atlas  0.6
Public Member Functions | Private Attributes | List of all members
atlas::tits::TitsCoset Class Reference

#include <tits.h>

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Public Member Functions

 TitsCoset (const InnerClass &G, Grading base_grading)
 
 TitsCoset (const InnerClass &G)
 
 TitsCoset (const InnerClass &G, tags::DualTag)
 
 ~TitsCoset ()
 
const TitsGrouptitsGroup () 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

TitsGroupmy_Tits_group
 
const TitsGroupTg
 
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

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