atlas::rootdata::RootSystem Class Reference

#include <rootdata.h>

Inheritance diagram for atlas::rootdata::RootSystem:

Collaboration diagram for atlas::rootdata::RootSystem:

Classes
struct	root_info

Public Member Functions
	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
typedef signed char	byte
typedef matrix::Vector< byte >	Byte_vector

Private Member Functions
byte &	Cartan_entry (weyl::Generator i, weyl::Generator j)
const byte &	Cartan_entry (weyl::Generator i, weyl::Generator j) const
Byte_vector &	root (RootNbr i)
Byte_vector &	coroot (RootNbr i)
const Byte_vector &	root (RootNbr i) const
const Byte_vector &	coroot (RootNbr i) const

Private Attributes
size_t	rk
Byte_vector	Cmat
std::vector< root_info >	ri
int_Vector	two_rho_in_simple_roots
std::vector< Permutation >	root_perm
	Root permutations induced by reflections in all positive roots. More...

Member Typedef Documentation

typedef signed char atlas::rootdata::RootSystem::byte

private

typedef matrix::Vector<byte> atlas::rootdata::RootSystem::Byte_vector

private

Constructor & Destructor Documentation

atlas::rootdata::RootSystem::RootSystem ( const int_Matrix &  Cartan_matrix )

explicit

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

Member Function Documentation

RankFlags atlas::rootdata::RootSystem::ascent_set ( RootNbr  alpha ) const

inline

int atlas::rootdata::RootSystem::bracket	(	RootNbr	alpha,
		RootNbr	beta
	)		const

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

inline

byte& atlas::rootdata::RootSystem::Cartan_entry ( weyl::Generator  i, weyl::Generator  j  )

inlineprivate

const byte& atlas::rootdata::RootSystem::Cartan_entry ( weyl::Generator  i, weyl::Generator  j  ) const

inlineprivate

int_Matrix atlas::rootdata::RootSystem::cartanMatrix

(

)

const

int_Matrix atlas::rootdata::RootSystem::cartanMatrix

(

const RootNbrList &

sub

)

const

Returns the Cartan matrix of the root subsystem with basis |rb|.

int atlas::rootdata::RootSystem::colevel

(

RootNbr

alpha

)

const

Byte_vector& atlas::rootdata::RootSystem::coroot ( RootNbr  i )

inlineprivate

const Byte_vector& atlas::rootdata::RootSystem::coroot ( RootNbr  i ) const

inlineprivate

int_Vector atlas::rootdata::RootSystem::coroot_expr

(

RootNbr

alpha

)

const

RankFlags atlas::rootdata::RootSystem::descent_set ( RootNbr  alpha ) const

inline

Permutation atlas::rootdata::RootSystem::extend_to_roots

(

const RootNbrList &

simple_images

)

const

size_t atlas::rootdata::RootSystem::find_descent ( RootNbr  alpha ) const

inline

RootNbrList atlas::rootdata::RootSystem::high_roots

(

)

const

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

inline

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

inline

bool atlas::rootdata::RootSystem::is_negroot ( RootNbr  alpha ) const

inline

bool atlas::rootdata::RootSystem::is_posroot ( RootNbr  alpha ) const

inline

bool atlas::rootdata::RootSystem::is_simple_root ( RootNbr  alpha ) const

inline

bool atlas::rootdata::RootSystem::isOrthogonal ( RootNbr  alpha, RootNbr  beta  ) const

inline

int atlas::rootdata::RootSystem::level

(

RootNbr

alpha

)

const

LieType atlas::rootdata::RootSystem::Lie_type

(

)

const

LieType atlas::rootdata::RootSystem::Lie_type

(

RootNbrList

sub

)

const

RootNbrSet atlas::rootdata::RootSystem::long_orthogonalize

(

const RootNbrSet &

rset

)

const

Makes the orthogonal system |rset| into an equaivalent (for |refl_prod|) one that is additively closed inside the full root system.

This can be achieved by repeatedly replacing a pair of short roots spanning a B2 subsystem by a pair of long roots of that subsystem. Although we record no information about relative root lengths it is easily seen that the long roots in some B2 subsystem can never be short roots in another subsystem, so there is no need to assure newly created roots are inspected subsequently.

unsigned long atlas::rootdata::RootSystem::numPosRoots ( ) const

inline

unsigned long atlas::rootdata::RootSystem::numRoots ( ) const

inline

RootNbr atlas::rootdata::RootSystem::permuted_root ( const WeylWord &  ww, RootNbr  r  ) const

inline

RootNbr atlas::rootdata::RootSystem::permuted_root ( RootNbr  r, const WeylWord &  ww  ) const

inline

matrix::Vector< int > atlas::rootdata::RootSystem::pos_system_vec

(

const RootNbrList &

Delta

)

const

RootNbr atlas::rootdata::RootSystem::posRootIndex ( size_t  alpha ) const

inline

RootNbr atlas::rootdata::RootSystem::posRootNbr ( size_t  alpha ) const

inline

RootNbrSet atlas::rootdata::RootSystem::posRootSet

(

)

const

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

inline

WeylWord atlas::rootdata::RootSystem::reflectionWord

(

RootNbr

r

)

const

Byte_vector& atlas::rootdata::RootSystem::root ( RootNbr  i )

inlineprivate

const Byte_vector& atlas::rootdata::RootSystem::root ( RootNbr  i ) const

inlineprivate

int_Vector atlas::rootdata::RootSystem::root_expr

(

RootNbr

alpha

)

const

const Permutation& atlas::rootdata::RootSystem::root_permutation ( RootNbr  alpha ) const

inline

Permutation atlas::rootdata::RootSystem::root_permutation

(

const Permutation &

twist

)

const

RootNbr atlas::rootdata::RootSystem::rootMinus ( RootNbr  alpha ) const

inline

RootNbr atlas::rootdata::RootSystem::rt_abs ( RootNbr  alpha ) const

inline

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

inline

RootNbr atlas::rootdata::RootSystem::simple_reflected_root ( weyl::Generator  i, RootNbr  r  ) const

inline

const Permutation& atlas::rootdata::RootSystem::simple_root_permutation ( weyl::Generator  i ) const

inline

RootNbrList atlas::rootdata::RootSystem::simpleBasis

(

RootNbrSet

rs

)

const

RootNbr atlas::rootdata::RootSystem::simpleRootIndex ( size_t  alpha ) const

inline

RootNbrList atlas::rootdata::RootSystem::simpleRootList

(

)

const

RootNbr atlas::rootdata::RootSystem::simpleRootNbr ( weyl::Generator  i ) const

inline

RootNbrSet atlas::rootdata::RootSystem::simpleRootSet

(

)

const

bool atlas::rootdata::RootSystem::sumIsRoot	(	RootNbr	alpha,
		RootNbr	beta,
		RootNbr &	gamma
	)		const

bool atlas::rootdata::RootSystem::sumIsRoot ( RootNbr  alpha, RootNbr  beta  ) const

inline

template<typename I , typename O >

void atlas::rootdata::RootSystem::toRootBasis	(	I	first,
		I	last,
		O	out
	)		const

In this template we assume that |I| is an |InputIterator| with |value_type| |RootNbr|, and that |O| is an |OutputIterator| with |value_type| |Weight|. We output via |out| the expressions of the roots numbered in |[first,last[| in the simple root basis.

template<typename I , typename O >

void atlas::rootdata::RootSystem::toRootBasis	(	I	first,
		I	last,
		O	out,
		const RootNbrList &	rb
	)		const

In this template, |I| is an Input_iterator with |value_type| |RootNbr|, and |O| is an OutputIterator with |value_type| |Weight|. We assume that |rb| contains a basis of some sub rootsystem of |rd|, and that |I| inputs |RootNbr|s corresponding to roots in that subsystem. Then we write to |out| the expression of the root in the basis |rb|.

The idea is to use the coroots of the vectors in |rb| to get the expression of both the input roots and those from |rb| itself in the simple weight basis for |rb| (this is done by |toSimpleWeights| below); the latter matrix (the Cartan matrix of |rb|) is square, so we can then apply |baseChange|, which amounts to left-multiplication by the inverse of that Cartan matrix.

template<typename I , typename O >

void atlas::rootdata::RootSystem::toSimpleWeights	(	I	first,
		I	last,
		O	out,
		const RootNbrList &	rb
	)		const

In this template we assume that |I| is an InputIterator with value type |RootNbr|, that |O| is an OutputIterator with |value_type| |Weight|, and that |rb| flags a basis for some sub rootsystem of our |RootSystem|. The range $[first,last)$ should access root numbers of roots in the subsystem. Then for each |v| in this range we output to |out| the expression of |v| in the simple weight basis of the root subsystem |rb|; this is obtained simply by taking scalar products with the coroots of the roots in |rb|.

Member Data Documentation

Byte_vector atlas::rootdata::RootSystem::Cmat

private

std::vector<root_info> atlas::rootdata::RootSystem::ri

private

size_t atlas::rootdata::RootSystem::rk

private

std::vector<Permutation> atlas::rootdata::RootSystem::root_perm

private

Root permutations induced by reflections in all positive roots.

int_Vector atlas::rootdata::RootSystem::two_rho_in_simple_roots

private

---

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