#include <cartanset.h>
Collaboration diagram for atlas::cartanset::CartanClassSet:
Public Member Functions | |
| CartanClassSet (const complexredgp::ComplexReductiveGroup &parent, const latticetypes::LatticeMatrix &distinguished) | |
| Entry #j is the sum of the positive real roots for d_twistedInvolution[j]. Entry #j is intended to flag the imaginary roots for d_twistedInvolution[j]. Entry #j is intended to flag the simple imaginary roots for d_twistedInvolution[j]. | |
| ~CartanClassSet () | |
| CartanClassSet (const complexredgp::ComplexReductiveGroup &parent, const CartanClassSet &) | |
| const cartanclass::CartanClass & | cartan (size_t cn) const |
| Returns data for stable conjugacy class #cn of Cartan subgroups. | |
| const latticetypes::LatticeMatrix & | distinguished () const |
| Recover the matrix of the involution for the fundamental Cartan. | |
| const latticetypes::LatticeMatrix & | dualDistinguished () const |
| Matrix of involution for the fundamental Cartan in the dual group. | |
| unsigned long | fiberSize (realform::RealForm rf, size_t cn) const |
| Returns the size of the fiber orbits corresponding to the strong real forms lying over (weak) real form #rf, in cartan #cn. | |
| const cartanclass::Fiber & | fundamental () const |
| Fiber class for the fundamental Cartan subgroup. | |
| unsigned long | dualFiberSize (realform::RealForm, size_t) const |
| Returns the size of the dual fiber orbits corresponding to the dual strong real forms lying over dual real form #rf, in Cartan #cn. | |
| const cartanclass::Fiber & | dualFundamental () const |
| Fiber class for the fundamental Cartan in the dual group. | |
| bool | isDefined (realform::RealForm rf, size_t cn) const |
| Tells whether Cartan #cn is defined in real form #rf. | |
| size_t | mostSplit (realform::RealForm rf) const |
| Entry #rf is the number of the most split Cartan for real form #rf. | |
| rootdata::RootSet | noncompactRoots (realform::RealForm rf) const |
| Returns the set of noncompact imaginary roots for (the representative in the adjoint fiber of) the real form #rf. | |
| size_t | numCartan () const |
| Returns the number of stable conjugacy classes of Cartans for G. | |
| size_t | numDualRealForms () const |
| Returns the number of weak real forms of the dual group of G. | |
| size_t | numDualRealForms (size_t cn) const |
| Returns the number of weak real forms of the dual group for which the dual of Cartan #cn is defined. | |
| size_t | numInvolutions () const |
| Returns the total number of involutions corresponding to the currently defined set of Cartans. | |
| size_t | numInvolutions (const bitmap::BitMap &Cartan_classes) const |
| Returns the total number of involutions corresponding to the indicated set of Cartans. | |
| size_t | numRealForms () const |
| Returns the number of weak real forms of G. | |
| size_t | numRealForms (size_t cn) const |
| Returns the number of weak real forms of G for which Cartan #cn is defined. | |
| const poset::Poset & | ordering () const |
| Returns the partial ordering of the set of Cartans. | |
| realform::RealForm | quasisplit () const |
| Returns the (inner) number of the quasisplit real form. | |
| const realform::RealFormList & | realFormLabels (size_t cn) const |
| Lists the real forms for which Cartan #cn is defined. | |
| const realform::RealFormList & | dualRealFormLabels (size_t cn) const |
| Entry #cn lists the dual real forms in which dual Cartan #cn is defined. | |
| cartanclass::adjoint_fiber_orbit | real_form_part (realform::RealForm rf, size_t cn) const |
| cartanclass::adjoint_fiber_orbit | dual_real_form_part (realform::RealForm drf, size_t cn) const |
| unsigned long | representative (realform::RealForm rf, size_t cn) const |
| Returns a representative for real form #rf in Cartan #cn. | |
| unsigned long | dualRepresentative (realform::RealForm, size_t) const |
| Returns a representative for dual real form #rf in Cartan #cn. | |
| const rootdata::RootDatum & | rootDatum () const |
| const bitmap::BitMap & | support (realform::RealForm rf) const |
| Entry #rf flags the Cartans defined in real form #rf. | |
| const bitmap::BitMap & | dualSupport (realform::RealForm rf) const |
| Entry #rf flags the dual Cartans defined in dual real form #rf. | |
| const weyl::WeylGroup & | weylGroup () const |
| const TwistedInvolution & | twistedInvolution (size_t cn) const |
| (Representative) twisted involutions for each class of Cartan subgroup. | |
| const weyl::WeylElt | canonicalize (TwistedInvolution &) const |
| Make |sigma| canonical and return Weyl group |w| element that twisted conjugates the canonical representative back to original |sigma|. | |
| latticetypes::LatticeMatrix | involutionMatrix (const TwistedInvolution &tw) const |
| matrix giving involution action of |tw| on weight lattice | |
| void | twistedAct (const weyl::TwistedInvolution &tw, latticetypes::LatticeElt &v) const |
| Modify |v| through through involution associated to |tw|. | |
| unsigned long | KGB_size (realform::RealForm rf, const bitmap::BitMap &Cartan_classes) const |
Returns the cardinality of the subset of  associated to |rf| whose twisted involutions belong to |Cartan_classes|. | |
| unsigned long | block_size (realform::RealForm, realform::RealForm, const bitmap::BitMap &Cartan_classes) const |
| latticetypes::LatticeElt | posRealRootSum (const TwistedInvolution &) const |
| Sum of the real roots. | |
| latticetypes::LatticeElt | posImaginaryRootSum (const TwistedInvolution &) const |
| Sum of the imaginary roots. | |
| size_t | cayley (size_t, size_t, weyl::WeylElt *) const |
| returns index of canonical form of the product of twisted involution |d_twistedInvolution[j]| with the reflection through its |i|-th imaginary simple root. If |conjugator| is non-null, the conjugating element as returnd by |canonicalize| is assigned to |conjugator|. | |
| size_t | classNumber (TwistedInvolution) const |
| find index of canonical representative of |sigma| in |d_twistedInvolution|, under the assumption that it is (already) present | |
| void | extend (realform::RealForm) |
| Extends the CartanClassSet structure so that it contains all Cartans for the real form |rf|. | |
Private Member Functions | |
| CartanClassSet (const CartanClassSet &) | |
| CartanClassSet & | operator= (const CartanClassSet &) |
| weyl::TwistedInvolution | reflection (rootdata::RootNbr rn, const TwistedInvolution &tw) const |
| Returns |tw| composed to the left with the reflection |s_rn| corresponding to root #rn. | |
| rootdata::RootSet | noncompactPosRootSet (realform::RealForm, size_t) const |
| Flags in rs the set of noncompact positive roots for Cartan #j. | |
| std::vector< weyl::WeylEltList > | expand () const |
| Returns the various conjugacy classes of twisted involutions for the currently known Cartans. | |
| void | addCartan (TwistedInvolution tw) |
| Adds a new cartan, with Cartan involution given by |tw|. | |
| void | addCartan (rootdata::RootNbr rn, size_t j) |
| Adds a new cartan to |d_cartan|, obtained from cartan #j by Cayley transform through root #rn. NO LONGER USED. | |
| void | correlateForms () |
| Adds a new real form label list to d_realFormLabels. | |
| void | correlateDualForms (const rootdata::RootDatum &rd, const weyl::WeylGroup &W) |
| void | updateStatus (size_t prev_Cartan_size) |
| Updates d_status. | |
| void | updateSupports (size_t last_Cartan_class_added) |
| Updates d_support and d_dualSupport. | |
| void | updateTwistedInvolutions (std::vector< weyl::WeylEltList > &known, const TwistedInvolution &tw) |
| Updates the known list by adding the twisted class of tw to it. | |
Private Attributes | |
| const complexredgp::ComplexReductiveGroup & | d_parent |
| The inner class to which we are associated (and accessed from). | |
| std::vector< cartanclass::CartanClass * > | d_cartan |
| List of stable conjugacy classes of Cartan subgroups. | |
| TwistedInvolutionList | d_twistedInvolution |
| (Representative) twisted involutions for each class of Cartan subgroup. Choice at |j| must match corresponding |cartan(j).involution()| | |
| poset::Poset | d_ordering |
| Partial order of Cartan subgroups. | |
| cartanclass::Fiber | d_fundamental |
| Fiber class for the fundamental Cartan subgroup. | |
| cartanclass::Fiber | d_dualFundamental |
| Fiber class for the fundamental Cartan in the dual group. | |
| std::vector< realform::RealFormList > | d_realFormLabels |
| Entry #n lists the real forms in which Cartan #n is defined. | |
| std::vector< realform::RealFormList > | d_dualRealFormLabels |
| Entry #n lists the dual real forms in which dual Cartan #n is defined. | |
| std::vector< bitmap::BitMap > | d_support |
| Entry #rf flags the Cartans defined in real form #rf. | |
| std::vector< bitmap::BitMap > | d_dualSupport |
| Entry #rf flags the dual Cartans defined in dual real form #rf. | |
| bitmap::BitMap | d_status |
| Flags the set of real forms for which the full set of Cartan classes is constructed. | |
| std::vector< size_t > | d_mostSplit |
| Entry #rf is the number of the most split Cartan for real form #rf. | |
Each stable conjugacy classes of Cartan subgroups corresponds to a W-conjugacy class of involutions in the Gamma-enlarged Weyl group (W semidirect <gamma>, where <gamma>=Z/2Z acts on W), contained in the complement of its subgroup W. Since such involutions are of the form (w,Gamma), they can be represented by their element w, which is called a twisted involution. The condition for being a twisted involution $t$ is $t(t)=e$ and "twisted conjugacy" of $t$ by 
is given by 
. The stable conjugacy classes of Cartan subgroups will each be represented by a canonical representative of the corresponding twisted conjugacy class of twisted involutions.
In addition to describing the set of Cartan classes, this class provides access (via the |d_cartan| array) to data for each individual one of them, and (via |d_ordering|) to the partial order relation between them. For the latter, let |tau_i| be involutions acting on the complex torus |H| for various classes of Cartan subgroups; (H,tau_1) is considered "more compact" than (H,tau_2) if the identity component of the fixed point set H^tau_2 is W-conjugate to a subtorus of H^tau_1.
The problem for the dual group of G is identical, the bijection taking the negative transpose of a twisted involution. This bijection reverses the partial order on Cartans. The class provides also access to Cartans in the dual group.
Definition at line 80 of file cartanset.h.
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Entry #j is the sum of the positive real roots for d_twistedInvolution[j]. Entry #j is intended to flag the imaginary roots for d_twistedInvolution[j]. Entry #j is intended to flag the simple imaginary roots for d_twistedInvolution[j]. [Not yet implemented. DV 8/5/06.] Definition at line 111 of file cartanset.cpp. References atlas::latticetypes::LatticeMatrix. |
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The only data explicitly allocated by the CartanClass is that for the CartanClass pointers. Definition at line 165 of file cartanset.cpp. References d_cartan. |
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Adds a new cartan to |d_cartan|, obtained from cartan #j by Cayley transform through root #rn. NO LONGER USED.
Definition at line 298 of file cartanset.cpp. References cartan(), d_cartan, atlas::cartanclass::CartanClass::involution(), atlas::latticetypes::LatticeMatrix, rootDatum(), and atlas::rootdata::RootDatum::rootReflection(). |
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Adds a new cartan, with Cartan involution given by |tw|.
Definition at line 497 of file cartanset.h. References atlas::weyl::TwistedInvolution. Referenced by extend(). |
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Definition at line 904 of file cartanset.cpp. References atlas::bitmap::BitMap::begin(), cartan(), dualFiberSize(), fiberSize(), and atlas::cartanclass::CartanClass::orbitSize(). Referenced by atlas::cartanset::blockSize(). |
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Make |sigma| canonical and return Weyl group |w| element that twisted conjugates the canonical representative back to original |sigma|. We find conjugating generators starting at the original `|sigma|' end, so these form the letters of |w| from left to right. Definition at line 755 of file cartanset.cpp. References atlas::bitset::BitSet< n >::begin(), atlas::weyl::WeylGroup::mult(), posImaginaryRootSum(), posRealRootSum(), atlas::bitset::RankFlags, atlas::rootdata::RootDatum::reflection(), atlas::bitset::BitSet< n >::reset(), rootDatum(), atlas::latticetypes::scalarProduct(), atlas::rootdata::RootDatum::semisimpleRank(), atlas::bitset::BitSet< n >::set(), atlas::rootdata::RootDatum::simpleCoroot(), atlas::rootdata::RootDatum::simpleRootNbr(), twistedAct(), atlas::weyl::WeylGroup::twistedConjugate(), atlas::weyl::TwistedInvolution, atlas::rootdata::RootDatum::twoRho(), and weylGroup(). Referenced by cayley(), classNumber(), and extend(). |
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Returns data for stable conjugacy class #cn of Cartan subgroups.
Definition at line 235 of file cartanset.h. Referenced by addCartan(), block_size(), atlas::complexredgp::ComplexReductiveGroup::cartan(), dualFiberSize(), dualRepresentative(), fiberSize(), KGB_size(), numInvolutions(), representative(), and updateStatus(). |
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returns index of canonical form of the product of twisted involution |d_twistedInvolution[j]| with the reflection through its |i|-th imaginary simple root. If |conjugator| is non-null, the conjugating element as returnd by |canonicalize| is assigned to |conjugator|. Note that the mentioned reflection twisted-commutes with |d_twistedInvolution[j]|, so that the product is again a twisted involution. Definition at line 867 of file cartanset.cpp. References canonicalize(), d_twistedInvolution, atlas::setutils::find_index(), involutionMatrix(), atlas::weyl::WeylGroup::leftMult(), atlas::rootdata::RootDatum::reflectionWord(), rootDatum(), atlas::weyl::TwistedInvolution, and weylGroup(). |
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find index of canonical representative of |sigma| in |d_twistedInvolution|, under the assumption that it is (already) present
Definition at line 851 of file cartanset.cpp. References canonicalize(), d_twistedInvolution, atlas::setutils::find_index(), and atlas::weyl::TwistedInvolution. |
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Adds a new real form label list to d_realFormLabels. Algorithm: the gradings of the imaginary root system corresponding to the various real forms for which the new cartan is defined are known. We find a cross-action followed by a composite Cayley transform, taking the fundamental cartan to the new one. Then for each real form, we take a representative grading, and compute a grading for the fundamental cartan transforming to it. This amounts to solving a system of linear equations mod 2. Explanation: this is called when a new cartan class has just been added to d_cartan. Then this function finds the labels corresponding to the real forms for which this cartan is defined (the labelling of real forms being defined by the adjoint orbit picture in the fundamental fiber.) Definition at line 361 of file cartanset.cpp. References atlas::cartanset::cayley_and_cross_part(), atlas::cartanset::checkDecomposition(), atlas::partition::Partition::classRep(), atlas::bits::copy(), atlas::cartanset::crossTransform(), d_cartan, d_realFormLabels, d_twistedInvolution, distinguished(), fundamental(), atlas::cartanclass::Fiber::grading(), atlas::gradings::Grading, atlas::cartanclass::Fiber::imaginaryRootSet(), atlas::bitmap::BitMap::isMember(), atlas::cartanset::makeRepresentative(), atlas::cartanclass::Fiber::numRealForms(), rootDatum(), atlas::rootdata::RootList, atlas::bitset::BitSet< n >::set(), atlas::cartanclass::Fiber::simpleImaginary(), atlas::cartanset::transformGrading(), atlas::weyl::TwistedInvolution, atlas::cartanclass::Fiber::weakReal(), weylGroup(), and atlas::weyl::WeylWord. Referenced by extend(). |
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Recover the matrix of the involution for the fundamental Cartan. This is the one permuting the simple roots, the distinguished one among the involutions in this inner class of G. Definition at line 245 of file cartanset.h. References atlas::cartanclass::Fiber::involution(), and atlas::latticetypes::LatticeMatrix. Referenced by correlateForms(), atlas::complexredgp::ComplexReductiveGroup::distinguished(), and involutionMatrix(). |
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get part in the |weakReal| partition of the dual fiber in Cartan #cn corresponding to dual real form |drf| Definition at line 394 of file cartanset.h. References atlas::setutils::find_index(). Referenced by dualFiberSize(), and dualRepresentative(). |
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Matrix of involution for the fundamental Cartan in the dual group. This is -w_0 times the transpose of the fundamental involution. Definition at line 255 of file cartanset.h. References atlas::cartanclass::Fiber::involution(), and atlas::latticetypes::LatticeMatrix. |
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Returns the size of the dual fiber orbits corresponding to the dual strong real forms lying over dual real form #rf, in Cartan #cn. Precondition: real form #rf is defined for cartan #cn. Definition at line 620 of file cartanset.cpp. References cartan(), atlas::cartanclass::Fiber::central_square_class(), atlas::partition::Partition::classSize(), dual_real_form_part(), atlas::cartanclass::CartanClass::dualFiber(), atlas::cartanclass::Fiber::strongReal(), and atlas::cartanclass::Fiber::strongRepresentative(). Referenced by block_size(). |
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Fiber class for the fundamental Cartan in the dual group. The fiber group here is the group of characters of the component group of the quasisplit Cartan. Definition at line 284 of file cartanset.h. Referenced by correlateDualForms(), and atlas::complexredgp::ComplexReductiveGroup::dualFundamental(). |
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Entry #cn lists the dual real forms in which dual Cartan #cn is defined.
Definition at line 377 of file cartanset.h. Referenced by atlas::complexredgp::ComplexReductiveGroup::dualRealFormLabels(), and updateSupports(). |
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Returns a representative for dual real form #rf in Cartan #cn. Precondition: cartan #cn is defined for rf. Definition at line 686 of file cartanset.cpp. References cartan(), atlas::partition::Partition::classRep(), dual_real_form_part(), atlas::cartanclass::CartanClass::dualFiber(), and atlas::cartanclass::Fiber::weakReal(). Referenced by atlas::complexredgp::ComplexReductiveGroup::dualRepresentative(). |
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Entry #rf flags the dual Cartans defined in dual real form #rf.
Definition at line 429 of file cartanset.h. Referenced by atlas::cartanset::blockSize(). |
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Returns the various conjugacy classes of twisted involutions for the currently known Cartans. This method is NO LONGER USED during constructiont of a |CartanClassSet|. Note: the lists come out sorted, to allow binary look-up; this is in fact dependent on the implementation of |weyl::WeylGroup::twistedConjugacyClass| Definition at line 281 of file cartanset.cpp. References d_cartan, d_twistedInvolution, atlas::weyl::WeylGroup::twistedConjugacyClass(), and weylGroup(). |
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Extends the CartanClassSet structure so that it contains all Cartans for the real form |rf|.
This is done by traversing all the known Cartan classes |j| defined for the real form |rf| (which includes at least the distinguished Cartan for the inner class), and all non-compact positive imaginary roots While generating the Cartan classes, the ordering is extended by a link from the Cartan class |j| to the Cartan class of any twisted involution |ti| obtained directly from it. Definition at line 195 of file cartanset.cpp. References addCartan(), atlas::bitmap::BitMap::andnot(), atlas::bitmap::BitMap::begin(), canonicalize(), correlateDualForms(), correlateForms(), d_cartan, d_dualRealFormLabels, d_dualSupport, d_mostSplit, d_ordering, d_realFormLabels, d_status, d_support, d_twistedInvolution, atlas::poset::Poset::extend(), atlas::setutils::find_index(), isDefined(), atlas::bitmap::BitMap::isMember(), noncompactPosRootSet(), numRealForms(), reflection(), atlas::poset::Poset::resize(), rootDatum(), atlas::rootdata::RootSet, atlas::bitmap::BitMap::swap(), atlas::weyl::TwistedInvolution, updateStatus(), updateSupports(), and weylGroup(). Referenced by atlas::complexredgp::ComplexReductiveGroup::fillCartan(). |
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Returns the size of the fiber orbits corresponding to the strong real forms lying over (weak) real form #rf, in cartan #cn. Precondition: Real form #rf is defined for cartan #cn. Definition at line 592 of file cartanset.cpp. References cartan(), atlas::cartanclass::Fiber::central_square_class(), atlas::partition::Partition::classSize(), atlas::cartanclass::CartanClass::fiber(), real_form_part(), atlas::cartanclass::Fiber::strongReal(), and atlas::cartanclass::Fiber::strongRepresentative(). Referenced by block_size(), atlas::complexredgp::ComplexReductiveGroup::fiberSize(), and KGB_size(). |
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Fiber class for the fundamental Cartan subgroup. The involution is delta, which preserves the simple roots. Definition at line 272 of file cartanset.h. Referenced by correlateForms(), and atlas::complexredgp::ComplexReductiveGroup::fundamental(). |
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matrix giving involution action of |tw| on weight lattice
Definition at line 695 of file cartanset.cpp. References distinguished(), atlas::latticetypes::LatticeMatrix, atlas::weyl::WeylGroup::out(), rootDatum(), atlas::rootdata::toMatrix(), atlas::weyl::TwistedInvolution, atlas::weyl::WeylElt::w(), and weylGroup(). Referenced by cayley(), atlas::complexredgp::ComplexReductiveGroup::involutionMatrix(), posImaginaryRootSum(), and posRealRootSum(). |
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Tells whether Cartan #cn is defined in real form #rf.
Definition at line 291 of file cartanset.h. Referenced by extend(). |
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Returns the cardinality of the subset of Precondition: the Cartan classes for this real form have been generated Definition at line 892 of file cartanset.cpp. References atlas::bitmap::BitMap::begin(), cartan(), fiberSize(), and atlas::cartanclass::CartanClass::orbitSize(). Referenced by atlas::kgb::KGB::KGB(), atlas::kgb::KGBHelp::KGBHelp(), atlas::cartanset::kgbSize(), atlas::small_dual_kgb_f(), and atlas::small_kgb_f(). |
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Entry #rf is the number of the most split Cartan for real form #rf.
Definition at line 298 of file cartanset.h. Referenced by atlas::complexredgp::ComplexReductiveGroup::mostSplit(). |
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Flags in rs the set of noncompact positive roots for Cartan #j.
Definition at line 345 of file cartanset.cpp. References atlas::cartanclass::Fiber::noncompactRoots(), and atlas::rootdata::RootSet. Referenced by extend(). |
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Returns the set of noncompact imaginary roots for (the representative in the adjoint fiber of) the real form #rf.
Definition at line 306 of file cartanset.h. References atlas::partition::Partition::classRep(), atlas::cartanclass::Fiber::noncompactRoots(), atlas::rootdata::RootSet, and atlas::cartanclass::Fiber::weakReal(). Referenced by atlas::complexredgp::ComplexReductiveGroup::noncompactRoots(). |
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Returns the number of stable conjugacy classes of Cartans for G.
Definition at line 315 of file cartanset.h. References atlas::bitmap::BitMap::size(). Referenced by atlas::complexredgp::ComplexReductiveGroup::numCartanClasses(). |
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Returns the number of weak real forms of the dual group for which the dual of Cartan #cn is defined.
Definition at line 330 of file cartanset.h. |
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Returns the number of weak real forms of the dual group of G.
Definition at line 322 of file cartanset.h. References atlas::cartanclass::Fiber::numRealForms(). Referenced by atlas::complexredgp::ComplexReductiveGroup::numDualRealForms(). |
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Returns the total number of involutions corresponding to the indicated set of Cartans.
Definition at line 655 of file cartanset.cpp. References atlas::bitmap::BitMap::begin(), cartan(), and atlas::cartanclass::CartanClass::orbitSize(). |
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Returns the total number of involutions corresponding to the currently defined set of Cartans.
Definition at line 641 of file cartanset.cpp. References cartan(), d_cartan, and atlas::cartanclass::CartanClass::orbitSize(). Referenced by atlas::complexredgp::ComplexReductiveGroup::numInvolutions(). |
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Returns the number of weak real forms of G for which Cartan #cn is defined.
Definition at line 348 of file cartanset.h. References atlas::cartanclass::Fiber::numRealForms(). |
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Returns the number of weak real forms of G.
Definition at line 340 of file cartanset.h. References atlas::cartanclass::Fiber::numRealForms(). Referenced by extend(), and atlas::complexredgp::ComplexReductiveGroup::numRealForms(). |
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Returns the partial ordering of the set of Cartans.
Definition at line 355 of file cartanset.h. Referenced by atlas::complexredgp::ComplexReductiveGroup::cartanOrdering(), and atlas::kgb::KGBHelp::KGBHelp(). |
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Sum of the imaginary roots.
Definition at line 742 of file cartanset.cpp. References atlas::cartanclass::InvolutionData::imaginary_roots(), involutionMatrix(), rootDatum(), atlas::weyl::TwistedInvolution, and atlas::rootdata::RootDatum::twoRho(). Referenced by canonicalize(). |
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Sum of the real roots.
Definition at line 732 of file cartanset.cpp. References involutionMatrix(), atlas::cartanclass::InvolutionData::real_roots(), rootDatum(), atlas::weyl::TwistedInvolution, and atlas::rootdata::RootDatum::twoRho(). Referenced by canonicalize(). |
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Returns the (inner) number of the quasisplit real form.
Definition at line 362 of file cartanset.h. References atlas::realform::RealForm. Referenced by atlas::complexredgp::ComplexReductiveGroup::quasisplit(). |
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get part in the |weakReal| partition of the fiber in Cartan #cn corresponding to real form |rf| Definition at line 385 of file cartanset.h. References atlas::setutils::find_index(). Referenced by fiberSize(), atlas::kgb::KGBHelp::grading_seed(), atlas::kgb::KGBHelp::naive_seed(), and representative(). |
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Lists the real forms for which Cartan #cn is defined.
Definition at line 369 of file cartanset.h. Referenced by atlas::complexredgp::ComplexReductiveGroup::realFormLabels(), updateStatus(), and updateSupports(). |
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Returns |tw| composed to the left with the reflection |s_rn| corresponding to root #rn. This is a twisted involution if |s_rn| twisted-commutes with |tw|; in practice root #rn will in fact be imaginary for |tw| Definition at line 330 of file cartanset.cpp. References atlas::weyl::WeylGroup::leftMult(), atlas::rootdata::RootDatum::reflectionWord(), rootDatum(), atlas::weyl::TwistedInvolution, and weylGroup(). Referenced by extend(). |
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Returns a representative for real form #rf in Cartan #cn. Precondition: cartan #cn is defined for rf. Definition at line 673 of file cartanset.cpp. References cartan(), atlas::partition::Partition::classRep(), atlas::cartanclass::CartanClass::fiber(), real_form_part(), and atlas::cartanclass::Fiber::weakReal(). Referenced by atlas::complexredgp::ComplexReductiveGroup::representative(). |
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Definition at line 415 of file cartanset.h. References atlas::complexredgp::ComplexReductiveGroup::rootDatum(). Referenced by addCartan(), canonicalize(), cayley(), correlateForms(), extend(), involutionMatrix(), posImaginaryRootSum(), posRealRootSum(), and reflection(). |
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Entry #rf flags the Cartans defined in real form #rf.
Definition at line 422 of file cartanset.h. Referenced by atlas::cartanset::blockSize(), atlas::complexredgp::ComplexReductiveGroup::cartanSet(), and atlas::cartanset::kgbSize(). |
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Modify |v| through through involution associated to |tw|.
Definition at line 710 of file cartanset.cpp. References atlas::matrix::Matrix< C >::apply(), and atlas::weyl::TwistedInvolution. Referenced by canonicalize(). |
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(Representative) twisted involutions for each class of Cartan subgroup.
Definition at line 441 of file cartanset.h. References atlas::weyl::TwistedInvolution. Referenced by atlas::kgb::KGBHelp::backtrack_seed(), atlas::kgb::KGBHelp::grading_seed(), atlas::kgb::KGBHelp::naive_seed(), atlas::cartan_io::printCartanClass(), and atlas::complexredgp::ComplexReductiveGroup::twistedInvolution(). |
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Updates d_status. Precondition: prev is the previous size of d_cartan; Explanation: d_status holds the subset of the set of real forms for which the full set of Cartan classes is constructed; equivalently, those for which the most split Cartan has been reached. Definition at line 489 of file cartanset.cpp. References cartan(), atlas::partition::Partition::classCount(), d_cartan, d_mostSplit, d_status, atlas::cartanclass::CartanClass::fiber(), atlas::bitmap::BitMap::insert(), atlas::cartanclass::CartanClass::isMostSplit(), realFormLabels(), and atlas::cartanclass::Fiber::weakReal(). Referenced by extend(). |
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Updates d_support and d_dualSupport. Explanation: for each real form rf, d_support[rf] contains the subset of the set of the currently defined cartans which are defined for rf; and analogously for d_dualSupport[rf]. This information is in fact a "cross-section" of the realFormLabels lists, which for each Cartan list the set of real forms for which the Cartan is defined. This function is called each time a new Cartan is added, and updates these lists. Definition at line 524 of file cartanset.cpp. References d_dualSupport, d_support, dualRealFormLabels(), atlas::bitmap::BitMap::insert(), realFormLabels(), and atlas::bitmap::BitMap::set_capacity(). Referenced by extend(). |
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Updates the known list by adding the twisted class of tw to it.
Definition at line 560 of file cartanset.cpp. References atlas::bitmap::BitMap::swap(), and atlas::weyl::TwistedInvolution. |
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Definition at line 433 of file cartanset.h. References atlas::complexredgp::ComplexReductiveGroup::weylGroup(). Referenced by canonicalize(), cayley(), correlateForms(), expand(), extend(), involutionMatrix(), and reflection(). |
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List of stable conjugacy classes of Cartan subgroups. The list includes only Cartans appearing in real forms considered so far. Definition at line 92 of file cartanset.h. Referenced by addCartan(), correlateDualForms(), correlateForms(), expand(), extend(), numInvolutions(), updateStatus(), and ~CartanClassSet(). |
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Fiber class for the fundamental Cartan in the dual group. The fiber group here is the group of characters (i.e., the dual group) of the component group of the quasisplit Cartan. Definition at line 133 of file cartanset.h. |
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Entry #n lists the dual real forms in which dual Cartan #n is defined. More precisely, d_dualRealFormLabels[n][i] is the inner number of the dual real form that corresponds to part i of the partition cartan(n).dualFiber().weakReal() Definition at line 152 of file cartanset.h. Referenced by correlateDualForms(), and extend(). |
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Entry #rf flags the dual Cartans defined in dual real form #rf.
Definition at line 162 of file cartanset.h. Referenced by extend(), and updateSupports(). |
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Fiber class for the fundamental Cartan subgroup. The involution is delta, which is stored here. It permutes the simple roots. Definition at line 125 of file cartanset.h. |
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Entry #rf is the number of the most split Cartan for real form #rf.
Definition at line 176 of file cartanset.h. Referenced by extend(), and updateStatus(). |
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Partial order of Cartan subgroups. This is the ordering by containment of H^theta up to conjugacy: (H,theta_1) precedes (H,theta_2) if (H^theta_2)_0 is W-conjugate to a subtorus of H^theta_1. Numbering of elements is as in d_twistedInvolution Definition at line 118 of file cartanset.h. Referenced by extend(). |
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The inner class to which we are associated (and accessed from).
Definition at line 85 of file cartanset.h. |
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Entry #n lists the real forms in which Cartan #n is defined. More precisely, d_realFormLabels[n][i] is the inner number of the real form that corresponds to part i of the partition cartan(n).fiber().weakReal() Definition at line 142 of file cartanset.h. Referenced by correlateForms(), and extend(). |
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Flags the set of real forms for which the full set of Cartan classes is constructed. Because of the way the construction proceeds, these are exactly the real forms for which the most split Cartan has been reached. Definition at line 171 of file cartanset.h. Referenced by extend(), and updateStatus(). |
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Entry #rf flags the Cartans defined in real form #rf.
Definition at line 157 of file cartanset.h. Referenced by extend(), and updateSupports(). |
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(Representative) twisted involutions for each class of Cartan subgroup. Choice at |j| must match corresponding |cartan(j).involution()| In this version the representative involutions are the canonical ones. They satisfy: (1) the sum of the positive real roots is dominant (call it $SR$) (2) in the subsystem of roots orthogonal to $SR$, which contains all the imaginary roots, the sum of the imaginary roots is dominant for the subsystem (call it $SI$) (3) in the subsystem of roots orthogonal to both $SR$ and $SI$, the involution corresponding to twisted involution fixes (globally) the dominant chamber of the subsystem (it permutes its simple roots). Definition at line 109 of file cartanset.h. Referenced by cayley(), classNumber(), correlateForms(), expand(), and extend(). |
1.3.9.1
for |(rf,j)|; the twisted involution |tw| for the Cartan class |j| is then left-multiplied by 
to obtain a new twisted involution |ti|, and if |ti| is not member of any of the known classes of twisted involutions, it starts a new Cartan class defined in |rf|, which will later itself be considered in the loop described here.