Base class for all Semi simple groups, i.e. groups that has a SemiSimpleAlgebra. More...

#include <group.h>

Inheritance diagram for mty::SemiSimpleGroup:

Public Member Functions
	SemiSimpleGroup (const std::string &name)
	Constructor with one parameter. More...

	SemiSimpleGroup (const SemiSimpleAlgebra &t_algebra, const std::string &t_name)
	Constructor with two parameters. More...

	~SemiSimpleGroup () override
	Destructor.

SemiSimpleAlgebra *	getAlgebra () const override
	Returns a pointer to the algebra of the group. More...

void	setAlgebra (const SemiSimpleAlgebra &t_algebra)
	Replaces the algebra owned by the group. More...

Irrep	highestWeightRep (const std::vector< int > &highestWeight) const override
	Returns an Irrep of highest weight highestWeight. More...

csl::Expr	getQuadraticCasimir (const AlgebraState &highest) const
	Returns the expression (symbolic, rational number) of the quadratic Casimir operator for a representation. More...

Irrep	getTrivialRep () const override
	Returns the trivial (dimension 1) representation of the group. More...

SumIrrep	tensorProduct (const Irrep &rep1, const Irrep &rep2) const
	Computes the product of two representations and returns its decomposition in a sum of irreducible representations (SumIrrep). See SemiSimpleAlgebra::tensorProduct(). More...

SumIrrep	tensorProduct (const std::initializer_list< int > &rep1, const std::initializer_list< int > &rep2) const
	Computes and the product of two representations and returns its decomposition in a sum of irreducible representations. See SemiSimpleAlgebra::tensorProduct(). More...

const csl::Space *	getVectorSpace (const Irrep &irrep) const override
	Returns the vector space (csl::Space) associated with a certain irreducible representation of the group. More...

AlgebraState	getHighestWeight (const csl::Space *t_space) const override
	Searches and returns the highest weight of the representation living in t_space (csl::Space). More...

std::vector< const csl::Space * >	getAllVectorSpace () const override
	Returns all vector spaces (csl::Space) created by the group. More...

csl::Index	getCorrespondingIndex (const std::string &index_name, const QuantumFieldParent &parent) override
	Returns a csl::Index of the vector space corresponding to a given mty::QuantumFieldParent's representation of the group. More...

Public Member Functions inherited from mty::AbstractGroup
	AbstractGroup (const std::string &t_name="")
	Constructor with one optional parameter. More...

	AbstractGroup (int t_dim, const std::string &t_name="")
	Constructor with two parameters. More...

virtual	~AbstractGroup ()
	Destructor.

virtual Irrep	getAdjointRep () const
	Virtual function. Raises an error if called. See reimplementations.

virtual group::Type	getType () const =0
	Returns the type of the group. See mty::group::Type. More...

mty::GaugedGroup *	getGaugedGroup () const

void	setGaugedGroup (mty::GaugedGroup *t_gauged)

std::string	getName () const
	Returns the name of the group. More...

int	getDim () const
	Returns the dimension (integer) of the group. More...

Protected Attributes
std::unique_ptr< SemiSimpleAlgebra >	algebra
	Owner std::unique_ptr to the SemiSimpleAlgebra associated with the group.

std::vector< std::pair< AlgebraState, const csl::Space * > >	vectorSpaces
	Association of AlgebraState (highest weights of irreps) and vector spaces. To each irrep (i.e. each highest weight) is associated a different and unique vector space (csl::Space). See getVectorSpace().

std::map< AlgebraState, csl::Expr >	quadraticCasimir
	Association of AlgebraState (highest weights of irreps) and quadratic Casimirs. To each irrep (i.e. each highest weight) is associated a quadratic casimir operator. It is computed the first time asked, and stored.

Protected Attributes inherited from mty::AbstractGroup
int	dim
	Dimension of the group.

std::string	name
	Name of the group.

mty::GaugedGroup *	gauged

Detailed Description

Base class for all Semi simple groups, i.e. groups that has a SemiSimpleAlgebra.

This class handles the different vector spaces for different representations. Each time a new representation is asked, a csl::Space is created. Then, a same representation in the same group will live in the same csl::Space. All computations of representations and products use the features of SemiSimpleAlgebra.

Note: Two identical representations of two instances of SemiSimpleGroup (with the same algebra) will live in different vector spaces. For example, spin and weak-isospin will not mix, the two csl::Space will be totally independant.

Constructor & Destructor Documentation

◆ SemiSimpleGroup() [1/2]

mty::SemiSimpleGroup::SemiSimpleGroup ( const std::string & name )

explicit

Constructor with one parameter.

Parameters

name	Name of the group.

◆ SemiSimpleGroup() [2/2]

mty::SemiSimpleGroup::SemiSimpleGroup	(	const SemiSimpleAlgebra &	t_algebra,
		const std::string &	t_name
	)

Constructor with two parameters.

Parameters

t_algebra	algebra corresponding to the group.
t_name	name of the group.

Member Function Documentation

◆ getAlgebra()

SemiSimpleAlgebra * mty::SemiSimpleGroup::getAlgebra ( ) const

overridevirtual

Returns a pointer to the algebra of the group.

Returns: A raw pointer to the algebra of the group.

Reimplemented from mty::AbstractGroup.

◆ getAllVectorSpace()

vector< const Space * > mty::SemiSimpleGroup::getAllVectorSpace ( ) const

overridevirtual

Returns all vector spaces (csl::Space) created by the group.

All vector spaces are put in a std::vector that is returned.

Returns: All vector spaces (const) created since the creation of the SemiSimpleGroup.

Implements mty::AbstractGroup.

◆ getCorrespondingIndex()

Index mty::SemiSimpleGroup::getCorrespondingIndex	(	const std::string &	index_name,
		const QuantumFieldParent &	parent
	)

overridevirtual

Returns a csl::Index of the vector space corresponding to a given mty::QuantumFieldParent's representation of the group.

The QuantumFieldParent is important to know which representation is wanted. For example in QCD, if 'g' is the gluon and 'q' a quark that are respectively in octet and triplet representations of a SemiSimpleGroup 'SU3_c',

SU3_c.getCorrespondingIndex("A", g);

will return an index in the adjoint representation (dimension 8), whereas

SU3_c.getCorrespondingIndex("a", q);

will return an index in the fundamental representation (dimension 3).

Parameters

index_name	Name of the csl::Index.
parent	QuantumFieldParent in which we search a representation of the group.

Returns: An index of name index_name living in the space of parent's representation of the group.

Implements mty::AbstractGroup.

◆ getHighestWeight()

AlgebraState mty::SemiSimpleGroup::getHighestWeight ( const csl::Space * t_space ) const

overridevirtual

Searches and returns the highest weight of the representation living in t_space (csl::Space).

If a representation living in t_space is found, its highest weight is returned. Else an error is raised.

Parameters

t_space Vector space in which the representation lives.

Returns: The highest weight of the representation if found.

Reimplemented from mty::AbstractGroup.

◆ getQuadraticCasimir()

csl::Expr mty::SemiSimpleGroup::getQuadraticCasimir ( const AlgebraState & highest ) const

Returns the expression (symbolic, rational number) of the quadratic Casimir operator for a representation.

See SemiSimpleAlgebra::getQuadraticCasimir().

Parameters

irrep Highest weight state of the representation from which we want the Casimir.

Returns: The symbolic expression of the Casimir, a rational number.

◆ getTrivialRep()

mty::Irrep mty::SemiSimpleGroup::getTrivialRep ( ) const

overridevirtual

Returns the trivial (dimension 1) representation of the group.

Returns: The trivial Irrep of the group (see SemiSimpleAlgebra::getTrivialRep()).

Implements mty::AbstractGroup.

◆ getVectorSpace()

const Space * mty::SemiSimpleGroup::getVectorSpace ( const Irrep & irrep ) const

overridevirtual

Returns the vector space (csl::Space) associated with a certain irreducible representation of the group.

If the vector space of a representation identical to irrep (same highest weight) is found, it is returned. Else, a new vector space is created (csl::Space) and returned. The spaces created in this function are specialized in color::ColorSpace in order to implement trace properties in simplifications. See documentation of color::ColorSpace.

Note: In this function only the highest weights of the irreps are compared. This means that a representation of \( F_4 \) may be successfully compared to a representation of \(A_4\) because they have the same number of labels.

Parameters

irrep Representation from which we want the vector space.

Returns: A pointer to the color::ColorSpace of the representation (creates the space if it had not already been created).

Implements mty::AbstractGroup.

◆ highestWeightRep()

mty::Irrep mty::SemiSimpleGroup::highestWeightRep ( const std::vector< int > & highestWeight ) const

overridevirtual

Returns an Irrep of highest weight highestWeight.

The group uses the function SemiSimpleAlgebra::highestWeightRep() of its algebra to create the representation and returns it.

Parameters

highestWeight Highest weight state of the representation.

Returns: The Irrep of highest weight highestWeight (see SemiSimpleAlgebra::highestWeightRep()).

Reimplemented from mty::AbstractGroup.

◆ setAlgebra()

void mty::SemiSimpleGroup::setAlgebra ( const SemiSimpleAlgebra & t_algebra )

Replaces the algebra owned by the group.

t_algebra may be an r-value. This function simply creates (allocated memory with a unique_ptr) a new algebra identical to t_algebra.

Note: This function does not verify that the algebra corresponds indeed to the group. This function should not be called by the user as once a group has been created, in particular if representations have been defined, there is no reason for the algebra to change.

Parameters

t_algebra New algebra of the group.

◆ tensorProduct() [1/2]

SumIrrep mty::SemiSimpleGroup::tensorProduct	(	const Irrep &	rep1,
		const Irrep &	rep2
	)		const

Computes the product of two representations and returns its decomposition in a sum of irreducible representations (SumIrrep). See SemiSimpleAlgebra::tensorProduct().

Parameters

rep1	First representation in the product.
rep2	Second representation in the product.

Returns: A SumIrrep containing decomposition of the product in a direct sum of irreducible representations.

◆ tensorProduct() [2/2]

SumIrrep mty::SemiSimpleGroup::tensorProduct	(	const std::initializer_list< int > &	rep1,
		const std::initializer_list< int > &	rep2
	)		const

Computes and the product of two representations and returns its decomposition in a sum of irreducible representations. See SemiSimpleAlgebra::tensorProduct().

Instead of taking representations, this function takes directly the dynkin labels of their highest weight state. It first call SemiSimpleAlgebra::highestWeightRep() to instantiate representations from their highest weight, and then computes the tensor product.

Parameters

rep1	First representation in the product, caracterized by its dinkin labels in a std::vector of integers.
rep2	Second representation in the product, caracterized by its dinkin labels in a std::vector of integers.

Returns: A SumIrrep containing decomposition of the product in a direct sum of irreducible representations.

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

include/group.h
src/group.cpp

Public Member Functions

Protected Attributes

Detailed Description

Constructor & Destructor Documentation

◆ SemiSimpleGroup() [1/2]

◆ SemiSimpleGroup() [2/2]

Member Function Documentation

◆ getAlgebra()

◆ getAllVectorSpace()

◆ getCorrespondingIndex()

◆ getHighestWeight()

◆ getQuadraticCasimir()

◆ getTrivialRep()

◆ getVectorSpace()

◆ highestWeightRep()

◆ setAlgebra()

◆ tensorProduct() [1/2]

◆ tensorProduct() [2/2]