Represents a block of representations of an inner form of G. More...

#include <blocks.h>

Inheritance diagram for atlas::blocks::Block:

Collaboration diagram for atlas::blocks::Block:

Public Member Functions
	~Block ()

	Block (const Block &b)

const TwistedWeylGroup &	twistedWeylGroup () const

const WeylGroup &	weylGroup () const

virtual KGBElt	xsize () const

virtual KGBElt	ysize () const

size_t	Cartan_class (BlockElt z) const

size_t	max_Cartan () const

virtual const TwistedInvolution &	involution (BlockElt z) const
	Returns the twisted involution corresponding to z. More...

const RankFlags &	involutionSupport (BlockElt z) const
	the simple roots occurring in reduced expression \|involution(z)\| More...

virtual std::ostream &	print (std::ostream &strm, BlockElt z, bool as_invol_expr) const

Public Member Functions inherited from atlas::blocks::Block_base
	Block_base (const KGB &kgb, const KGB &dual_kgb)

	Block_base (unsigned int rank)

virtual	~Block_base ()

	Block_base (const Block_base &b)

size_t	rank () const

size_t	folded_rank () const

size_t	size () const

const DynkinDiagram &	Dynkin () const

ext_gen	orbit (weyl::Generator s) const

const std::vector< ext_gen > &	fold_orbits () const

KGBElt	x (BlockElt z) const

KGBElt	y (BlockElt z) const

virtual BlockElt	element (KGBElt x, KGBElt y) const
	Look up element by \|x\|, \|y\| coordinates. More...

size_t	length (BlockElt z) const

BlockElt	length_first (size_t l) const

BlockElt	cross (weyl::Generator s, BlockElt z) const

BlockEltPair	cayley (weyl::Generator s, BlockElt z) const

BlockEltPair	inverseCayley (weyl::Generator s, BlockElt z) const

const DescentStatus &	descent (BlockElt z) const

DescentStatus::Value	descentValue (weyl::Generator s, BlockElt z) const

bool	isWeakDescent (weyl::Generator s, BlockElt z) const

bool	isStrictAscent (weyl::Generator, BlockElt) const
	Tells if s is a strict ascent generator for z. More...

bool	isStrictDescent (weyl::Generator, BlockElt) const
	Tells if s is a strict descent generator for z. More...

weyl::Generator	firstStrictDescent (BlockElt z) const
	Returns the first descent for z (the number of a simple root) that is not imaginary compact, or rank() if there is no such descent. More...

weyl::Generator	firstStrictGoodDescent (BlockElt z) const
	Returns the first descent for z (the number of a simple root) that is either complex or real type I; if there is no such descent returns \|rank()\|. More...

BlockElt	Hermitian_dual (BlockElt z) const

BlockEltPair	link (weyl::Generator alpha, weyl::Generator beta, BlockElt y) const

std::ostream &	print_to (std::ostream &strm, bool as_invol_expr) const

BruhatOrder &	bruhatOrder ()

kl::KLContext &	klc (BlockElt last_y, bool verbose)

Static Public Member Functions
static Block	build (InnerClass &, RealFormNbr rf, RealFormNbr drf)

static Block	build (RealReductiveGroup &G_R, RealReductiveGroup &dG_R)

Private Member Functions
	Block (const KGB &kgb, const KGB &dual_kgb)

Block &	operator= (const Block &b)

void	compute_supports ()

Private Attributes
const TwistedWeylGroup &	tW

size_t	xrange

size_t	yrange

std::vector< size_t >	d_Cartan

TwistedInvolutionList	d_involution

std::vector< RankFlags >	d_involutionSupport
	Flags the generators occurring in reduced expression for \|d_involution\|. More...

Additional Inherited Members
Protected Member Functions inherited from atlas::blocks::Block_base
KGBElt	renumber_x (const std::vector< KGBElt > &new_x)

void	compute_first_zs ()

Protected Attributes inherited from atlas::blocks::Block_base
std::vector< EltInfo >	info

std::vector< std::vector< block_fields > >	data

std::vector< ext_gen >	orbits

std::vector< BlockElt >	d_first_z_of_x

DynkinDiagram	dd

BruhatOrder *	d_bruhat

kl::KLContext *	klc_ptr

Detailed Description

Represents a block of representations of an inner form of G.

For our fixed inner form, orbits of $K$ on $G/B$ are parametrized by classes of elements $x$ in $N_G(H).$ (the normalizer in the non-identity component $G.$ of the extended group $G^Gamma=G disju G.$, where $$ is (i.e., acts on $G$ as) an involution that itself normalises $H$), modulo the {conjugation} action of $H$. (Dangerous bend: this $H$ conjugacy class of $x$ is a subset, usually proper, of the coset $xH$. The collection of all $x$ is therefore NOT a subset of the extended Weyl group $N(H)/H$, but something more subtle.) The requirement on $x$ is that it belong to the $G$-conjugacy class of strong involutions defining the inner form.

Each $x$ therefore defines an involution $$ of $H$. Data pertaining to the subset of $x$ with a fixed $$ is stored in the |Fiber| class.

A block is characterized by specifying also an inner form of the dual group $G^vee$. For this inner form, $K^vee$ orbits on $G^vee/B^vee$ are parametrized by elements $y$. The basic theorem is that the block of representations is parametrized by pairs $(x,y)$ as above, subject to the requirement that $theta_y$ is the negative transpose of $theta_x$.

Constructor & Destructor Documentation

atlas::blocks::Block::Block	(	const KGB &	kgb,
		const KGB &	dual_kgb
	)

private

atlas::blocks::Block::~Block ( )

inline

atlas::blocks::Block::Block ( const Block & b )

Member Function Documentation

Block atlas::blocks::Block::build	(	InnerClass &	G,
		RealFormNbr	rf,
		RealFormNbr	drf
	)

static

Block atlas::blocks::Block::build	(	RealReductiveGroup &	G_R,
		RealReductiveGroup &	dG_R
	)

static

size_t atlas::blocks::Block::Cartan_class ( BlockElt z ) const

inline

void atlas::blocks::Block::compute_supports ( )

private

virtual const TwistedInvolution& atlas::blocks::Block::involution ( BlockElt z ) const

inlinevirtual

Returns the twisted involution corresponding to z.

This is the corresponding Weyl group element w, such that w.delta is the root datum involution tau corresponding to z

Implements atlas::blocks::Block_base.

const RankFlags& atlas::blocks::Block::involutionSupport ( BlockElt z ) const

inline

the simple roots occurring in reduced expression |involution(z)|

size_t atlas::blocks::Block::max_Cartan ( ) const

inline

Block& atlas::blocks::Block::operator= ( const Block & b )

private

std::ostream & atlas::blocks::Block::print	(	std::ostream &	strm,
		BlockElt	z,
		bool	as_invol_expr
	)		const

virtual

Implements atlas::blocks::Block_base.

const TwistedWeylGroup& atlas::blocks::Block::twistedWeylGroup ( ) const

inline

const WeylGroup& atlas::blocks::Block::weylGroup ( ) const

inline

virtual KGBElt atlas::blocks::Block::xsize ( ) const

inlinevirtual

Implements atlas::blocks::Block_base.

virtual KGBElt atlas::blocks::Block::ysize ( ) const

inlinevirtual

Implements atlas::blocks::Block_base.

Member Data Documentation

std::vector<size_t> atlas::blocks::Block::d_Cartan

private

TwistedInvolutionList atlas::blocks::Block::d_involution

private

std::vector<RankFlags> atlas::blocks::Block::d_involutionSupport

private

Flags the generators occurring in reduced expression for |d_involution|.

const TwistedWeylGroup& atlas::blocks::Block::tW

private

size_t atlas::blocks::Block::xrange

private

size_t atlas::blocks::Block::yrange

private

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

/home/ran/atlas_project/latest_branch_07182016/sources/gkmod/blocks.h
/home/ran/atlas_project/latest_branch_07182016/sources/gkmod/blocks.cpp
/home/ran/atlas_project/latest_branch_07182016/sources/io/block_io.cpp

Public Member Functions

Static Public Member Functions

Private Member Functions

Private Attributes

Additional Inherited Members

Detailed Description

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation