Based root datum for a complex reductive group. More...

#include <rootdata.h>

Inheritance diagram for atlas::rootdata::RootDatum:

Collaboration diagram for atlas::rootdata::RootDatum:

Public Member Functions
	RootDatum ()

	RootDatum (const PreRootDatum &)

	RootDatum (const RootDatum &, tags::DualTag)

	RootDatum (int_Matrix &projector, const RootDatum &, tags::DerivedTag)

	RootDatum (int_Matrix &injector, const RootDatum &, tags::AdjointTag)

RootDatum	sub_datum (const RootNbrList &generators) const

const RootSystem &	root_system () const

size_t	rank () const

size_t	semisimpleRank () const

WeightList::const_iterator	beginRoot () const

WeightList::const_iterator	endRoot () const

CoweightList::const_iterator	beginCoroot () const

CoweightList::const_iterator	endCoroot () const

CoweightList::const_iterator	beginRadical () const

CoweightList::const_iterator	endRadical () const

WeightList::const_iterator	beginCoradical () const

WeightList::const_iterator	endCoradical () const

WRootIterator	beginSimpleRoot () const

WRootIterator	endSimpleRoot () const

WRootIterator	beginPosRoot () const

WRootIterator	endPosRoot () const

WRootIterator	beginSimpleCoroot () const

WRootIterator	endSimpleCoroot () const

WRootIterator	beginPosCoroot () const

WRootIterator	endPosCoroot () const

bool	isRoot (const Weight &v) const

bool	isSemisimple () const

const Weight &	root (RootNbr i) const

const Weight &	simpleRoot (weyl::Generator i) const

const Weight &	posRoot (size_t i) const

RootNbr	root_index (const Root &r) const

const Coweight &	coroot (RootNbr i) const

const Coweight &	simpleCoroot (weyl::Generator i) const

const Coweight &	posCoroot (size_t i) const

RootNbr	coroot_index (const Root &r) const

RatWeight	fundamental_weight (weyl::Generator i) const

RatCoweight	fundamental_coweight (weyl::Generator i) const

bool	isAdjoint () const
	Tells whether the rootdatum is the rootdatum of an adjoint group. More...

bool	isSimplyConnected () const
	Tells whether the rootdatum is the rootdatum of a simply connected group. More...

const Weight &	twoRho () const

const Coweight &	dual_twoRho () const

int	scalarProduct (const Weight &v, RootNbr j) const

bool	isOrthogonal (const Weight &v, RootNbr j) const

int	cartan (weyl::Generator i, weyl::Generator j) const

template<typename C >
void	reflect (RootNbr alpha, matrix::Vector< C > &lambda) const

template<typename C >
void	coreflect (matrix::Vector< C > &co_lambda, RootNbr alpha) const

void	reflect (RootNbr alpha, LatticeMatrix &M) const

void	reflect (LatticeMatrix &M, RootNbr alpha) const

template<typename C >
matrix::Vector< C >	reflection (matrix::Vector< C > lambda, RootNbr alpha) const

template<typename C >
matrix::Vector< C >	coreflection (matrix::Vector< C > co_lambda, RootNbr alpha) const

template<typename C >
void	simple_reflect (weyl::Generator i, matrix::Vector< C > &v) const

template<typename C >
void	simple_coreflect (matrix::Vector< C > &v, weyl::Generator i) const

void	simple_reflect (weyl::Generator i, LatticeMatrix &M) const

void	simple_reflect (LatticeMatrix &M, weyl::Generator i) const

template<typename C >
matrix::Vector< C >	simple_reflection (weyl::Generator i, matrix::Vector< C > lambda) const

template<typename C >
matrix::Vector< C >	simple_coreflection (matrix::Vector< C > ell, weyl::Generator i) const

WeylWord	to_dominant (Weight lambda) const
	Returns a reduced expression of the shortest \|w\| making \|w.v\| dominant. More...

void	act (const WeylWord &ww, Weight &lambda) const

Weight	image_by (const WeylWord &ww, Weight lambda) const

void	act_inverse (Weight &lambda, const WeylWord &ww) const

Weight	image_by_inverse (Weight lambda, const WeylWord &ww) const

void	dual_act (Coweight &ell, const WeylWord &ww) const

Weight	dual_image_by (Coweight ell, const WeylWord &ww) const

Permutation	rootPermutation (const WeightInvolution &q) const
	Returns the permutation of the roots induced by \|q\|. More...

WeightInvolution	root_reflection (RootNbr r) const
	Returns the reflection for root #alpha. More...

WeightInvolution	simple_reflection (weyl::Generator i) const

WeightInvolution	matrix (const WeylWord &ww) const
	Writes in q the matrix represented by ww. More...

WeylWord	reflectionWord (RootNbr r) const

int_Vector	inSimpleRoots (RootNbr alpha) const

int_Vector	inSimpleCoroots (RootNbr alpha) const

Weight	twoRho (const RootNbrList &) const
	Returns the sum of the positive roots in rl. More...

Weight	twoRho (const RootNbrSet &) const
	Returns the sum of the positive roots in rs. More...

Coweight	dual_twoRho (const RootNbrList &) const

Coweight	dual_twoRho (const RootNbrSet &) const

WeylWord	word_of_inverse_matrix (const WeightInvolution &) const
	Returns the expression of $q^{-1}$ as a product of simple reflections. More...

void	swap (RootDatum &)

Public Member Functions inherited from atlas::rootdata::RootSystem
	RootSystem (const int_Matrix &Cartan_matrix)

	RootSystem (const RootSystem &rs, tags::DualTag)

size_t	rank () const

unsigned long	numPosRoots () const

unsigned long	numRoots () const

int	cartan (weyl::Generator i, weyl::Generator j) const

int_Matrix	cartanMatrix () const

LieType	Lie_type () const

int_Matrix	cartanMatrix (const RootNbrList &sub) const
	Returns the Cartan matrix of the root subsystem with basis \|rb\|. More...

LieType	Lie_type (RootNbrList sub) const

int_Vector	root_expr (RootNbr alpha) const

int_Vector	coroot_expr (RootNbr alpha) const

int	level (RootNbr alpha) const

int	colevel (RootNbr alpha) const

template<typename I , typename O >
void	toRootBasis (I, I, O) const

template<typename I , typename O >
void	toRootBasis (I, I, O, const RootNbrList &) const

template<typename I , typename O >
void	toSimpleWeights (I, I, O, const RootNbrList &) const

bool	is_simple_root (RootNbr alpha) const

bool	is_posroot (RootNbr alpha) const

bool	is_negroot (RootNbr alpha) const

RootNbr	simpleRootNbr (weyl::Generator i) const

RootNbr	posRootNbr (size_t alpha) const

RootNbr	simpleRootIndex (size_t alpha) const

RootNbr	posRootIndex (size_t alpha) const

RootNbr	rootMinus (RootNbr alpha) const

RootNbr	rt_abs (RootNbr alpha) const

RootNbrSet	simpleRootSet () const

RootNbrList	simpleRootList () const

RootNbrSet	posRootSet () const

const Permutation &	simple_root_permutation (weyl::Generator i) const

RankFlags	descent_set (RootNbr alpha) const

RankFlags	ascent_set (RootNbr alpha) const

size_t	find_descent (RootNbr alpha) const

bool	is_descent (weyl::Generator i, RootNbr alpha) const

bool	is_ascent (weyl::Generator i, RootNbr alpha) const

RootNbr	simple_reflected_root (weyl::Generator i, RootNbr r) const

void	simple_reflect_root (weyl::Generator i, RootNbr &r) const

RootNbr	permuted_root (const WeylWord &ww, RootNbr r) const

RootNbr	permuted_root (RootNbr r, const WeylWord &ww) const

const Permutation &	root_permutation (RootNbr alpha) const

bool	isOrthogonal (RootNbr alpha, RootNbr beta) const

int	bracket (RootNbr alpha, RootNbr beta) const

Permutation	root_permutation (const Permutation &twist) const

Permutation	extend_to_roots (const RootNbrList &simple_images) const

WeylWord	reflectionWord (RootNbr r) const

matrix::Vector< int >	pos_system_vec (const RootNbrList &Delta) const

RootNbrList	simpleBasis (RootNbrSet rs) const

bool	sumIsRoot (RootNbr alpha, RootNbr beta, RootNbr &gamma) const

bool	sumIsRoot (RootNbr alpha, RootNbr beta) const

RootNbrSet	long_orthogonalize (const RootNbrSet &rest) const
	Makes the orthogonal system \|rset\| into an equaivalent (for \|refl_prod\|) one that is additively closed inside the full root system. More...

RootNbrList	high_roots () const

Private Types
enum	StatusFlagNames { IsAdjoint, IsSimplyConnected, numFlags }
	Names describing the bits of the bitset d_status. More...

typedef BitSet< numFlags >	Status

Private Member Functions
void	fillStatus ()

Private Attributes
size_t	d_rank
	Rank of the root datum. More...

WeightList	d_roots
	Full list of roots. More...

CoweightList	d_coroots
	Full list of coroots. More...

WeightList	weight_numer
	Fundamental weight numerators. More...

CoweightList	coweight_numer
	Fundamental coweight numerators. More...

CoweightList	d_radicalBasis
	Basis for orthogonal to coroots. More...

WeightList	d_coradicalBasis
	Basis for orthogonal to roots. More...

Weight	d_2rho
	Sum of the positive roots. More...

Coweight	d_dual_2rho

int	Cartan_denom
	Denominator for (co)\|weight_numer\|. More...

Status	d_status
	BitSet recording whether the root datum is adjoint/simply connected. More...

Detailed Description

Based root datum for a complex reductive group.

What we call a root datum in this program is what is usually called a based root datum, in other words a fixed choice of positive roots is always assumed.

The root datum defines the complex reductive group entirely. It consists of a |RootSystem| that describes the roots and coroots in the lattices they span themselves, and of additional data that correspond to embeddings of these lattices into mutually dual free abelian groups (weight and coweight lattices).

The rank |d_rank| is that of the weight and coweight lattices, the root system itself has rank |semisimpleRank()| which may be smaller. The roots and coroots are stored in compact form in the |RootSystem|, and again as represented in the weight and coweight lattices, for efficiency of retrieval under this form. Also constructed are various useful auxiliary things, like d_twoRho (the sum of the positive roots).

The code is designed to make it preferable always to refer to a root by its number (index in the root system), for which we use the type name |RootNbr|.

Member Typedef Documentation

typedef BitSet<numFlags> atlas::rootdata::RootDatum::Status

private

Member Enumeration Documentation

enum atlas::rootdata::RootDatum::StatusFlagNames

private

Names describing the bits of the bitset d_status.

The last enum numFlags is there as a standard programming trick. Its value - in this case 2 - is one-past-the-last meaningful bit, for use by accessors.

Enumerator
IsAdjoint
IsSimplyConnected
numFlags

Constructor & Destructor Documentation

atlas::rootdata::RootDatum::RootDatum ( )

inline

atlas::rootdata::RootDatum::RootDatum ( const PreRootDatum & prd )

explicit

atlas::rootdata::RootDatum::RootDatum	(	const RootDatum &	rd,
		tags::DualTag
	)

atlas::rootdata::RootDatum::RootDatum	(	int_Matrix &	projector,
		const RootDatum &	rd,
		tags::DerivedTag
	)

atlas::rootdata::RootDatum::RootDatum	(	int_Matrix &	injector,
		const RootDatum &	rd,
		tags::AdjointTag
	)

Member Function Documentation

void atlas::rootdata::RootDatum::act	(	const WeylWord &	ww,
		Weight &	lambda
	)		const

inline

void atlas::rootdata::RootDatum::act_inverse	(	Weight &	lambda,
		const WeylWord &	ww
	)		const

inline

WeightList::const_iterator atlas::rootdata::RootDatum::beginCoradical ( ) const

inline

CoweightList::const_iterator atlas::rootdata::RootDatum::beginCoroot ( ) const

inline

WRootIterator atlas::rootdata::RootDatum::beginPosCoroot ( ) const

inline

WRootIterator atlas::rootdata::RootDatum::beginPosRoot ( ) const

inline

CoweightList::const_iterator atlas::rootdata::RootDatum::beginRadical ( ) const

inline

WeightList::const_iterator atlas::rootdata::RootDatum::beginRoot ( ) const

inline

WRootIterator atlas::rootdata::RootDatum::beginSimpleCoroot ( ) const

inline

WRootIterator atlas::rootdata::RootDatum::beginSimpleRoot ( ) const

inline

int atlas::rootdata::RootDatum::cartan	(	weyl::Generator	i,
		weyl::Generator	j
	)		const

inline

template<typename C >

void atlas::rootdata::RootDatum::coreflect	(	matrix::Vector< C > &	co_lambda,
		RootNbr	alpha
	)		const

inline

template<typename C >

matrix::Vector<C> atlas::rootdata::RootDatum::coreflection	(	matrix::Vector< C >	co_lambda,
		RootNbr	alpha
	)		const

inline

const Coweight& atlas::rootdata::RootDatum::coroot ( RootNbr i ) const

inline

RootNbr atlas::rootdata::RootDatum::coroot_index ( const Root & r ) const

inline

void atlas::rootdata::RootDatum::dual_act	(	Coweight &	ell,
		const WeylWord &	ww
	)		const

inline

Weight atlas::rootdata::RootDatum::dual_image_by	(	Coweight	ell,
		const WeylWord &	ww
	)		const

inline

const Coweight& atlas::rootdata::RootDatum::dual_twoRho ( ) const

inline

Coweight atlas::rootdata::RootDatum::dual_twoRho ( const RootNbrList & rl ) const

Coweight atlas::rootdata::RootDatum::dual_twoRho ( const RootNbrSet & rs ) const

WeightList::const_iterator atlas::rootdata::RootDatum::endCoradical ( ) const

inline

CoweightList::const_iterator atlas::rootdata::RootDatum::endCoroot ( ) const

inline

WRootIterator atlas::rootdata::RootDatum::endPosCoroot ( ) const

inline

WRootIterator atlas::rootdata::RootDatum::endPosRoot ( ) const

inline

CoweightList::const_iterator atlas::rootdata::RootDatum::endRadical ( ) const

inline

WeightList::const_iterator atlas::rootdata::RootDatum::endRoot ( ) const

inline

WRootIterator atlas::rootdata::RootDatum::endSimpleCoroot ( ) const

inline

WRootIterator atlas::rootdata::RootDatum::endSimpleRoot ( ) const

inline

void atlas::rootdata::RootDatum::fillStatus ( )

private

RatWeight atlas::rootdata::RootDatum::fundamental_coweight ( weyl::Generator i ) const

RatWeight atlas::rootdata::RootDatum::fundamental_weight ( weyl::Generator i ) const

Weight atlas::rootdata::RootDatum::image_by	(	const WeylWord &	ww,
		Weight	lambda
	)		const

inline

Weight atlas::rootdata::RootDatum::image_by_inverse	(	Weight	lambda,
		const WeylWord &	ww
	)		const

inline

int_Vector atlas::rootdata::RootDatum::inSimpleCoroots ( RootNbr alpha ) const

inline

int_Vector atlas::rootdata::RootDatum::inSimpleRoots ( RootNbr alpha ) const

inline

bool atlas::rootdata::RootDatum::isAdjoint ( ) const

inline

Tells whether the rootdatum is the rootdatum of an adjoint group.

NOTE: we define a reductive group to be adjoint if its center is connected. An equivalent condition is that the derived group of the dual group is simply connected.

bool atlas::rootdata::RootDatum::isOrthogonal	(	const Weight &	v,
		RootNbr	j
	)		const

inline

bool atlas::rootdata::RootDatum::isRoot ( const Weight & v ) const

inline

bool atlas::rootdata::RootDatum::isSemisimple ( ) const

inline

bool atlas::rootdata::RootDatum::isSimplyConnected ( ) const

inline

Tells whether the rootdatum is the rootdatum of a simply connected group.

NOTE: this is the dual condition to being adjoint: it means that the derived group is simply connected. An equivalent condition is that the center of the dual group is connected.

WeightInvolution atlas::rootdata::RootDatum::matrix ( const WeylWord & ww ) const

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 InnerClass to do, as it is it that has access to both the Weyl group and the root datum.

const Coweight& atlas::rootdata::RootDatum::posCoroot ( size_t i ) const

inline

const Weight& atlas::rootdata::RootDatum::posRoot ( size_t i ) const

inline

size_t atlas::rootdata::RootDatum::rank ( ) const

inline

template<typename C >

void atlas::rootdata::RootDatum::reflect	(	RootNbr	alpha,
		matrix::Vector< C > &	lambda
	)		const

inline

void atlas::rootdata::RootDatum::reflect	(	RootNbr	alpha,
		LatticeMatrix &	M
	)		const

void atlas::rootdata::RootDatum::reflect	(	LatticeMatrix &	M,
		RootNbr	alpha
	)		const

template<typename C >

matrix::Vector<C> atlas::rootdata::RootDatum::reflection	(	matrix::Vector< C >	lambda,
		RootNbr	alpha
	)		const

inline

WeylWord atlas::rootdata::RootDatum::reflectionWord ( RootNbr r ) const

const Weight& atlas::rootdata::RootDatum::root ( RootNbr i ) const

inline

RootNbr atlas::rootdata::RootDatum::root_index ( const Root & r ) const

inline

WeightInvolution atlas::rootdata::RootDatum::root_reflection ( RootNbr alpha ) const

Returns the reflection for root #alpha.

NOTE: this is not intended for heavy use. If that is envisioned, it would be better to construct the matrices once and for all and return const references.

const RootSystem& atlas::rootdata::RootDatum::root_system ( ) const

inline

Permutation atlas::rootdata::RootDatum::rootPermutation ( const WeightInvolution & q ) const

Returns the permutation of the roots induced by |q|.

Precondition: |q| permutes the roots;

int atlas::rootdata::RootDatum::scalarProduct	(	const Weight &	v,
		RootNbr	j
	)		const

inline

size_t atlas::rootdata::RootDatum::semisimpleRank ( ) const

inline

template<typename C >

void atlas::rootdata::RootDatum::simple_coreflect	(	matrix::Vector< C > &	v,
		weyl::Generator	i
	)		const

inline

template<typename C >

matrix::Vector<C> atlas::rootdata::RootDatum::simple_coreflection	(	matrix::Vector< C >	ell,
		weyl::Generator	i
	)		const

inline

template<typename C >

void atlas::rootdata::RootDatum::simple_reflect	(	weyl::Generator	i,
		matrix::Vector< C > &	v
	)		const

inline

void atlas::rootdata::RootDatum::simple_reflect	(	weyl::Generator	i,
		LatticeMatrix &	M
	)		const

inline

void atlas::rootdata::RootDatum::simple_reflect	(	LatticeMatrix &	M,
		weyl::Generator	i
	)		const

inline

template<typename C >

matrix::Vector<C> atlas::rootdata::RootDatum::simple_reflection	(	weyl::Generator	i,
		matrix::Vector< C >	lambda
	)		const

inline

WeightInvolution atlas::rootdata::RootDatum::simple_reflection ( weyl::Generator i ) const

inline

const Coweight& atlas::rootdata::RootDatum::simpleCoroot ( weyl::Generator i ) const

inline

const Weight& atlas::rootdata::RootDatum::simpleRoot ( weyl::Generator i ) const

inline

RootDatum atlas::rootdata::RootDatum::sub_datum ( const RootNbrList & generators ) const

void atlas::rootdata::RootDatum::swap ( RootDatum & other )

WeylWord atlas::rootdata::RootDatum::to_dominant ( Weight v ) const

Returns a reduced expression of the shortest |w| making |w.v| dominant.

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 dominant chamber.

const Weight& atlas::rootdata::RootDatum::twoRho ( ) const

inline

Weight atlas::rootdata::RootDatum::twoRho ( const RootNbrList & rl ) const

Returns the sum of the positive roots in rl.

Precondition: rl holds the roots in a sub-rootsystem of the root system of rd;

Weight atlas::rootdata::RootDatum::twoRho ( const RootNbrSet & rs ) const

Returns the sum of the positive roots in rs.

Precondition: rs holds the roots in a sub-rootsystem of the root system of rd;

WeylWord atlas::rootdata::RootDatum::word_of_inverse_matrix ( const WeightInvolution & q ) const

Returns the expression of $q^{-1}$ as a product of simple reflections.

Precondition: $q$ gives the action on the weight lattice of some Weyl group element

Algorithm: apply $q$ to |twoRho|, then use |toPositive| to find a Weyl word making it dominant again, which by assumption expresses $q^{-1}$.

Member Data Documentation

int atlas::rootdata::RootDatum::Cartan_denom

private

Denominator for (co)|weight_numer|.

CoweightList atlas::rootdata::RootDatum::coweight_numer

private

Fundamental coweight numerators.

Weight atlas::rootdata::RootDatum::d_2rho

private

Sum of the positive roots.

WeightList atlas::rootdata::RootDatum::d_coradicalBasis

private

Basis for orthogonal to roots.

CoweightList atlas::rootdata::RootDatum::d_coroots

private

Full list of coroots.

Coweight atlas::rootdata::RootDatum::d_dual_2rho

private

CoweightList atlas::rootdata::RootDatum::d_radicalBasis

private

Basis for orthogonal to coroots.

size_t atlas::rootdata::RootDatum::d_rank

private

Rank of the root datum.

WeightList atlas::rootdata::RootDatum::d_roots

private

Full list of roots.

Status atlas::rootdata::RootDatum::d_status

private

BitSet recording whether the root datum is adjoint/simply connected.

"Adjoint" here means that the center of the complex group determined by the root datum is connected. "Simply connected" means that the derived group of that complex group is simply connected. These two properties are exchanged by passage to the dual root datum.

WeightList atlas::rootdata::RootDatum::weight_numer

private

Fundamental weight numerators.

The documentation for this class was generated from the following files:

/home/ran/atlas_project/latest_branch_07182016/sources/structure/rootdata.h
/home/ran/atlas_project/latest_branch_07182016/sources/structure/rootdata.cpp

Public Member Functions

Private Types

Private Member Functions

Private Attributes

Detailed Description

Member Typedef Documentation

Member Enumeration Documentation

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation