SW_Maker Class Reference

#include <sw_maker.H>

Inheritance diagram for SW_Maker:

List of all members.

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Detailed Description

SW_Maker class: computes Stiefel-Whitney classes.

This is an Exponentiator<F_2> which knows:

• that the prefix is to be "w" and the factor 1.
• that the SW classes of a complex resp quaternion representation are zero in degrees not divisible by 2 resp 4.
• that *target is really an unstable algebra, and how to use Wu's formula to compute the Steenrod operations
• if given a nonzero pointer to a GradedAlgebra<F_2> (typically created with a Chern_Maker) giving relations between Chern classes, the SW_Maker class knows how to translate these into relations between SW classes.

These changes are implemented by modifying the create_regular_variables() method (which is virtual). So it is only required from the user to call start() (same as for any Exponentiator).

If a pointer to a chern ring has been provided, then 'chern_sw' will contain the homomorphism between this ring and the sw ring just created by SW_Maker. It is public.

Definition at line 33 of file sw_maker.H.

Public Member Functions
void	start (RealRepresentationRing *thesource, GradedAlgebra< F_2 > *thetarget, GradedAlgebra< F_2 > *the_chern_ring=0)
	start() with a slightly different prototype
virtual void	create_regular_variables (string prefix)
	new version, which takes the schur indices into account
void	add_complex_relations ()
	translate relations between Chern classes into SW relations
AffineHomomorphism< F_2 >	translate_homomorphism (const AffineHomomorphism< QQ > &r, SW_Maker &sw)
	turn a homomorphism of representation rings into a homomorphism between the exponential algebras
Public Attributes
GradedAlgebra< F_2 > *	chern_ring
	a pointer to an algebra created by Chern_Maker
AbstractHomomorphism< F_2 >	chern_sw
	the homomorphism from the Chern ring to the SW ring
Private Member Functions
string	name_of_rep (const string &name_chern)
long	index_of_chern_class (const string &name_chern)
F_2	comb (long n, long i)

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Member Function Documentation

void SW_Maker::add_complex_relations (   )

translate relations between Chern classes into SW relations This is called at the end of the new create_regular_variables() when the pointer chern_ring is nonzero. This method looks for the variables in *chern_ring and expects them to have names of the form c_i(r_j). They are all converted into the corresponding SW classes (using info in *source such as source->nameof() but also source->conjugates). Then, the relations in *chern_ring are converted into SW relations and added to total_relations. Definition at line 96 of file sw_maker.cpp. References Polynomial< K >::alphabet, chern_ring, chern_sw, RealRepresentationRing::conjugates, Ideal< K >::generators, SimpleAlphabet::nameof(), AbstractHomomorphism< K >::of(), AffineAlgebra< K >::relations, RepresentationRing::schur_indices, AbstractHomomorphism< K >::set_image(), Exponentiator< F_2 >::var, SimpleAlphabet::variables_in_use(), and AffineAlgebra< K >::very_verbose. Referenced by create_regular_variables().

void SW_Maker::create_regular_variables (  string  prefix  )  [virtual]

new version, which takes the schur indices into account Some of the var(i,j) will be 0, depending on the Schur indices. Then the Steenrod operations are computed using Wu's formula. If chern_ring is not the zero pointer, then add_complex_relations() is called. Reimplemented from Exponentiator< F_2 >. Definition at line 16 of file sw_maker.cpp. References add_complex_relations(), Polynomial< K >::alphabet, chern_ring, AugmentedAlgebra< K >::epsilon, SimpleAlphabet::nameof(), GradedAlgebra< K >::new_variable(), AffineAlgebra< K >::one(), Table< T >::resize(), RepresentationRing::schur_indices, Polynomial< K >::sets_to_zero(), UnstableAlgebra::steenrod, Exponentiator< F_2 >::var, SimpleAlphabet::variables_in_use(), and UnstableAlgebra::write_trivial_steenrod_operations().

void SW_Maker::start (  RealRepresentationRing *  thesource, GradedAlgebra< F_2 > *  thetarget, GradedAlgebra< F_2 > *  the_chern_ring = 0 )

start() with a slightly different prototype Definition at line 7 of file sw_maker.cpp. References chern_ring, and Exponentiator< K >::start().

AffineHomomorphism< F_2 > SW_Maker::translate_homomorphism (  const AffineHomomorphism< QQ > &  r, SW_Maker &  sw )

turn a homomorphism of representation rings into a homomorphism between the exponential algebras This is the first attempt to make the Exponentiators functorial. It is a bit ad hoc at the moment, so we do not integrate this as a member of Exponentiator. What this is doing: we have two Exponentiators, this and sw, and we assume that they have both done their job (cf start()). Given a homomorphism r between their respective representation rings (r for restriction, the most common type of homomorphism that we shall need), the method translates r into the corresponding homomorphism between the exponential algebras. Limitation: we assume that r is of the form r(rep)= sum of reps with nonnegative, integer coefficents. Note: the sw is not 'const' here, as we may need to compute more universal polynomials in the process. These will be stored. Definition at line 225 of file sw_maker.cpp. References Polynomial< K >::coeffs, AugmentedAlgebra< K >::epsilon, Exponentiator< K >::exp_classes_after_tensoring(), GradedAlgebra< K >::hom_part_of(), Polynomial< K >::is_zero(), AbstractHomomorphism< K >::of_generator(), AffineAlgebra< K >::one(), Exponentiator< K >::target, Exponentiator< F_2 >::var, and SimpleAlphabet::variables_in_use().

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Member Data Documentation

GradedAlgebra<F_2>* SW_Maker::chern_ring

a pointer to an algebra created by Chern_Maker Definition at line 44 of file sw_maker.H. Referenced by add_complex_relations(), create_regular_variables(), and start().

AbstractHomomorphism<F_2> SW_Maker::chern_sw

the homomorphism from the Chern ring to the SW ring Definition at line 46 of file sw_maker.H. Referenced by add_complex_relations().

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

• include/sw_maker.H
• src/sw_maker.cpp

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