/home/r0/dav/atlas.dir/atlas3/sources/gkmod/kgb.cpp File Reference

Implementation of the class KGB representing orbits of K on G/B. More...

#include "kgb.h"
#include <cassert>
#include <map>
#include <memory>
#include <set>
#include <stdexcept>
#include <iostream>
#include "bitmap.h"
#include "bruhat.h"
#include "cartanset.h"
#include "cartanclass.h"
#include "complexredgp.h"
#include "error.h"
#include "gradings.h"
#include "hashtable.h"
#include "latticetypes.h"
#include "realredgp.h"
#include "rootdata.h"
#include "tits.h"
#include "tori.h"
#include "weyl.h"

Include dependency graph for kgb.cpp:

Go to the source code of this file.

Namespaces
namespace	atlas
namespace	atlas::kgb
Classes
class	atlas::kgb::FiberData
	A |FiberData| object associates to each twisted involution a subspace describing how corresponding Tits elements should be normalized. More...
class	atlas::kgb::KGBHelp
class	atlas::kgb::InvolutionCompare
class	atlas::kgb::IndexCompare
Functions
gradings::Grading	square_class_grading_offset (const complexredgp::ComplexReductiveGroup &G, cartanclass::square_class csc)
gradings::Grading	grading_offset_for (const realredgp::RealReductiveGroup &GR)
void	makeHasse (std::vector< set::SetEltList > &, const KGB &)
gradings::Grading	grading_offset_for (const realredgp::RealReductiveGroup &G)
	Returns the grading offset (on simple roots) adapted to |G|. This flags the simple roots that are noncompact imaginary at the fundamental Cartan in G.
gradings::Grading	square_class_grading_offset (const complexredgp::ComplexReductiveGroup &G, cartanclass::square_class csc)
	Returns the grading offset for the base real form of |csc|.
void	makeHasse (std::vector< set::SetEltList > &hd, const KGB &kgb)
	Puts in hd the hasse diagram data for the Bruhat ordering on KGB.

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Detailed Description

Implementation of the class KGB representing orbits of K on G/B.

This module contains code for the construction of a block in the one-sided parameter set (in other words, the subset of the one-sided parameter set corresponding to a single real form.) As explained in my Palo Alto III notes, this is equivalent to parametrizing the set K\G/B of (K,B)-orbits in G; hence the provocative title.

Definition in file kgb.cpp.

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Function Documentation

gradings::Grading grading_offset_for (  const realredgp::RealReductiveGroup &  G  )  [static]

Returns the grading offset (on simple roots) adapted to |G|. This flags the simple roots that are noncompact imaginary at the fundamental Cartan in G. Algorithm: the variable |rset| is first made to flag, among the imaginary roots of the fundamental Cartan, those that are noncompact for the chosen representative (in the adjoint fiber) of the real form of |G|. The result is formed by extracting only the information concerning the presence of the {simple} roots in |rset|. Definition at line 1266 of file kgb.cpp. References atlas::gradings::Grading, atlas::realredgp::RealReductiveGroup::noncompactRoots(), atlas::cartanclass::restrictGrading(), atlas::realredgp::RealReductiveGroup::rootDatum(), atlas::rootdata::RootSet, and atlas::rootdata::RootDatum::simpleRootList().

gradings::Grading grading_offset_for (  const realredgp::RealReductiveGroup &  GR  )  [static]

Referenced by atlas::kgb::KGBHelp::KGBHelp().

void makeHasse (  std::vector< set::SetEltList > &  hd, const KGB &  kgb )  [static]

Puts in hd the hasse diagram data for the Bruhat ordering on KGB. Explanation: this is the closure ordering of orbits. We use the algorithm from Richardson and Springer. Definition at line 1289 of file kgb.cpp. References atlas::kgb::KGB::cayley(), atlas::kgb::KGB::cross(), atlas::kgb::KGB::descent(), atlas::kgb::Descent, atlas::bitset::BitSet< n >::firstBit(), atlas::kgb::KGB::inverseCayley(), atlas::kgb::KGB::isAscent(), atlas::kgb::KGBElt, atlas::kgb::KGBEltPair, atlas::bitset::BitSet< n >::none(), atlas::kgb::KGB::size(), and atlas::kgb::KGB::status().

void makeHasse (  std::vector< set::SetEltList > &  , const KGB &  )  [static]

Referenced by atlas::kgb::KGB::fillBruhat().

gradings::Grading square_class_grading_offset (  const complexredgp::ComplexReductiveGroup &  G, cartanclass::square_class  csc )  [static]

Returns the grading offset for the base real form of |csc|. Precondition: |csc| specifies a square class (coset in adjoint fiber group) Definition at line 1280 of file kgb.cpp. References atlas::cartanclass::Fiber::class_base(), atlas::complexredgp::ComplexReductiveGroup::fundamental(), atlas::cartanclass::Fiber::noncompactRoots(), atlas::cartanclass::restrictGrading(), atlas::complexredgp::ComplexReductiveGroup::rootDatum(), atlas::rootdata::RootSet, and atlas::rootdata::RootDatum::simpleRootList().

gradings::Grading square_class_grading_offset (  const complexredgp::ComplexReductiveGroup &  G, cartanclass::square_class  csc )  [static]

Referenced by atlas::kgb::KGBHelp::KGBHelp().

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