Represents a torus defined over R. More...

#include <tori.h>

Collaboration diagram for atlas::tori::RealTorus:

Public Member Functions
	RealTorus (const WeightInvolution &)

	RealTorus (const RealTorus &, tags::DualTag)

const WeightInvolution &	involution () const

size_t	rank () const

size_t	complexRank () const

size_t	compactRank () const

size_t	splitRank () const

size_t	plusRank () const

size_t	minusRank () const

size_t	twoRank () const

bool	isCompact () const

bool	isSplit () const

BinaryMap	componentMap (const LatticeMatrix &, const RealTorus &) const

const WeightList &	plusLattice () const

const WeightList &	minusLattice () const

void	toPlus (Weight &dest, const Weight &source) const

void	toMinus (Weight &dest, const Weight &source) const

const SmallSubquotient &	topology () const

Private Attributes
size_t	d_rank

size_t	d_complexRank

WeightInvolution	d_involution

WeightList	d_plus

WeightList	d_minus

LatticeMatrix	d_toPlus

LatticeMatrix	d_toMinus

SmallSubquotient	d_topology

Detailed Description

Represents a torus defined over R.

This is equivalent to the datum of a lattice with an involution; to be consistent with the rest of the program, we use the Cartan involution tau which is the negative of the Galois involution on the character lattice.

The fundamental data are the rank d_rank (allowing us to identify the character lattice X with d_rank-tuples of integers) and the integer matrix d_involution of tau.

The software constructs two sublattices X_+ and X_-, the +1 and -1 eigenspaces for the involution. The datum d_plus is a basis of X_+, and d_minus is a basis of X_-. Both X_+ and X_- are supplementable in X, but in general X is not equal to the direct sum X_+ + X_-.

The most delicate invariant we shall have to deal with is the component group of the group of real points of T. This is an elementary abelian 2-group; we shall rather consider its dual dpi0(T). To describe its rank is fairly easy. Indeed, T may be decomposed as a product of compact, split and complex factors; denote r_u, r_s and r_c the number of factors of each type, so that the rank n of T is r_u + r_s + 2 r_c, then the rank of the component group is r_s. We have moreover : rk(X_+) = r_u + r_c; rk(X_-) = r_s + r_c. Denote V = X/2X, a vector space over the two-element field F_2, and denote V_+,V_- the images of X_+, X_- in V. Then it is not hard to show that r_c = dim(V_+ cap V_-), which allows computing r_s once rk(X_-) is known. One may prove that V_+- := V_+ cap V_- is also the image of tau - 1 in V.

It is a little bit harder to describe dpi0(T) functorially as a vector space. The group T(2)(R) of real points of the group of elements of order 2 in T is in natural duality with V/V_+-. There is a natural surjection from T(2)(R) to pi0(T), so dpi0(T) is a sub-vector space of V/V_+-, which may in fact be described as the image of V_-. This is how we will consider it.

Constructor & Destructor Documentation

atlas::tori::RealTorus::RealTorus ( const WeightInvolution & i )

explicit

atlas::tori::RealTorus::RealTorus	(	const RealTorus &	T,
		tags::DualTag
	)

Member Function Documentation

size_t atlas::tori::RealTorus::compactRank ( ) const

inline

size_t atlas::tori::RealTorus::complexRank ( ) const

inline

BinaryMap atlas::tori::RealTorus::componentMap	(	const LatticeMatrix &	m,
		const RealTorus &	T_dest
	)		const

const WeightInvolution& atlas::tori::RealTorus::involution ( ) const

inline

bool atlas::tori::RealTorus::isCompact ( ) const

inline

bool atlas::tori::RealTorus::isSplit ( ) const

inline

const WeightList& atlas::tori::RealTorus::minusLattice ( ) const

inline

size_t atlas::tori::RealTorus::minusRank ( ) const

inline

const WeightList& atlas::tori::RealTorus::plusLattice ( ) const

inline

size_t atlas::tori::RealTorus::plusRank ( ) const

inline

size_t atlas::tori::RealTorus::rank ( ) const

inline

size_t atlas::tori::RealTorus::splitRank ( ) const

inline

void atlas::tori::RealTorus::toMinus	(	Weight &	dest,
		const Weight &	source
	)		const

inline

void atlas::tori::RealTorus::toPlus	(	Weight &	dest,
		const Weight &	source
	)		const

inline

const SmallSubquotient& atlas::tori::RealTorus::topology ( ) const

inline

size_t atlas::tori::RealTorus::twoRank ( ) const

inline

Member Data Documentation

size_t atlas::tori::RealTorus::d_complexRank

private

number of C^x-factors

WeightInvolution atlas::tori::RealTorus::d_involution

private

matrix of the Cartan involution

WeightList atlas::tori::RealTorus::d_minus

private

basis for -1 eigenlattice of the Cartan involution

WeightList atlas::tori::RealTorus::d_plus

private

basis for +1 eigenlattice of the Cartan involution

size_t atlas::tori::RealTorus::d_rank

private

rank of torus

LatticeMatrix atlas::tori::RealTorus::d_toMinus

private

coordinate transformation from standard basis of $X$ to basis |d_minus| of $X_-$; should be applied only to elements of $X_-$

LatticeMatrix atlas::tori::RealTorus::d_toPlus

private

coordinate transformation from standard basis of $X$ to basis |d_plus| of $X_+$; should be applied only to elements of $X_+$

SmallSubquotient atlas::tori::RealTorus::d_topology

private

dual component group of real torus (a vector space over $Z/2Z$), realised as the subquotient $(V_+ + V_-)/V_+$ of the $Z/2Z$ vector space $X/2X$

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

/home/ran/atlas_project/latest_branch_07182016/sources/structure/tori.h
/home/ran/atlas_project/latest_branch_07182016/sources/structure/tori.cpp

Public Member Functions

Private Attributes

Detailed Description

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation