RepresentationRing Class Reference

#include <repring.H>

Inheritance diagram for RepresentationRing:

List of all members.

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Detailed Description

RepresentationRing class.

This represents the (complex) representation ring of a finite group -- or so we think of it. In practice it is an augmented algebra over QQ with the following extra features:

• It is a lambda-ring. There seemed to be no use for a distinct class for lambda-rings, so we have added here the extra information, namely for each variable x of augmentation d, the value of lambda i(x) is stored for 2 <= i <= d.
• The "Schur indices" are stored for each variable: 1 for real type, 2 for complex type, 4 for quaternion type. WARNING: this is not the standard use of the term "Schur index" ! I apologize for this confusion. In the sequel the Schur index will always be as above.
• The class knows how to read a file "file.gaprep", produced by GAP, which contains a description of the algebra in terms of generators and relations. (Such files are created by our GAP script gaprep.g; see the comments to read_from_GAP_file to find out about their structure). Note: since a representation ring is torsion free, there is no harm in viewing it as an algebra over QQ rather than ZZ, and this allows the use of Grobner basis methods.

Definition at line 41 of file repring.H.

Public Member Functions
	RepresentationRing ()
	a constuctor setting base_field to "complex"
Polynomial< QQ >	new_variable (const string &name, const QQ &aug, long index)
virtual void	swap_variables_order (long i, long j, list< Polynomial< QQ > > &polys)
	changes the order of two variables
virtual void	swap_variables_order (long i, long j)
virtual void	kill_top_variable (const Polynomial< QQ > &replacement, list< Polynomial< QQ > > &polys)
	replaces the top variable by a polynomial in the others
virtual void	kill_top_variable (const Polynomial< QQ > &replacement)
virtual long	read_variables (ifstream &thefile, Tuple< Polynomial< QQ > > &thevars)
	read information on the variables in .gaprep file
bool	read_from_GAP_file (const string &name)
	read information on the algebra from .gaprep file
Public Attributes
string	base_field
	for display purposes. A string containing "complex". Replaced in RealRepresentationRing.
map< long, long >	schur_indices
	the schur indices
map< long, Tuple< Polynomial< QQ > > >	lambda_operations
	the lambda operations
Friends
ostream &	operator<< (ostream &os, const RepresentationRing &R)

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Constructor & Destructor Documentation

RepresentationRing::RepresentationRing (   )

a constuctor setting base_field to "complex" Definition at line 8 of file repring.cpp. References base_field.

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Member Function Documentation

void RepresentationRing::kill_top_variable (  const Polynomial< QQ > &  replacement, list< Polynomial< QQ > > &  polys )  [virtual]

replaces the top variable by a polynomial in the others This modifies the list polys so its members take the changes into account. Reimplemented from AugmentedAlgebra< QQ >. Definition at line 57 of file repring.cpp. References AugmentedAlgebra< K >::kill_top_variable(), lambda_operations, and schur_indices.

bool RepresentationRing::read_from_GAP_file (  const string &  name  )

read information on the algebra from .gaprep file This opens a .gaprep file. First, read_variables is called -- see corresponding comments. So n-1 variables have been created. Call them x_i. Next, the file contains a collection of integers, which in fact should be seen as a sequence of n-tuples of integers. The first n-tuple indicates how to express x_1^2 as a linear combination of the n variables 1, x_1, ..., x_n-1. Then the second n-tuple expresses x_1x_2 as such a combination. Follow x_1x_3, x_1x_4, ..., x_1x_n. Then this repeats starting from x_2, so: x_2^2, x_2x_3, x_2x_4, ..., x_2x_n. And so on until x_n^2. Finally, the lambda operations are described in the file in the same fashion: so for each variable x, call d its augmentation, and for each i with 2 <= i <= d, there is an n-tuple of integers expressing lambda i(x) as a linear combination of 1, x_1, ..., x_n-1. Definition at line 106 of file repring.cpp. References Ideal< K >::generators, lambda_operations, read_variables(), and Polynomial< K >::sets_to_zero().

long RepresentationRing::read_variables (  ifstream &  thefile, Tuple< Polynomial< QQ > > &  thevars )  [virtual]

read information on the variables in .gaprep file This is called by read_from_GAP_file, which has opened a file with gaprep extension. The .gaprep file contains first an integer, call it n. This is the number of (complex) representations. Then the file contains a sequence of n-1 pairs of integers. The first is the Schur index, the second is the (complex) dimension. Using this, n-1 variables are created (dropping the first one which is the trivial rep, and which will correspond to 1 in the rep ring). They are stored in thevars[i] (2<= i <=n) as polynomials. thevars[1] contains this->one(). thevars[0] is not used. The variables are automatically named r_1, r_2, ..., r_{n-1}. Returns: the number of variables created + 1, ie n. Reimplemented in RealRepresentationRing. Definition at line 79 of file repring.cpp. References AffineAlgebra< QQ >::new_variable(), AffineAlgebra< QQ >::one(), and Tuple< T >::resize(). Referenced by read_from_GAP_file().

void RepresentationRing::swap_variables_order (  long  i, long  j, list< Polynomial< QQ > > &  polys )  [virtual]

changes the order of two variables This corrects all the relations generators so they take the change of order into account. Likewise all polynomials created with this before should be modified. Parameters:   polys, a list of polynomials to update. Reimplemented from AugmentedAlgebra< QQ >. Definition at line 26 of file repring.cpp. References lambda_operations, schur_indices, Tuple< T >::steal(), and AugmentedAlgebra< K >::swap_variables_order().

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Member Data Documentation

string RepresentationRing::base_field

for display purposes. A string containing "complex". Replaced in RealRepresentationRing. Definition at line 44 of file repring.H. Referenced by RepresentationRing().

map< long, Tuple< Polynomial<QQ> > > RepresentationRing::lambda_operations

the lambda operations Definition at line 48 of file repring.H. Referenced by Exponentiator< F_2 >::compute_exp_relations(), kill_top_variable(), read_from_GAP_file(), and swap_variables_order().

map<long, long> RepresentationRing::schur_indices

the schur indices Definition at line 46 of file repring.H. Referenced by SW_Maker::add_complex_relations(), SW_Maker::create_regular_variables(), Chern_Maker::create_regular_variables(), kill_top_variable(), and swap_variables_order().

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

• include/repring.H
• src/repring.cpp

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