atlas::tits::TitsCoset Class Reference

#include <tits.h>

Inheritance diagram for atlas::tits::TitsCoset:

Collaboration diagram for atlas::tits::TitsCoset:

Public Member Functions
	TitsCoset (const InnerClass &G, Grading base_grading)
	TitsCoset (const InnerClass &G)
	TitsCoset (const InnerClass &G, tags::DualTag)
	~TitsCoset ()
const TitsGroup &	titsGroup () const
const WeylGroup &	weylGroup () const
bool	hasTwistedCommutation (weyl::Generator s, const TwistedInvolution &tw) const
Grading	base_grading () const
bool	is_valid (TitsElt a) const
bool	simple_grading (const TitsElt &a, size_t s) const
bool	simple_imaginary_grading (TorusPart x, RootNbr alpha) const
bool	grading (TitsElt a, RootNbr alpha) const
void	basedTwistedConjugate (TitsElt &a, size_t s) const
void	strict_based_twisted_conjugate (TitsElt &a, size_t s) const
void	basedTwistedConjugate (TitsElt &a, const WeylWord &w) const
void	basedTwistedConjugate (const WeylWord &w, TitsElt &a) const
void	Cayley_transform (TitsElt &a, size_t s) const
void	inverse_Cayley_transform (TitsElt &a, size_t s, const SmallSubspace &mod_space) const
TitsElt	twisted (const TitsElt &a) const
TorusPart	twisted (const TorusPart &t) const

Private Attributes
TitsGroup *	my_Tits_group
const TitsGroup &	Tg
Grading	grading_offset
const RootSystem &	rs

Detailed Description

The class |TitsCoset| augments the |TitsGroup| class with the choice of a basic strong involution $$, which is needed to define the additional methods |simple_grading|, |basedTwistedConjugate| and |Cayley_transform| (although the latter happens to have an implementation independent of ).

All |TitsElement| values $t.$ get reinterpreted as $t..$ We are no longer in a group, but in a coset of the group. Also the images of the operations |basedTwistedConjugate| and |Cayley_transform| are only defined modulo the equivalence relation generated by conjugation by torus elements. If conjugation by $w.$ acts as $$ on the Lie algebra $h$ of $H$, then the equivalence class of $a=t..$ is obtained by modifying $t$ by any value in the direction of the image of $-1$ (the $-1$ eigenspace of $$; in the concrete representation |TorusPart|, by any reduction modulo 2 of a $-$ fixed vector). In practice this means most of all that any torus element $x$ the elements $x.a$ and $(x).a=a.x$ and are equivalent.

Constructor & Destructor Documentation

atlas::tits::TitsCoset::TitsCoset	(	const InnerClass &	G,
		Grading	base_grading
	)

atlas::tits::TitsCoset::TitsCoset

(

const InnerClass &

G

)

atlas::tits::TitsCoset::TitsCoset	(	const InnerClass &	G,
		tags::DualTag
	)

atlas::tits::TitsCoset::~TitsCoset ( )

inline

Member Function Documentation

Grading atlas::tits::TitsCoset::base_grading ( ) const

inline

void atlas::tits::TitsCoset::basedTwistedConjugate ( TitsElt &  a, size_t  s  ) const

inline

void atlas::tits::TitsCoset::basedTwistedConjugate	(	TitsElt &	a,
		const WeylWord &	w
	)		const

void atlas::tits::TitsCoset::basedTwistedConjugate	(	const WeylWord &	w,
		TitsElt &	a
	)		const

void atlas::tits::TitsCoset::Cayley_transform ( TitsElt &  a, size_t  s  ) const

inline

bool atlas::tits::TitsCoset::grading	(	TitsElt	a,
		RootNbr	alpha
	)		const

bool atlas::tits::TitsCoset::hasTwistedCommutation ( weyl::Generator  s, const TwistedInvolution &  tw  ) const

inline

void atlas::tits::TitsCoset::inverse_Cayley_transform	(	TitsElt &	a,
		size_t	s,
		const SmallSubspace &	mod_space
	)		const

bool atlas::tits::TitsCoset::is_valid

(

TitsElt

a

)

const

bool atlas::tits::TitsCoset::simple_grading ( const TitsElt &  a, size_t  s  ) const

inline

bool atlas::tits::TitsCoset::simple_imaginary_grading	(	TorusPart	x,
		RootNbr	alpha
	)		const

void atlas::tits::TitsCoset::strict_based_twisted_conjugate	(	TitsElt &	a,
		size_t	s
	)		const

const TitsGroup& atlas::tits::TitsCoset::titsGroup ( ) const

inline

TitsElt atlas::tits::TitsCoset::twisted

(

const TitsElt &

a

)

const

TorusPart atlas::tits::TitsCoset::twisted ( const TorusPart &  t ) const

inline

const WeylGroup& atlas::tits::TitsCoset::weylGroup ( ) const

inline

Member Data Documentation

Grading atlas::tits::TitsCoset::grading_offset

private

TitsGroup* atlas::tits::TitsCoset::my_Tits_group

private

const RootSystem& atlas::tits::TitsCoset::rs

private

const TitsGroup& atlas::tits::TitsCoset::Tg

private

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

• /home/ran/atlas_project/latest_branch_07182016/sources/structure/tits.h
• /home/ran/atlas_project/latest_branch_07182016/sources/structure/tits.cpp