts in q the matrix of the action of w_0;.
void	makeOrthogonal (RootList &orth, const RootList &rl, const RootList &rs, const RootDatum &rd)
	Puts in orth the elements of rs which are orthogonal to all elements of rl.
void	reflectionMatrix (LT::LatticeMatrix &qs, RootNbr n, const RootDatum &rd)
	Writes in qs the matrix of the reflection through root #n.
void	rootBasis (RootList &rb, const RootList &rl, const RootDatum &rd)
	Puts in rb the canonical basis (set of simple roots) of rl.
void	rootBasis (RootList &rb, RootSet rs, const RootDatum &rd)
	puts in rb the canonical basis (set of simple roots) of rs.
void	strongOrthogonalize (RootList &rl, const RootDatum &rd)
	Makes rl into a strongly orthogonal system.
bool	sumIsRoot (const Weight &a, const Weight &b, const RootDatum &rd)
	Tells if a+b is a root.
void	toDistinguished (LatticeMatrix &q, const RootDatum &rd)
	Transforms q into w.q, where w is the unique element such that w.q fixes the positive Weyl chamber.
void	toMatrix (LatticeMatrix &q, const weyl::WeylWord &ww, const RootDatum &rd)
	Writes in q the matrix represented by ww.
void	toMatrix (LatticeMatrix &q, const RootList &rl, const RootDatum &rd)
	Writes in q the matrix represented by rl (the product of the root reflections for the various roots in rl.).
void	toPositive (weyl::WeylWord &ww, const Weight &d_v, const RootDatum &rd)
	Puts in ww a list of simple reflections constituting a reduced expression of the unique shortest element w in the Weyl group such that w.v is in the positive chamber.
void	toWeylWord (weyl::WeylWord &ww, RootNbr rn, const RootDatum &rd)
	Writes in ww the expression of the reflection about root #rn as a product of simple reflections.
void	cartanMatrix (LT::LatticeMatrix &, const RootDatum &)
void	cartanMatrix (LT::LatticeMatrix &, const RootList &, const RootDatum &)
void	reflectionMatrix (LT::LatticeMatrix &, RootNbr, const RootDatum &, RootBasisTag)
bool	sumIsRoot (const LT::Weight &, const LT::Weight &, const RootDatum &)
void	toDistinguished (LT::LatticeMatrix &, const RootDatum &)
void	toMatrix (LT::LatticeMatrix &, const weyl::WeylWord &, const RootDatum &)
void	toMatrix (LT::LatticeMatrix &, const RootList &, const RootDatum &)
void	toPositive (weyl::WeylWord &, const LT::Weight &, const RootDatum &)
template<typename I>
RootIterator< I >	operator+ (typename RootIterator< I >::difference_type n, RootIterator< I > i)

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Typedef Documentation

typedef latticetypes::LatticeElt atlas::rootdata::Root

Type for a root, coroot, character, or cocharacter: an element of the lattice Z^n. Definition at line 42 of file rootdata_fwd.h. Referenced by atlas::rootdata::RootDatum::RootDatum(), and atlas::rootdata::RootDatum::rootPermutation().

typedef std::vector<RootNbr> atlas::rootdata::RootList

List of _numbers_ of roots, referring to the list of roots in RootDatum. According to Fokko, this type should therefore have been called RootNbrList. Definition at line 50 of file rootdata_fwd.h. Referenced by atlas::compactEquations(), atlas::cartanset::CartanClassSet::correlateDualForms(), atlas::cartanset::CartanClassSet::correlateForms(), atlas::gradings::findGrading(), atlas::gradings::gradingType(), atlas::realform_io::Interface::Interface(), atlas::realredgp::RealReductiveGroup::RealReductiveGroup(), atlas::rootdata::RootDatum::simpleBasis(), and atlas::rootdata::RootDatum::simpleRootList().

typedef unsigned long atlas::rootdata::RootNbr

Integer representing the location of a root on the list maintained by a RootDatum. The roots in a root datum should almost always be referred to by their numbers. Definition at line 33 of file rootdata_fwd.h. Referenced by atlas::gradings::gradingType(), atlas::rootdata::RootDatum::posRootNbr(), atlas::rootdata::RootDatum::reflectedRoot(), atlas::rootdata::RootDatum::RootDatum(), atlas::rootdata::RootDatum::rootMinus(), atlas::rootdata::RootDatum::rootNbr(), atlas::rootdata::RootDatum::simpleRootNbr(), and atlas::cartanclass::toMostSplit().

typedef bitmap::BitMap atlas::rootdata::RootSet

BitMap representing a set of roots (such as the set of positive roots, or noncompact roots). Definition at line 56 of file rootdata_fwd.h. Referenced by atlas::cartanclass::Fiber::compactRoots(), atlas::cartanclass::InvolutionData::complex_roots(), atlas::cartanclass::Fiber::complexRootSet(), atlas::cartanset::CartanClassSet::extend(), atlas::gradings::findGrading(), atlas::kgb::grading_offset_for(), atlas::cartanclass::InvolutionData::imaginary_roots(), atlas::cartanclass::CartanClass::imaginaryRootSet(), atlas::cartanclass::Fiber::imaginaryRootSet(), atlas::cartanclass::Fiber::makeBaseGrading(), atlas::cartanclass::Fiber::makeGradingShifts(), atlas::cartanset::makeRepresentative(), atlas::cartanset::CartanClassSet::noncompactPosRootSet(), atlas::realredgp::RealReductiveGroup::noncompactRoots(), atlas::cartanset::CartanClassSet::noncompactRoots(), atlas::cartanclass::Fiber::noncompactRoots(), atlas::orthogonalMAlpha(), atlas::rootdata::RootDatum::posRootSet(), atlas::prettyprint::printInRootBasis(), atlas::cartanclass::InvolutionData::real_roots(), atlas::cartanclass::CartanClass::realRootSet(), atlas::cartanclass::Fiber::realRootSet(), atlas::cartanclass::restrictGrading(), rootBasis(), atlas::rootdata::RootDatum::simpleBasis(), atlas::rootdata::RootDatum::simpleRootSet(), atlas::kgb::square_class_grading_offset(), atlas::cartanclass::toMostSplit(), and atlas::rootdata::RootDatum::twoRho().

typedef std::vector<RootSet> atlas::rootdata::RootSetList

Definition at line 57 of file rootdata_fwd.h.

typedef RootIterator<RootList::const_iterator> atlas::rootdata::WRootIterator

Definition at line 51 of file rootdata_fwd.h. Referenced by atlas::rootdata::RootDatum::beginPosCoroot(), atlas::rootdata::RootDatum::beginPosRoot(), atlas::rootdata::RootDatum::beginSimpleCoroot(), atlas::rootdata::RootDatum::beginSimpleRoot(), atlas::interpreter::check_involution(), atlas::rootdata::RootDatum::endPosCoroot(), atlas::rootdata::RootDatum::endPosRoot(), atlas::rootdata::RootDatum::endSimpleCoroot(), atlas::rootdata::RootDatum::endSimpleRoot(), atlas::fillRoots(), atlas::poscoroots_f(), atlas::posroots_f(), atlas::rootdata::RootDatum::RootDatum(), atlas::simplecoroots_f(), and atlas::simpleroots_f().

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Function Documentation

void cartanMatrix (  LT::LatticeMatrix &  , const RootList &  , const RootDatum &  )

Referenced by atlas::interpreter::Cartan_matrix_wrapper(), atlas::interpreter::check_involution(), atlas::cmatrix_f(), atlas::testrun::CoveringIterator::CoveringIterator(), atlas::interpreter::datum_Cartan_wrapper(), atlas::complexredgp::lieType(), atlas::cartanclass::CartanClass::makeSimpleComplex(), atlas::interpreter::print_gradings_wrapper(), atlas::cartan_io::printGradings(), atlas::interpreter::root_datum_wrapper(), atlas::interactive_lattice::smithBasis(), atlas::interpreter::smithBasis(), atlas::interpreter::type_of_datum(), and atlas::involutions::helper::Helper::weylCorrelation().

void cartanMatrix (  LT::LatticeMatrix &  , const RootDatum &  )

void cartanMatrix (  latticetypes::LatticeMatrix &  c, const RootList &  rb, const RootDatum &  rd )

Puts in c the Cartan matrix of the root basis rb. Precondition: rb contains a basis of a sub-root system of the root system of rd. Definition at line 720 of file rootdata.cpp. References atlas::rootdata::RootDatum::coroot(), atlas::latticetypes::LatticeMatrix, atlas::matrix::Matrix< C >::resize(), atlas::rootdata::RootDatum::root(), and atlas::latticetypes::scalarProduct().

void cartanMatrix (  latticetypes::LatticeMatrix &  c, const RootDatum &  rd )

Puts in c the Cartan matrix of the root datum. Definition at line 702 of file rootdata.cpp. References atlas::rootdata::RootDatum::cartan(), atlas::latticetypes::LatticeMatrix, atlas::matrix::Matrix< C >::resize(), and atlas::rootdata::RootDatum::semisimpleRank(). Referenced by lieType().

LT::LatticeMatrix atlas::rootdata::dualBasedInvolution (  const LT::LatticeMatrix &  i, const RootDatum &  rd )

Definition at line 773 of file rootdata.cpp. References dualBasedInvolution(), and atlas::latticetypes::LatticeMatrix.

void atlas::rootdata::dualBasedInvolution (  LT::LatticeMatrix &  di, const LT::LatticeMatrix &  i, const RootDatum &  rd )

Puts i^vee in di. Precondition: i is an involution of rd as a _based_ root datum; Postcondition: di is an involution of the dual based root datum; Algorithm: di is (w_0^t)(-i^t). (The first factor is the long element of the Weyl group, acting on the dual root datum, and the second factor is the negative transpose of i.) [DV 13 July 2006: since RootDatum no longer contains the involution defining the inner class, this function seems not to belong in the rootdata namespace. Not sure where would be a better home.] Definition at line 742 of file rootdata.cpp. References atlas::latticetypes::LatticeMatrix, longest(), atlas::matrix::Matrix< C >::negate(), and atlas::matrix::Matrix< C >::transpose(). Referenced by atlas::complexredgp::ComplexReductiveGroup::ComplexReductiveGroup(), and dualBasedInvolution().

void atlas::rootdata::lieType (  lietype::LieType &  lt, const RootList &  rb, const RootDatum &  rd )

puts in lt the Lie type of the root system spanned by rb. Precondition: rb contains a basis of a sub-rootsystem of the root system of rd. Definition at line 785 of file rootdata.cpp. References cartanMatrix(), and atlas::latticetypes::LatticeMatrix. Referenced by atlas::interpreter::check_involution(), atlas::complexredgp::lieType(), atlas::cartanclass::CartanClass::makeOrbitSize(), atlas::interpreter::print_Cartan_info_wrapper(), atlas::interpreter::print_gradings_wrapper(), atlas::testprint::printBlockData(), atlas::cartan_io::printCartanClass(), atlas::realweyl::RealWeyl::RealWeyl(), atlas::interpreter::type_of_Cartan_matrix_wrapper(), and atlas::interpreter::type_of_datum().

void atlas::rootdata::longest (  LT::LatticeMatrix &  q, const RootDatum &  rd )

Puts in q the matrix of the action of w_0;. Algorithm: w_0 is the element making -twoRho positive. NOTE: not intended for heavy use. Should be precomputed if that is the case. Definition at line 803 of file rootdata.cpp. References atlas::matrix::identityMatrix(), atlas::latticetypes::LatticeMatrix, atlas::rootdata::RootDatum::rank(), atlas::rootdata::RootDatum::rootReflection(), atlas::rootdata::RootDatum::simpleRootNbr(), toPositive(), atlas::rootdata::RootDatum::twoRho(), atlas::latticetypes::Weight, and atlas::weyl::WeylWord. Referenced by dualBasedInvolution().

void atlas::rootdata::makeOrthogonal (  RootList &  orth, const RootList &  rl, const RootList &  rs, const RootDatum &  rd )

Puts in orth the elements of rs which are orthogonal to all elements of rl. Definition at line 825 of file rootdata.cpp. References atlas::rootdata::RootDatum::isOrthogonal(), and atlas::matrix::Matrix< C >::resize(). Referenced by atlas::interactive_lattice::getLattice().

template<typename I> RootIterator<I> operator+ (  typename RootIterator< I >::difference_type  n, RootIterator< I >  i )  [inline]

Definition at line 574 of file rootdata.h.

void reflectionMatrix (  LT::LatticeMatrix &  , RootNbr  , const RootDatum &  , RootBasisTag  )

void atlas::rootdata::reflectionMatrix (  LT::LatticeMatrix &  qs, RootNbr  n, const RootDatum &  rd )

Writes in qs the matrix of the reflection through root #n. Definition at line 846 of file rootdata.cpp. References atlas::rootdata::RootDatum::coroot(), atlas::latticetypes::LatticeCoeff, atlas::latticetypes::LatticeMatrix, atlas::rootdata::RootDatum::rank(), atlas::matrix::Matrix< C >::resize(), atlas::rootdata::RootDatum::root(), and atlas::latticetypes::Weight.

void atlas::rootdata::rootBasis (  RootList &  rb, RootSet  rs, const RootDatum &  rd )

puts in rb the canonical basis (set of simple roots) of rs. Precondition: rs holds the roots in a sub-root system of rd; Algorithm: compute the sum of the positive roots in rs; extract the positive roots whose scalar product with that sum is 2. Definition at line 890 of file rootdata.cpp. References atlas::bitmap::BitMap::begin(), atlas::rootdata::RootDatum::posRootSet(), atlas::rootdata::RootDatum::rank(), atlas::rootdata::RootDatum::root(), RootSet, and atlas::rootdata::RootDatum::scalarProduct(). Referenced by atlas::compactEquations(), atlas::gradings::findGrading(), and atlas::realweyl::RealWeyl::RealWeyl().

void atlas::rootdata::rootBasis (  RootList &  rb, const RootList &  rl, const RootDatum &  rd )

Puts in rb the canonical basis (set of simple roots) of rl. Precondition: rl holds the roots in a sub-root system of rd; Forwarded to the RootSet version. Therefore the order of the roots in |rl| is ignored! The roots of |rb| are by order of occurrence in |rd.roots()| Definition at line 874 of file rootdata.cpp. References atlas::rootdata::RootDatum::numRoots(), and RootSet.

void atlas::rootdata::strongOrthogonalize (  RootList &  rl, const RootDatum &  rd )

Makes rl into a strongly orthogonal system. Precondition: rl is a set of pairwise orthogonal roots; Algorithm: for i,j in rl, the sum can be a root iff i,j span a subsystem of type B2, and are short there; we can then replace i and j by their sum and difference. Note that this will not create new bad pairs, as bad pairs are made up of short roots. Definition at line 908 of file rootdata.cpp. References atlas::rootdata::RootDatum::root(), atlas::rootdata::RootDatum::rootNbr(), atlas::rootdata::RootDatum::sumIsRoot(), and atlas::latticetypes::Weight. Referenced by atlas::cartanset::cayley_and_cross_part(), atlas::cartanset::cayleyPart(), and atlas::cartanclass::toMostSplit().

bool sumIsRoot (  const LT::Weight &  , const LT::Weight &  , const RootDatum &  )

Referenced by atlas::rootdata::RootDatum::sumIsRoot().

bool sumIsRoot (  const Weight &  a, const Weight &  b, const RootDatum &  rd )

Tells if a+b is a root. Precondition: a and b are roots; Definition at line 941 of file rootdata.cpp. References atlas::rootdata::RootDatum::isRoot(), and atlas::latticetypes::Weight.

void toDistinguished (  LT::LatticeMatrix &  , const RootDatum &  )

void toDistinguished (  LatticeMatrix &  q, const RootDatum &  rd )

Transforms q into w.q, where w is the unique element such that w.q fixes the positive Weyl chamber. Note that wq is then automatically an involution; w_0.w.q permutes the simple roots. [DV: I don't understand this last remark. Probably q is meant to be an involutive automorphism of order 2 of the (unbased) root datum. This would make w.q an involution of the based root datum, but then w_0.w.q would send the simple roots to their negatives.] Definition at line 959 of file rootdata.cpp. References atlas::matrix::Matrix< C >::apply(), atlas::latticetypes::LatticeMatrix, atlas::rootdata::RootDatum::rank(), atlas::matrix::Matrix< C >::swap(), toMatrix(), toPositive(), atlas::rootdata::RootDatum::twoRho(), and atlas::latticetypes::Weight.

void toMatrix (  LT::LatticeMatrix &  , const RootList &  , const RootDatum &  )

Referenced by atlas::cartanset::CartanClassSet::involutionMatrix(), and atlas::realredgp::RealReductiveGroup::RealReductiveGroup().

void toMatrix (  LT::LatticeMatrix &  , const weyl::WeylWord &  , const RootDatum &  )

void toMatrix (  LatticeMatrix &  q, const RootList &  rl, const RootDatum &  rd )

Writes in q the matrix represented by rl (the product of the root reflections for the various roots in rl.). NOTE: the products are written in the same order as the appearance of the roots in rl. In practice this will be used for sequences of orthogonal roots, so that the order doesn't matter. Definition at line 1008 of file rootdata.cpp. References atlas::matrix::identityMatrix(), atlas::latticetypes::LatticeMatrix, atlas::rootdata::RootDatum::rank(), and atlas::rootdata::RootDatum::rootReflection().

void toMatrix (  LatticeMatrix &  q, const weyl::WeylWord &  ww, const RootDatum &  rd )

Writes in q the matrix represented by ww. NOTE: not intended for heavy use. If that were the case, it would be better to use the decomposition of ww into pieces, and multiply those together. However, that would be for the ComplexGroup to do, as it is it that has access to both the Weyl group and the root datum. Definition at line 989 of file rootdata.cpp. References atlas::matrix::identityMatrix(), atlas::latticetypes::LatticeMatrix, atlas::rootdata::RootDatum::rank(), atlas::rootdata::RootDatum::rootReflection(), and atlas::rootdata::RootDatum::simpleRootNbr(). Referenced by toDistinguished().

void toPositive (  weyl::WeylWord &  , const LT::Weight &  , const RootDatum &  )

void toPositive (  weyl::WeylWord &  ww, const Weight &  d_v, const RootDatum &  rd )

Puts in ww a list of simple reflections constituting a reduced expression of the unique shortest element w in the Weyl group such that w.v is in the positive chamber. Algorithm: the greedy algorithm -- if v is not positive, there is a simple coroot alpha^v such that <v,alpha^v> is < 0; then s_alpha.v takes v closer to the positive chamber. Definition at line 1029 of file rootdata.cpp. References atlas::latticetypes::scalarProduct(), atlas::rootdata::RootDatum::semisimpleRank(), atlas::rootdata::RootDatum::simpleCoroot(), atlas::rootdata::RootDatum::simpleReflection(), and atlas::latticetypes::Weight. Referenced by longest(), atlas::rootdata::RootDatum::reflectionWord(), toDistinguished(), toWeylWord(), and atlas::rootdata::RootDatum::word_of_inverse_matrix().

void atlas::rootdata::toWeylWord (  weyl::WeylWord &  ww, RootNbr  rn, const RootDatum &  rd )

Writes in ww the expression of the reflection about root #rn as a product of simple reflections. Definition at line 1064 of file rootdata.cpp. References atlas::rootdata::RootDatum::reflection(), toPositive(), atlas::rootdata::RootDatum::twoRho(), and atlas::latticetypes::Weight.

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