atlas::tits::TitsGroup Class Reference

#include <tits.h>

Inheritance diagram for atlas::tits::TitsGroup:

Collaboration diagram for atlas::tits::TitsGroup:

Public Member Functions
	TitsGroup (const RootDatum &, const WeylGroup &W, const WeightInvolution &d)
	TitsGroup (const int_Matrix &Cartan_matrix, const WeylGroup &W, const weyl::Twist &twist)
	TitsGroup (const TitsGroup &Tg, const WeylGroup &W)
const size_t	rank () const
const size_t	semisimple_rank () const
TorusPart	m_alpha (size_t j) const
TorusPart	dual_m_alpha (size_t j) const
void	reflect (TorusPart &x, weyl::Generator s) const
TorusPart	push_across (TorusPart x, const WeylElt &w) const
	find torus part $x'$ so that $x.w=w.x'$ More...
TorusPart	pull_across (const WeylElt &w, TorusPart y) const
TorusPart	twisted (const TorusPart x) const
TorusPart	dual_twisted (const TorusPart x) const
TitsElt	twisted (const TitsElt &te) const
TitsElt	dual_twisted (const TitsElt &te, const TorusPart shift) const
void	twist (TorusPart x) const
BinaryMap	involutionMatrix (const WeylWord &tw) const
unsigned long	length (const TitsElt &a) const
TorusPart	left_torus_part (const TitsElt &a) const
TorusPart	right_torus_part (const TitsElt &a) const
void	sigma_mult (weyl::Generator s, TitsElt &a) const
	Left multiplication of |a| by the canonical generator $$. More...
void	sigma_inv_mult (weyl::Generator s, TitsElt &a) const
void	mult_sigma (TitsElt &, weyl::Generator) const
	Right multiplication of |a| by the canonical generator $sigma_s$. More...
void	mult_sigma_inv (TitsElt &, weyl::Generator) const
TitsElt	prod (const TitsElt &a, TitsElt b) const
	Product of general Tits group elements. More...
void	mult (TitsElt &a, const TitsElt &b) const
void	right_add (TitsElt &a, TorusPart t) const
void	left_add (TorusPart t, TitsElt &a) const
void	sigma_conjugate (TitsElt &a, weyl::Generator s) const
void	sigma_inv_conjugate (TitsElt &a, weyl::Generator s) const
void	twistedConjugate (TitsElt &a, weyl::Generator s) const
void	inverseTwistedConjugate (TitsElt &a, weyl::Generator s) const

Private Member Functions
	TitsGroup (const TitsGroup &)
TitsGroup &	operator= (const TitsGroup &)

Private Attributes
size_t	d_rank
std::vector< TorusPart >	d_simpleRoot
std::vector< TorusPart >	d_simpleCoroot
BinaryMap	d_involution
BinaryMap	dual_involution

Constructor & Destructor Documentation

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

private

atlas::tits::TitsGroup::TitsGroup	(	const RootDatum &	rd,
		const WeylGroup &	W,
		const WeightInvolution &	delta
	)

Constructs the Tits group corresponding to the root datum |rd|, and the fundamental involution |d| (which also defines the twist).

atlas::tits::TitsGroup::TitsGroup	(	const int_Matrix &	Cartan_matrix,
		const WeylGroup &	W,
		const weyl::Twist &	twist
	)

Constructs the Tits group corresponding to the adjoint group of type given by |Cartan_matrix| and distinguished involution given by |twist|

atlas::tits::TitsGroup::TitsGroup ( const TitsGroup &  Tg, const WeylGroup &  W  )

inline

Member Function Documentation

TorusPart atlas::tits::TitsGroup::dual_m_alpha ( size_t  j ) const

inline

TorusPart atlas::tits::TitsGroup::dual_twisted ( const TorusPart  x ) const

inline

TitsElt atlas::tits::TitsGroup::dual_twisted	(	const TitsElt &	te,
		const TorusPart	shift
	)		const

void atlas::tits::TitsGroup::inverseTwistedConjugate ( TitsElt &  a, weyl::Generator  s  ) const

inline

BinaryMap atlas::tits::TitsGroup::involutionMatrix

(

const WeylWord &

tw

)

const

void atlas::tits::TitsGroup::left_add ( TorusPart  t, TitsElt &  a  ) const

inline

TorusPart atlas::tits::TitsGroup::left_torus_part ( const TitsElt &  a ) const

inline

unsigned long atlas::tits::TitsGroup::length ( const TitsElt &  a ) const

inline

TorusPart atlas::tits::TitsGroup::m_alpha ( size_t  j ) const

inline

void atlas::tits::TitsGroup::mult ( TitsElt &  a, const TitsElt &  b  ) const

inline

void atlas::tits::TitsGroup::mult_sigma	(	TitsElt &	a,
		weyl::Generator	s
	)		const

Right multiplication of |a| by the canonical generator $sigma_s$.

Algorithm: similar to above, but omitting the |reflect| (since the torus part is at the left), and using |weyl::mult| instead of |weyl::leftMult|, and |right_add| to add the possible contribution $m_s$ instead of |left_add|; the latter implicitly involves a call to |pull_across|.

void atlas::tits::TitsGroup::mult_sigma_inv	(	TitsElt &	a,
		weyl::Generator	s
	)		const

TitsGroup& atlas::tits::TitsGroup::operator= ( const TitsGroup &  )

private

TitsElt atlas::tits::TitsGroup::prod	(	const TitsElt &	a,
		TitsElt	b
	)		const

Product of general Tits group elements.

Algorithm: The algorithm is to multiply the second factors successively by the generators in a reduced expression for |a.w()|, then left-add |a.t()|.

Since we have torus parts on the left, we prefer left-multiplication here.

TorusPart atlas::tits::TitsGroup::pull_across	(	const WeylElt &	w,
		TorusPart	y
	)		const

TorusPart atlas::tits::TitsGroup::push_across	(	TorusPart	x,
		const WeylElt &	w
	)		const

find torus part $x'$ so that $x.w=w.x'$

Algorithm: for simple reflections this is |reflect|; repeat left-to-right

Note: a more sophisticated algorithm would involve precomputing the conjugation matrices for the various pieces of w. Don't do it unless it turns out to really matter.

const size_t atlas::tits::TitsGroup::rank ( ) const

inline

void atlas::tits::TitsGroup::reflect ( TorusPart &  x, weyl::Generator  s  ) const

inline

void atlas::tits::TitsGroup::right_add ( TitsElt &  a, TorusPart  t  ) const

inline

TorusPart atlas::tits::TitsGroup::right_torus_part ( const TitsElt &  a ) const

inline

const size_t atlas::tits::TitsGroup::semisimple_rank ( ) const

inline

void atlas::tits::TitsGroup::sigma_conjugate ( TitsElt &  a, weyl::Generator  s  ) const

inline

void atlas::tits::TitsGroup::sigma_inv_conjugate ( TitsElt &  a, weyl::Generator  s  ) const

inline

void atlas::tits::TitsGroup::sigma_inv_mult	(	weyl::Generator	s,
		TitsElt &	a
	)		const

void atlas::tits::TitsGroup::sigma_mult	(	weyl::Generator	s,
		TitsElt &	a
	)		const

Left multiplication of |a| by the canonical generator $$.

This is the basic case defining the group structure in the Tits group (since left-multiplication by an element of $T(2)$ just adds to the torus part).

Algorithm: This is surprisingly simple: multiplying by $$ just amounts to reflecting the torus part through |s|, then left-multiplying the Weyl group part $w$ by |s| in the Weyl group, and if the length goes down in the latter step, add a factor of $()^2=m_s$ to the torus part. (To see this, write in the length-decreasing case $w=s.w'$ with $w'$ reduced; then $=m_s{w'}$, so we need to add $m_s$ to the reflected left torus part in that case.

The upshot is a multiplication algorithm almost as fast as in the Weyl group!

void atlas::tits::TitsGroup::twist ( TorusPart  x ) const

inline

TorusPart atlas::tits::TitsGroup::twisted ( const TorusPart  x ) const

inline

TitsElt atlas::tits::TitsGroup::twisted

(

const TitsElt &

te

)

const

void atlas::tits::TitsGroup::twistedConjugate ( TitsElt &  a, weyl::Generator  s  ) const

inline

Member Data Documentation

BinaryMap atlas::tits::TitsGroup::d_involution

private

size_t atlas::tits::TitsGroup::d_rank

private

std::vector<TorusPart> atlas::tits::TitsGroup::d_simpleCoroot

private

std::vector<TorusPart> atlas::tits::TitsGroup::d_simpleRoot

private

BinaryMap atlas::tits::TitsGroup::dual_involution

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