atlas::blocks::Block Class Reference

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

#include <blocks.h>

Collaboration diagram for atlas::blocks::Block:

[legend]List of all members.


Public Member Functions
	Block (complexredgp::ComplexReductiveGroup &, realform::RealForm rf, realform::RealForm drf, bool select_Cartans=false)
	Constructor for the block class.
	~Block ()
const descents::DescentStatus &	descent (BlockElt z) const
	vector of descent statuses of all simple roots
descents::DescentStatus::Value	descentValue (size_t s, BlockElt z) const
	descent type of s at z
bool	isWeakDescent (size_t s, BlockElt z) const
	whether \|s\| is a weak descent for \|z\|
realform::RealForm	dualForm () const
size_t	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.
size_t	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()\|.
BlockElt	cross (size_t s, BlockElt z) const
	cross action
BlockEltPair	cayley (size_t s, BlockElt z) const
	Cayley transform.
BlockEltPair	inverseCayley (size_t s, BlockElt z) const
	inverse Cayley transform
const bitset::RankFlags &	involutionSupport (size_t z) const
	the simple roots occurring in $\theta_z$ .
bool	isStrictAscent (size_t, BlockElt) const
	Tells if s is a strict ascent generator for z.
bool	isStrictDescent (size_t, BlockElt) const
	Tells if s is a strict descent generator for z.
size_t	length (BlockElt z) const
	length of block element
size_t	Cartan_class (BlockElt z) const
	Cartan class of block element.
BlockEltPair	link (size_t alpha, size_t beta, BlockElt y) const
	the functor $T_{\alpha,\beta}$
size_t	rank () const
	semisimple rank of the group this block is constructed for
realform::RealForm	realForm () const
size_t	size () const
	size of the block
const weyl::TwistedInvolution &	involution (BlockElt z) const
	Returns the twisted involution corresponding to z.
const weyl::WeylGroup &	weylGroup () const
kgb::KGBElt	x (BlockElt z) const
kgb::KGBElt	y (BlockElt z) const
BlockElt	element (kgb::KGBElt x, kgb::KGBElt y) const
	Look up element by \|x\|, \|y\| coordinates.
std::pair< BlockElt, BlockElt >	R_packet (BlockElt z) const
size_t	xsize () const
size_t	ysize () const
bruhat::BruhatOrder &	bruhatOrder ()
Private Types
enum	State { BruhatConstructed, NumStates }
Private Member Functions
weyl::TwistedInvolution	dualInvolution (const weyl::TwistedInvolution &tw, weyl::WeylInterface to_dual) const
	Returns the twisted involution dual to tw.
void	generate (realredgp::RealReductiveGroup &G, realredgp::RealReductiveGroup &dG, bool select_Cartans)
void	fillBruhat ()
	Constructs the BruhatOrder.
Private Attributes
realform::RealForm	d_realForm
	Number (in the list maintained by the complex reductive group) of the real form of G where the block lives.
realform::RealForm	d_dualForm
	Number of the real form of G^vee defining the block.
size_t	d_rank
	Semisimple rank of G.
const weyl::WeylGroup &	d_weylGroup
size_t	d_xrange
	Number of K orbits on G/B.
size_t	d_yrange
	Number of K^vee orbits on G^vee/B^vee.
kgb::KGBEltList	d_x
	Element d_x[z] identifies the K orbit x on G/B for z.
kgb::KGBEltList	d_y
	Element d_y[z] identifies the K^vee orbit y on G^vee/B^vee for z.
std::vector< BlockElt >	d_first_z_of_x
	maps KGB element \|x\| to the first block element \|z\| with \|d_x[z]>=x\|.
std::vector< BlockEltList >	d_cross
	d_cross[s][z] is $s * z$ (for s a simple root, z a BlockElt).
std::vector< BlockEltPairList >	d_cayley
	For $s$ a simple root and $z$ a \|BlockElt\|, \|d_cayley[s][z]\| is the Cayley transform $c_s(z)$ (noncompact imaginary case) or the inverse Cayley transform $c^s(z)$ (real type 1 or 2) or undefined (otherwise).
descents::DescentStatusList	d_descent
	Entry z flags the descent status of the simple roots for block element z.
std::vector< size_t >	d_length
	Entry z is the length of block element z.
std::vector< size_t >	d_Cartan
	Entry z is the Cartan class of block element z.
weyl::TwistedInvolutionList	d_involution
	Entry z (multiplied by the fixed outer automorphism ) is the involution $\tau_z$ of H attached to z (in other words, d_involution[z] is the twisted involution attached to z; MvL).
std::vector< bitset::RankFlags >	d_involutionSupport
	Entry z flags the simple roots occurring in $\theta_z$ .
bitset::BitSet< NumStates >	d_state
	Records state bits (in fact one: whether the Bruhat order is computed).
bruhat::BruhatOrder *	d_bruhat
	Bruhat order on the block.

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 $N_G(H).$ (which is the normalizer in the second half $G.$ 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 to the $G$-conjugacy class of strong involutions defining the inner form.

Each $x$ therefore defines an involution $theta_x$ of $H$. Describing the set of $x$ with a fixed involution is accomplished by 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 $theta_x$.

Definition at line 81 of file blocks.h.

Member Enumeration Documentation

enum atlas::blocks::Block::State [private]

Enumeration values:

BruhatConstructed

NumStates

Definition at line 83 of file blocks.h.

Constructor & Destructor Documentation

atlas::blocks::Block::Block ( complexredgp::ComplexReductiveGroup & G,

realform::RealForm rf,

realform::RealForm df,

bool select_Cartans = false

)

Constructor for the block class.
Constructs a block from the datum of a real form rf for G and a real form df for G^vee (_not_ strong real forms: up to isomorphism, the result depends only on the underlying real forms!).
Definition at line 171 of file blocks.cpp.
References atlas::blocks::BlockElt, cross(), d_involution, d_involutionSupport, d_length, d_rank, descentValue(), firstStrictDescent(), generate(), inverseCayley(), atlas::bitset::RankFlags, atlas::lietype::semisimpleRank(), atlas::bitset::BitSet< n >::set(), size(), weylGroup(), and atlas::weyl::WeylGroup::word().

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

Definition at line 239 of file blocks.cpp.

Member Function Documentation

bruhat::BruhatOrder& atlas::blocks::Block::bruhatOrder ( ) [inline]

Definition at line 322 of file blocks.h.
Referenced by atlas::blockorder_f(), atlas::kl_io::printAllKL(), and atlas::kl_io::printPrimitiveKL().

size_t atlas::blocks::Block::Cartan_class ( BlockElt z ) const [inline]

Cartan class of block element.

Definition at line 256 of file blocks.h.
Referenced by atlas::block_io::printBlock(), atlas::block_io::printBlockD(), and atlas::block_io::printBlockU().

BlockEltPair atlas::blocks::Block::cayley ( size_t s,

BlockElt z

) const [inline]

Cayley transform.

Definition at line 223 of file blocks.h.
References atlas::blocks::BlockEltPair, and atlas::blocks::UndefBlock.
Referenced by atlas::klsupport::KLSupport::cayley(), atlas::blocks::insertAscents(), link(), atlas::klsupport::KLSupport::primitivize(), atlas::block_io::printBlock(), atlas::block_io::printBlockD(), atlas::block_io::printBlockU(), and atlas::filekl::write_block_file().

BlockElt atlas::blocks::Block::cross ( size_t s,

BlockElt z

) const [inline]

cross action

Definition at line 216 of file blocks.h.
References atlas::blocks::BlockElt.
Referenced by Block(), atlas::klsupport::KLSupport::cross(), atlas::blocks::insertAscents(), link(), atlas::blocks::makeHasse(), atlas::klsupport::KLSupport::primitivize(), atlas::block_io::printBlock(), atlas::block_io::printBlockD(), atlas::block_io::printBlockU(), and atlas::filekl::write_block_file().

const descents::DescentStatus& atlas::blocks::Block::descent ( BlockElt z ) const [inline]

vector of descent statuses of all simple roots

Definition at line 192 of file blocks.h.
Referenced by link(), atlas::block_io::printBlock(), atlas::block_io::printBlockD(), and atlas::block_io::printBlockU().

descents::DescentStatus::Value atlas::blocks::Block::descentValue ( size_t s,

BlockElt z

) const [inline]

descent type of s at z

Definition at line 198 of file blocks.h.
Referenced by Block(), atlas::klsupport::KLSupport::descentValue(), firstStrictGoodDescent(), atlas::blocks::insertAscents(), isStrictAscent(), isStrictDescent(), atlas::blocks::makeHasse(), and atlas::filekl::write_block_file().

realform::RealForm atlas::blocks::Block::dualForm ( ) const [inline]

Definition at line 208 of file blocks.h.

weyl::TwistedInvolution atlas::blocks::Block::dualInvolution ( const weyl::TwistedInvolution & tw,

weyl::WeylInterface to_dual

) const [private]

Returns the twisted involution dual to tw.

Definition at line 563 of file blocks.cpp.
References atlas::weyl::WeylGroup::inverse(), atlas::weyl::WeylGroup::longest(), atlas::weyl::WeylGroup::prod(), atlas::weyl::WeylGroup::translation(), atlas::weyl::WeylGroup::twisted(), and atlas::weyl::TwistedInvolution.
Referenced by generate().

BlockElt atlas::blocks::Block::element ( kgb::KGBElt x,

kgb::KGBElt y

) const

Look up element by |x|, |y| coordinates.
Precondition: |x| and |y| should be compatible: such a block element exists
This uses the |d_first_z_of_x| table to locate the range where the |x| coordinates are correct; then comparing the given |y| value with the first one present for |x| (there must be at least one) we can predict the value directly, since for each fixed |x| value the values of |y| are consecutive.
Definition at line 258 of file blocks.cpp.
References atlas::blocks::BlockElt, d_first_z_of_x, d_x, d_y, and size().
Referenced by atlas::blocks::dual_map(), and generate().

void atlas::blocks::Block::fillBruhat ( ) [private]

Constructs the BruhatOrder.
NOTE: may throw a MemoryOverflow error. Commit-or-rollback is guaranteed.
Definition at line 497 of file blocks.cpp.
References BruhatConstructed, d_bruhat, d_state, atlas::blocks::makeHasse(), atlas::bitset::BitSet< n >::set(), and atlas::bitset::BitSet< n >::test().

size_t atlas::blocks::Block::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.

Definition at line 298 of file blocks.cpp.
References isStrictDescent(), and rank().
Referenced by Block().

size_t atlas::blocks::Block::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()|.

Definition at line 311 of file blocks.cpp.
References descentValue(), isStrictDescent(), and rank().
Referenced by atlas::blocks::makeHasse().

void atlas::blocks::Block::generate ( realredgp::RealReductiveGroup & G,

realredgp::RealReductiveGroup & dG,

bool select_Cartans

) [private]

Definition at line 388 of file blocks.cpp.
References atlas::blocks::BlockElt, atlas::blocks::BlockEltPair, atlas::complexredgp::ComplexReductiveGroup::blockSize(), atlas::kgb::KGB::Cartan_class(), atlas::kgb::KGB::cayley(), atlas::blocks::common_Cartans(), atlas::blocks::correlation(), atlas::kgb::KGB::cross(), d_Cartan, d_cayley, d_cross, d_descent, d_dualForm, d_first_z_of_x, d_involution, d_length, d_realForm, d_weylGroup, d_x, d_xrange, d_y, d_yrange, atlas::blocks::descents(), dualInvolution(), element(), atlas::kgb::KGB::inverseCayley(), atlas::kgb::KGB::involution(), atlas::kgb::KGB::length(), size(), atlas::kgb::KGB::size(), atlas::kgb::KGB::tauPacket(), atlas::weyl::TwistedInvolution, atlas::blocks::UndefBlock, x(), and y().
Referenced by Block().

BlockEltPair atlas::blocks::Block::inverseCayley ( size_t s,

BlockElt z

) const [inline]

inverse Cayley transform

Definition at line 232 of file blocks.h.
References atlas::blocks::BlockEltPair, and atlas::blocks::UndefBlock.
Referenced by Block(), link(), atlas::blocks::makeHasse(), atlas::block_io::printBlock(), atlas::block_io::printBlockD(), and atlas::block_io::printBlockU().

const weyl::TwistedInvolution& atlas::blocks::Block::involution ( BlockElt z ) const [inline]

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
Definition at line 284 of file blocks.h.
References atlas::weyl::TwistedInvolution.
Referenced by atlas::block_io::printBlock(), atlas::block_io::printBlockD(), and atlas::block_io::printBlockU().

const bitset::RankFlags& atlas::blocks::Block::involutionSupport ( size_t z ) const [inline]

the simple roots occurring in $\theta_z$ .

Definition at line 241 of file blocks.h.
References atlas::bitset::RankFlags.
Referenced by atlas::block_io::printBlockU().

bool atlas::blocks::Block::isStrictAscent ( size_t s,

BlockElt z

) const

Tells if s is a strict ascent generator for z.
Explanation: this means that d_descent[z][s] is one of ComplexAscent, ImaginaryTypeI or ImaginaryTypeII.
Definition at line 272 of file blocks.cpp.
References atlas::blocks::descents(), and descentValue().

bool atlas::blocks::Block::isStrictDescent ( size_t s,

BlockElt z

) const

Tells if s is a strict descent generator for z.
Explanation: this means that d_descent[z][s] is one of ComplexDescent, RealTypeI or RealTypeII.
Definition at line 286 of file blocks.cpp.
References descentValue().
Referenced by firstStrictDescent(), and firstStrictGoodDescent().

bool atlas::blocks::Block::isWeakDescent ( size_t s,

BlockElt z

) const [inline]

whether |s| is a weak descent for |z|

Definition at line 205 of file blocks.h.
Referenced by link(), atlas::block_io::printBlock(), atlas::block_io::printBlockD(), atlas::block_io::printBlockU(), and atlas::filekl::write_block_file().

size_t atlas::blocks::Block::length ( BlockElt z ) const [inline]

length of block element

Definition at line 251 of file blocks.h.
Referenced by atlas::fillLengthLess(), atlas::klsupport::KLSupport::length(), atlas::block_io::printBlock(), atlas::block_io::printBlockD(), atlas::block_io::printBlockU(), and atlas::filekl::write_block_file().

BlockEltPair atlas::blocks::Block::link ( size_t alpha,

size_t beta,

BlockElt y

) const

the functor $T_{\alpha,\beta}$
Precondition: alpha and beta are adjacent roots, of which alpha is a (weak) descent for y, while beta is not a descent for y.
In fact if this is not satisfied, we return a pair of UndefBlock elements
Definition at line 330 of file blocks.cpp.
References atlas::blocks::BlockElt, atlas::blocks::BlockEltPair, cayley(), cross(), descent(), inverseCayley(), isWeakDescent(), and atlas::blocks::UndefBlock.

std::pair<BlockElt,BlockElt> atlas::blocks::Block::R_packet ( BlockElt z ) const [inline]

Definition at line 307 of file blocks.h.
References atlas::blocks::BlockElt.
Referenced by atlas::kl::helper::Helper::completePacket(), atlas::kl::helper::Helper::fillKLRow(), and atlas::kl::helper::Helper::fillThickets().

size_t atlas::blocks::Block::rank ( ) const [inline]

semisimple rank of the group this block is constructed for

Definition at line 265 of file blocks.h.
Referenced by firstStrictDescent(), firstStrictGoodDescent(), atlas::blocks::makeHasse(), atlas::block_io::printBlock(), atlas::block_io::printBlockD(), atlas::block_io::printBlockU(), and atlas::filekl::write_block_file().

realform::RealForm atlas::blocks::Block::realForm ( ) const [inline]

Definition at line 269 of file blocks.h.

size_t atlas::blocks::Block::size ( ) const [inline]

size of the block

Definition at line 274 of file blocks.h.
References atlas::bitmap::BitMap::size().
Referenced by Block(), atlas::blockorder_f(), atlas::blocks::dual_map(), element(), atlas::klsupport::KLSupport::fillDownsets(), atlas::fillLengthLess(), generate(), atlas::blocks::makeHasse(), atlas::block_io::printBlock(), atlas::block_io::printBlockD(), atlas::block_io::printBlockU(), and atlas::filekl::write_block_file().

const weyl::WeylGroup& atlas::blocks::Block::weylGroup ( ) const [inline]

Definition at line 290 of file blocks.h.
Referenced by Block(), atlas::block_io::printBlock(), atlas::block_io::printBlockD(), and atlas::block_io::printBlockU().

kgb::KGBElt atlas::blocks::Block::x ( BlockElt z ) const [inline]

Definition at line 294 of file blocks.h.
Referenced by atlas::blocks::dual_map(), generate(), atlas::block_io::printBlock(), atlas::block_io::printBlockD(), and atlas::block_io::printBlockU().

size_t atlas::blocks::Block::xsize ( ) const [inline]

Definition at line 314 of file blocks.h.
Referenced by atlas::block_io::printBlock(), atlas::block_io::printBlockD(), and atlas::block_io::printBlockU().

kgb::KGBElt atlas::blocks::Block::y ( BlockElt z ) const [inline]

Definition at line 299 of file blocks.h.
Referenced by atlas::blocks::dual_map(), generate(), atlas::block_io::printBlock(), atlas::block_io::printBlockD(), and atlas::block_io::printBlockU().

size_t atlas::blocks::Block::ysize ( ) const [inline]

Definition at line 317 of file blocks.h.
Referenced by atlas::block_io::printBlock(), atlas::block_io::printBlockD(), and atlas::block_io::printBlockU().

Member Data Documentation

bruhat::BruhatOrder* atlas::blocks::Block::d_bruhat [private]

Bruhat order on the block.
Definition now corrected mathematically from the bad definition of Vogan's Park City notes to one equivalent to the transitive closure of non-vanishing KL polynomial.
Definition at line 178 of file blocks.h.
Referenced by fillBruhat().

std::vector<size_t> atlas::blocks::Block::d_Cartan [private]

Entry z is the Cartan class of block element z.

Definition at line 151 of file blocks.h.
Referenced by generate().

std::vector<BlockEltPairList> atlas::blocks::Block::d_cayley [private]

For $s$ a simple root and $z$ a |BlockElt|, |d_cayley[s][z]| is the Cayley transform $c_s(z)$ (noncompact imaginary case) or the inverse Cayley transform $c^s(z)$ (real type 1 or 2) or undefined (otherwise).

Definition at line 134 of file blocks.h.
Referenced by generate().

std::vector<BlockEltList> atlas::blocks::Block::d_cross [private]

d_cross[s][z] is $s * z$ (for s a simple root, z a BlockElt).

Definition at line 128 of file blocks.h.
Referenced by generate().

descents::DescentStatusList atlas::blocks::Block::d_descent [private]

Entry z flags the descent status of the simple roots for block element z.

Definition at line 141 of file blocks.h.
Referenced by generate().

realform::RealForm atlas::blocks::Block::d_dualForm [private]

Number of the real form of G^vee defining the block.

Definition at line 94 of file blocks.h.
Referenced by generate().

weyl::TwistedInvolutionList atlas::blocks::Block::d_involution [private]

Entry z (multiplied by the fixed outer automorphism ) is the involution $\tau_z$ of H attached to z (in other words, d_involution[z] is the twisted involution attached to z; MvL).

Definition at line 159 of file blocks.h.
Referenced by Block(), and generate().

std::vector<bitset::RankFlags> atlas::blocks::Block::d_involutionSupport [private]

Entry z flags the simple roots occurring in $\theta_z$ .

Definition at line 164 of file blocks.h.
Referenced by Block().

std::vector<size_t> atlas::blocks::Block::d_length [private]

Entry z is the length of block element z.

Definition at line 146 of file blocks.h.
Referenced by Block(), and generate().

size_t atlas::blocks::Block::d_rank [private]

Semisimple rank of G.

Definition at line 97 of file blocks.h.
Referenced by Block().

realform::RealForm atlas::blocks::Block::d_realForm [private]

Number (in the list maintained by the complex reductive group) of the real form of G where the block lives.

Definition at line 89 of file blocks.h.
Referenced by generate().

bitset::BitSet<NumStates> atlas::blocks::Block::d_state [private]

Records state bits (in fact one: whether the Bruhat order is computed).

Definition at line 169 of file blocks.h.
Referenced by fillBruhat().

const weyl::WeylGroup& atlas::blocks::Block::d_weylGroup [private]

Definition at line 99 of file blocks.h.
Referenced by generate().

kgb::KGBEltList atlas::blocks::Block::d_x [private]

Element d_x[z] identifies the K orbit x on G/B for z.

Definition at line 114 of file blocks.h.
Referenced by element(), and generate().

size_t atlas::blocks::Block::d_xrange [private]

Number of K orbits on G/B.

Definition at line 104 of file blocks.h.
Referenced by generate().

kgb::KGBEltList atlas::blocks::Block::d_y [private]

Element d_y[z] identifies the K^vee orbit y on G^vee/B^vee for z.

Definition at line 119 of file blocks.h.
Referenced by element(), and generate().

size_t atlas::blocks::Block::d_yrange [private]

Number of K^vee orbits on G^vee/B^vee.

Definition at line 109 of file blocks.h.
Referenced by generate().

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

/home/r0/dav/atlas.dir/atlas3/sources/gkmod/blocks.h
/home/r0/dav/atlas.dir/atlas3/sources/gkmod/blocks.cpp

Generated on Wed Mar 26 16:52:21 2008 for atlas by

1.3.9.1

atlas::blocks::Block Class Reference

Public Member Functions

Private Types

Private Member Functions

Private Attributes

Detailed Description

Member Enumeration Documentation

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation