Classes
Typedefs
determined by value of 
. square_class >
Classes | |
| class | atlas::cartanclass::InvolutionData |
| class | atlas::cartanclass::Fiber |
| Describes "the fiber" (over a fixed involution of a torus) in twisted Tits group. More... | |
| class | atlas::cartanclass::CartanClass |
| Represents a single stable conjugacy class of Cartan subgroups. More... | |
Typedefs | |
| typedef unsigned int | AdjointFiberElt |
| Element of the adjoint fiber or adjoint fiber group. | |
| typedef unsigned int | FiberElt |
| Element of the fiber group. | |
| typedef unsigned short | adjoint_fiber_orbit |
| Number of a W_imag orbit on the adjoint fiber group. | |
| typedef unsigned short | fiber_orbit |
| Number of a W_imag orbit on the fiber group. | |
| typedef unsigned short int | square_class |
| Identification of a class of real forms determined by value of . | |
| typedef std::pair< fiber_orbit, square_class > | StrongRealFormRep |
| Second number is a possible square in Z(G) of a strong real form. First is a W_im orbit on the coset of the fiber group corresponding to that square. | |
Functions | |
| Fiber | dualFiber (const rootdata::RootDatum &rd, const latticetypes::LatticeMatrix &q) |
| Constructs the dual fiber for the Cartan class determined by the involution |q| of the root datum |rd|. | |
| latticetypes::Weight | compactTwoRho (AdjointFiberElt x, const Fiber &f, const rootdata::RootDatum &rd) |
| Returns the sum of positive compact imaginary roots for |x| in |f|. | |
| gradings::Grading | restrictGrading (const rootdata::RootSet &rs, const rootdata::RootList &rl) |
| Returns the restriction of the grading in |rs| to |rl|. | |
| gradings::Grading | specialGrading (const cartanclass::Fiber &f, realform::RealForm rf, const rootdata::RootDatum &rd) |
| Returns a grading in the orbit corresponding to |rf|, with the smallest possible number of noncompact roots. | |
| rootdata::RootList | toMostSplit (const cartanclass::Fiber &fundf, realform::RealForm rf, const rootdata::RootDatum &rd) |
| Returns a set of strongly orthogonal roots, leading from the fundamental Cartan to the most split one for the strong real form |rf|. | |
| gradings::Grading | specialGrading (const Fiber &, realform::RealForm, const rootdata::RootDatum &) |
| rootdata::RootList | toMostSplit (const Fiber &, realform::RealForm, const rootdata::RootDatum &) |
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Number of a W_imag orbit on the adjoint fiber group.
Definition at line 37 of file cartanclass_fwd.h. Referenced by atlas::cartanclass::CartanClass::toWeakReal(), and atlas::cartanclass::Fiber::toWeakReal(). |
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Element of the adjoint fiber or adjoint fiber group.
Definition at line 29 of file cartanclass_fwd.h. Referenced by atlas::cartanclass::Fiber::class_base(), atlas::cartanclass::Fiber::gradingRep(), atlas::cartanclass::CartanClass::isMostSplit(), atlas::cartanclass::CartanClass::toAdjoint(), atlas::cartanclass::Fiber::toAdjoint(), and atlas::cartanclass::Fiber::toWeakReal(). |
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Number of a W_imag orbit on the fiber group.
Definition at line 41 of file cartanclass_fwd.h. |
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Element of the fiber group.
Definition at line 32 of file cartanclass_fwd.h. Referenced by atlas::cartanclass::Fiber::makeStrongRepresentatives(). |
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Identification of a class of real forms
Definition at line 47 of file cartanclass_fwd.h. Referenced by atlas::cartanclass::Fiber::central_square_class(), and atlas::cartanclass::Fiber::makeStrongReal(). |
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Second number is a possible square in Z(G) of a strong real form. First is a W_im orbit on the coset of the fiber group corresponding to that square.
Definition at line 54 of file cartanclass_fwd.h. Referenced by atlas::cartanclass::Fiber::strongRepresentative(). |
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Returns the sum of positive compact imaginary roots for |x| in |f|.
Definition at line 1013 of file cartanclass.cpp. References atlas::cartanclass::Fiber::compactRoots(), and atlas::rootdata::RootDatum::twoRho(). Referenced by atlas::orthogonalMAlpha(). |
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Constructs the dual fiber for the Cartan class determined by the involution |q| of the root datum |rd|. This is the fiber for the dual root datum and the negative transpose matrix. This used to be a separate constructor, but it is more practical to make it a function external to the class. This gives us the opportunity to prepare the dual root datum and involution locally, and let the main constructor do the real work; making this an official constructor would have given us the obligation to initialise the member data to irrelevant values, before being able to compute the dual root datum and matrix and start the construction. Definition at line 233 of file cartanclass.cpp. References atlas::latticetypes::LatticeMatrix, atlas::matrix::Matrix< C >::negate(), and atlas::matrix::Matrix< C >::transpose(). Referenced by atlas::cartanclass::CartanClass::CartanClass(). |
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Returns the restriction of the grading in |rs| to |rl|.
Definition at line 1022 of file cartanclass.cpp. References atlas::gradings::Grading, atlas::bitmap::BitMap::isMember(), atlas::rootdata::RootSet, and atlas::bitset::BitSet< n >::set(). Referenced by atlas::kgb::grading_offset_for(), atlas::cartanset::makeRepresentative(), specialGrading(), and atlas::kgb::square_class_grading_offset(). |
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Referenced by atlas::realform_io::Interface::Interface(). |
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Returns a grading in the orbit corresponding to |rf|, with the smallest possible number of noncompact roots. Precondition: |f| is the fundamental fiber; Explanation: for each noncompact noncomplex irreducible real form, there is at least one grading with exactly one noncompact simple root. Our choice amounts to a grading which induces one of the aforementioned ones on each noncompact noncomplex simple factor. The function of this is to enable easy type recognition. NOTE : the grading is represented in terms of simple roots for the root system |rd|. This is OK; knowledge of the gradings of those roots is enough to define the real form. Definition at line 1048 of file cartanclass.cpp. References atlas::cartanclass::Fiber::adjointFiberSize(), atlas::cartanclass::Fiber::noncompactRoots(), restrictGrading(), atlas::rootdata::RootDatum::simpleRootList(), and atlas::cartanclass::Fiber::weakReal(). |
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Referenced by atlas::realform_io::Interface::Interface(), and atlas::realredgp::RealReductiveGroup::RealReductiveGroup(). |
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Returns a set of strongly orthogonal roots, leading from the fundamental Cartan to the most split one for the strong real form |rf|. Algorithm: this is quite simple: as long as there are noncompact imaginary roots, use one of them to get to a less compact Cartan. Definition at line 1079 of file cartanclass.cpp. References atlas::bitmap::BitMap::begin(), atlas::partition::Partition::classRep(), atlas::bitmap::BitMap::empty(), atlas::bitmap::BitMap::flip(), atlas::bitmap::BitMap::front(), atlas::cartanclass::Fiber::imaginaryRootSet(), atlas::rootdata::RootDatum::isOrthogonal(), atlas::cartanclass::Fiber::noncompactRoots(), atlas::bitmap::BitMap::remove(), atlas::rootdata::RootNbr, atlas::rootdata::RootSet, atlas::rootdata::strongOrthogonalize(), atlas::rootdata::RootDatum::sumIsRoot(), and atlas::cartanclass::Fiber::weakReal(). |
1.3.9.1